Effective thermal conductivity and thermal diffusivity of ...

Experimental Thermal and Fluid Science 31 (2007) 593–599 www.elsevier.com/locate/etfs

Effective thermal conductivity and thermal diffusivity of nanofluids containing spherical and cylindrical nanoparticles Xing Zhang a

a,b,* ,

Hua Gu c, Motoo Fujii

b

Key Laboratory for Thermal Science and Power Engineering of Ministry of Education, Tsinghua University, Beijing 100084, PR China b Institute for Materials Chemistry and Engineering, Kyushu University, Kasuga 816-8580, Japan c Interdisciplinary Graduate School of Engineering Sciences, Kyushu University, Kasuga 816-8580, Japan Received 20 December 2005; accepted 2 June 2006

Abstract The effective thermal conductivity and thermal diffusivity of Au/toluene, Al2O3/water, TiO2/water, CuO/water and CNT/water nanofluids have been measured by using the transient short-hot-wire technique. The average diameters of Au, Al2O3, TiO2 and CuO spherical particles are 1.65, 20, 40 and 33 nm, respectively. The average length and diameter of CNFs are 10 lm and 150 nm, respectively. The uncertainty of the present measurements is estimated to be within 1% for the thermal conductivity and 5% for the thermal diffusivity. The measured results demonstrate that the effective thermal conductivities of the nanofluids show no anomalous enhancements and can be predicted accurately by the model equation of Hamilton and Crosser for the spherical nanoparticles, and by the unit-cell model equation of Yamada and Ota for carbon nanofibers. Ó 2006 Elsevier Inc. All rights reserved. Keywords: Nanofluid; Thermal conductivity; Thermal diffusivity; Transient short-hot-wire technique

1. Introduction The thermophysical properties of fluids containing various solid particles have been investigated for several decades [1–5], and excellent predicting formulae [1,3] on the effective thermal conductivity of dispersed materials have been proposed theoretically and experimentally. Since a large enhanced thermal conductivity for a dispersion of metallic or non-metallic nanoparticles or nanotubes in conventional fluids was reported and the term of nanofluids was coined by Choi [6] in 1995, many researchers [7–10] have reported their theoretical, numerical, and experimental results on the thermophysical properties of nanofluids. Recently, Kumar et al. [11] reported an enhanced thermal conductivity of about 20% for a nanofluid of only * Corresponding author. Address: Key Laboratory for Thermal Science and Power Engineering of Ministry of Education, Tsinghua University, Beijing 100084, PR China. Tel.: + 86 10 62773776; fax: +86 10 62781610. E-mail address: [email protected] (X. Zhang).

0894-1777/$ - see front matter Ó 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.expthermflusci.2006.06.009

0.00013% Au nanoparticles in water. Since such an anomalous enhancement is expected to have wide applications in thermal engineering, nanofluids have received considerable attention in thermal science and engineering. However, it is very difficult to understand why nanofluids would have such a high thermal conductivity. Meanwhile, there are large differences among the thermal conductivities reported by different researchers. Keblinski et al. [12] further pointed out that the most exciting experimental results have not been reproducible. Therefore, it is necessary to reconsider the reliability of the measurements reported so far. This paper reports on accurate measurements of the effective thermal conductivity and thermal diffusivity of various nanofluids by using a transient short-hot-wire (SHW) technique. To remove the effects of the static charge and electrical conductance of the nanoparticles on measurement accuracy, the SHW probes are coated by a pure Al2O3 thin film of about 1 lm thick with a sputtering apparatus. The electrical leakage of the coated SHW probes is carefully checked, and only those probes that are coated

594

X. Zhang et al. / Experimental Thermal and Fluid Science 31 (2007) 593–599

Nomenclature a A b B C dp Fo I K l lp n qv r t T

slope of measured temperature rise, K/s slope of dimensionless temperature rise, dimensionless intercept of measured temperature rise, K intercept of dimensionless temperature rise, dimensionless specific heat capacity, J/(kg K) diameter of the cylindrical particle, m Fourier number, dimensionless current, A shape factor, dimensionless length of the hot wire, m length of the cylindrical particle, m constant, dimensionless heating rate generated per unit volume, W/m3 radius of the hot wire, m time, s temperature, °C

DT V

measured temperature rise, K voltage, V

Greek symbols a thermal diffusivity, m2/s h dimensionless temperature rise, dimensionless k thermal conductivity, W/(m K) q density, kg/m3 /v volume fraction of the particle, dimensionless /w mass fraction of the particle, dimensionless Subscripts eff effective f fluid i initial p particle v volume-averaged

well are used for measurements. The tested nanofluids are Au/toluene, Al2O3/water, TiO2/water, CuO/water and CNF/water and measurements are carried out for various volume fractions and temperatures. The Au, Al2O3, TiO2 and CuO nanoparticles are spherical and their corresponding average diameters are 1.65, 20, 40 and 33 nm, respectively. The average length and diameter of CNFs are 10 lm and 150 nm, respectively. The measured results show that the effective thermal conductivity and thermal diffusivity increase generally as the volume fraction of particles increases. However, the effective thermal conductivities of the nanofluids do not show any anomalous enhancement for the cases of dilute dispersions of these spherical nanoparticles and cylindrical CNFs, and can be accurately predicted by the model equation of Hamilton and Crosser [1] for the spherical cases, and by the unit-cell model equation of Yamada and Ota [3] for the carbon nanofibers.

the logarithm of the Fourier number Fo [= (at)/r2], where Ti and Tv are the initial liquid temperature and volumeaveraged hot wire temperature, respectively, qv is the heating rate generated per unit volume, r is the radius of the SHW, t is the time, k and a are the thermal conductivity and thermal diffusivity of liquid, respectively. The coefficients A and B, the slope and intercept, are determined by the least-squares method for a relevant range of Fourier number corresponding to the measuring periods. The measured temperature rise DTv [= Tv Ti] of the wire is also approximated by a linear equation with coefficients a and b, which are also determined by the least-squares method for the time range before the onset of natural convection. A comparison of the above two equations allows us to evaluate the thermal conductivity (k) and thermal diffusivity (a) of nanofluids from

2. Experiments

and

The effective thermal conductivity and thermal diffusivity of nanofluids are simultaneously measured by the transient SHW technique. Since the principle and procedures of the SHW technique have been described in detail previously [13–15], only a brief description is given here. The SHW technique was developed from the conventional transient hot-wire (THW) technique and was based on the numerical solution of two-dimensional transient heat conduction for a short wire with the same length-to-diameter ratio and boundary conditions as those used in the actual measurements. The numerical results for the dimensionless volume-averaged hot wire temperature rise hv [= (Tv Ti)/ (qvr2/k)] are approximated by a linear equation in terms of

a ¼ r2 exp

k¼

VI A pl a

ð1Þ

b B a A

ð2Þ

where l is the length of the hot wire, and V and I are the voltage and current supplied to the wire. A SHW probe and a Teflon cell with a volume of about 30 cm3 were designed and fabricated. Fig. 1a shows the dimensions of the SHW probe used in the present measurements. The SHW probe is mounted on the Teflon cap of the cell. A short platinum (Pt) wire with a length of 14.5 mm and a diameter of 20 lm is welded at both the ends to Pt lead wires of 1.5 mm in diameter. Using distilled water and toluene as standard liquids of known thermophysical properties, the effective length and radius of the


a

b

φ 40

47

74.2

40

60

10

6

φ 30

14.5

Thermocouple

Lead wires (φ 1.5 mm) 10

10

Hot wire (φ 20 μm)

Fig. 1. Schematic of the short-hot-wire probe and experimental cell (dimensions in mm).

hot wire and the thickness of the Al2O3 insulation film are calibrated before and after application of the Al2O3 film coating. Fig. 1b shows the dimensions of the Teflon cell used for nanofluid measurements. It has an inner diameter of 30 mm and a height of 47 mm so that the total inner volume is 33.2 cm3. Two thermocouples are located at the same heights to the upper and lower welding spots of the hot wire and lead wires, respectively, to monitor the temperature homogeneity. In order to minimize temperature fluctuations, the hot wire cell was placed in a thermostatic bath at the control temperature for which the thermal conductivity measurements were performed. Based on the uncertainty analysis equations given by Zhang and Fujii [15], the uncertainties of the thermal conductivity and thermal diffusivity measurements in the present study are estimated to be within 1.0% and 5.0%, respectively. The uncertainties have further been confirmed by comparing the measured values with the standard reference data for pure water and toluene.

595

thermal conductivity and thermal diffusivity did not change over a period of 48 h for the nanofluids at lower volume fractions. Fig. 2 shows the transmission electron microscope (TEM) photographs of Au, Al2O3, TiO2 and CuO particles and CNFs used in the present measurements. It is noted that because the TEM must work in a vacuum, these photographs were taken only after drying the nanofluids. Therefore, we can clearly know the shape and size of the nanoparticles by TEM photographs, but not the intrinsic dispersion of nanoparticles in the fluids. Fig. 3a and b show the measured effective thermal conductivity and thermal diffusivity of Au/toluene nanofluids for the volume fraction of particles /v = 0.003%. Since the volume fraction of particles is very low, the effect of the particles on the thermal conductivity and thermal diffusivity is negligible. The present results agree well with those of pure toluene [16,17] for temperatures ranging from 5 to 40 °C. However, Kumar et al. [11] showed over 2% thermal

3. Results and discussion The samples of Au/toluene nanofluids were directly made by chemical reaction method. The Al2O3/water and TiO2/water nanofluids were made by Sigma-Aldrich Co., where the powders were directly dispersed into the deionized water with a sonic method. The CuO particles were made by Sigma-Aldrich Co., but the CuO/water nanofluids were made by the present authors with the sonic method, where the surfactants were not used. The CNF/water nanofluids were also made by the present authors with the sonic method, where the CNFs were made by Showa Denko Co. and a small amount of surfactants (SDS 1.5 mass%) were used to disperse the CNFs into pure water. Generally, the nanoparticles have an inherent static charge. The absolute value is very small, but due to the extremely small size of the nanoparticles, the charge is large enough to keep them in suspension without settling and aggregation. Our measurements have shown that the values of the effective

Fig. 2. TEM photographs of Au, Al2O3, TiO2 and CuO particles and carbon nanofibers.

596


a

0.15 Toluene-Au φ v=0.003%

λ eff [W/m K]

0.14

0.8

0.13 Measured results Pure toluene [16]

0.12

0.11

0.9

λeff [W/m K]

a

0

10

20

30

0.6

10

20

Toluene-Au φv=0.003%

9

8

Measured results Pure toluene [17]

7 0

10

20

40

50

60

2.5

b αeff [10-7m2/s]

α eff [10-8m2/s]

11

10

30

T [°C]

T [°C]

b

Water-Al2O3

0.7

0.5 0

40

Pure water [16] Distilled water φw = 10% φw = 20% φw = 40%

30

2

10

20

T [°C]

40

50

60

Fig. 4. Effective (a) thermal conductivity and (b) thermal diffusivity of Al2O3/water nanofluids.

a

2 H-C model [1] Measured at 10 °C Measured at 30 °C Measured at 50 °C Masuda et al. [5] Lee et al. [8] Xie et al. [9]

1.8

λ eff /λ f

conductivity enhancement for such a nanofluid. We have also measured the Au/toluene nanofluids with a bare Pt SHW probe, and then about 8% thermal conductivity enhancement was observed because of the electrical leakage between the probe and the nanofluids. Fig. 4a and b show the measured effective thermal conductivity and thermal diffusivity of Al2O3/water nanofluids for mass fractions of particles /w = 0%, 10%, 20% and 40%. For distilled water (/w = 0) the present results in the temperature range of 5–50 °C agree well with the standard values recommended in Refs. [16,17]. As shown in Fig. 4, the effective thermal conductivity and thermal diffusivity of the nanofluids increase with an increase in the particle concentration and in the temperature. Furthermore, as for the temperature dependence, the slopes are the same as those for pure water. This means that the temperaturedependent thermal conductivity and thermal diffusivity of nanoparticles do not affect the temperature dependence of the effective thermal conductivity and thermal diffusivity of nanofluids in the range of the present concentrations. Fig. 5a and b show the effective thermal conductivity and thermal diffusivity of Al2O3/water nanofluids for different particle concentrations and temperatures. In these figures, the thermal conductivity and thermal diffusivity are normalized by using the values of pure water. The present thermal conductivities are close to those measured by Lee et al. [8] and Xie et al. [9], but lower than ones obtained by Masuda et al. [5]. The solid line further represents the values predicted by the Hamilton and Crosser model equa-

30

T [°C]

1.6

Water-Al2O3

1.4 1.2 1 0

b 1.6

5

10

φ v [%]

H-C model [1] Measured at 10 °C Measured at 30 °C

1.4

αeff /α f

Fig. 3. Effective (a) thermal conductivity and (b) thermal diffusivity of Au/toluene nanofluids.

Water-Al2O3

1.5

1 0

40

Pure water [17] Distilled water φw = 10% φw = 20% φw = 40%

15

Water-Al2O3

Measured at 50 °C

1.2

1

0

5

φ v [%]

10

15

Fig. 5. Normalized (a) thermal conductivity and (b) thermal diffusivity of Al2O3/water nanofluids.

tion (H–C model) [1], where the effective thermal conductivity, keff, is expressed as


Here, kf and kp are the thermal conductivity of the fluid and particles, respectively. And n is a constant that depends on the shape of the dispersed particles and on the ratio of the conductivities of the two phases. When the particles are spherical, the theoretical result shows that n is equal to 3 and independent of both the particle size and the ratio of the conductivities, and the Hamilton and Crosser equation becomes the same as the Maxwell’s equation [18]. Therefore, in the present paper, n is taken to be 3. /v is the volume fraction of the particles and is calculated by /w qf /v ¼ qp þ /w qf /w qp

aeff

Water-TiO2

1.04 1.02 1 0

1

φ v [%]

2

3

1.1 H-C model [1]

1.08 1.06

Water-TiO2

Measured at 10 °C Measured at 30 °C Measured at 40 °C

1.04 1.02 1 0

1

2

φ v [%]

3

Fig. 6. Normalized (a) thermal conductivity and (b) thermal diffusivity of TiO2/water nanofluids.

ð6Þ 1.6

and ð7Þ

1.4

λ eff /λ f

and Cf and Cp are the specific heat capacities of the fluid and particles, respectively. As shown in Fig. 5, the present effective thermal conductivity and thermal diffusivity in a lower volume fraction range agree well with the values predicted by the H–C model equation. On the other hand, in the higher volume fraction range, the measured results are lower than the predicted values. This is due to the settling of some fraction of particles that occurred at the higher volume fraction. Furthermore, the normalized thermal conductivity and thermal diffusivity are almost independent of the temperature. This confirms that the temperature dependence of these properties of nanofluids is not dominated by the solid phase but by the fluid phase. Figs. 6 and 7 show the normalized effective thermal conductivity and thermal diffusivity of TiO2/water and CuO/ water nanofluids. The present thermal conductivities agree well with those obtained by Masuda et al. [5] for TiO2/ water nanofluids shown in Fig. 6a, but are lower than ones obtained by Lee et al. [8] and Das et al. [19] for CuO/water nanofluids shown in Fig. 7a. All of the present values in a range of lower volume fractions agree well with those predicted by the H–C model, although the absolute values are dependent on the thermal properties of nanoparticles. Fig. 8 shows the normalized effective thermal conductivity and thermal diffusivity of CNT/water nanofluids.

H-C model[1] Measured at 10 °C Measured at 23 °C Measured at 30 °C Lee et al. [8] Das et al. [19]

Water-CuO

1.2

1 0

2

4

φ v [%]

6

8

10

1.6 1.5 1.4

αeff /α f

C eff ¼ C p /w þ C f ð1 /w Þ

1.06

H-C model [1] Measured at 10 °C Measured at 30 °C Measured at 40 °C Masuda et al. [5]

ð5Þ

where qeff and Ceff can be calculated by qeff ¼ qp /v þ qf ð1 /v Þ

1.08

ð4Þ

where, qf and qp are the density of the fluid and particles, respectively. /w is the mass fraction of the particles dispersed in the nanofluids. Based on the definition of the thermal diffusivity, the effective thermal diffusivity of nanofluids can be calculated by keff ¼ qeff C eff

1.1

ð3Þ

λ eff /λ f

½kp þ ðn 1Þkf ðn 1Þ/v ðkf kp Þ ½kp þ ðn 1Þkf þ /v ðkf kp Þ

αeff /α f

keff ¼ kf

597

H-C model [1] Measured at 10 °C Measured at 23 °C Measured at 30 °C

Water-CuO

1.3 1.2 1.1 1 0

2

4

6

8

10

φ v [%] Fig. 7. Normalized (a) thermal conductivity and (b) thermal diffusivity of CuO/water nanofluids.

Fig. 8a further shows the measured thermal conductivities reported by Assael et al. [20] at the volume fraction of 0.6%

598


3.5

2

λ eff /λ f

1.8 1.6

Water-CNT

Yamada-Ota model [3] Measured at T=23 °C (SDS 1.5 mass%) Assaelet al. [20] (SDS 2.0 mass%) Xie et al. [21]

Water-CNT (λp/ λ f=1000)

3

λ eff /λ f

a

1.4

l/d=200

2.5

2

l/d=66.7

1.5

1.2

l/d=6.7

1 0

1 0.2

0.4

0.6

φ v [%]

0.8

1

0

0.5

φ v [%]

Water-CNT Yamada-Ota model [3] Measured at 23 °C

1.4

1.6

αeff /α f

(SDS 1.5 mass %)

Water-CNT (l/d=66.7)

φv=1.0 %

1.4

λ eff/λ f

1.2

0

0.2

0.4

0.6

φ v [%]

0.8

and by Xie et al. [21] at the volume fraction of 1.0%. The present results of CNT/water nanofluids are much higher than those of the nanofluids dispersed with spherical nanoparticles. Furthermore, our results agree well with the predictions of the unit-cell model proposed by Yamada and Ota [3], where the normalized effective thermal conductivity is calculated by ½kp =kf þ K K/v ð1 kp =kf Þ ½kp =kf þ K þ /v ð1 kp =kf Þ

φv=0.6 %

1.2

1

Fig. 8. Normalized (a) thermal conductivity and (b) thermal diffusivity of CNT/water nanofluids.

keff =kf ¼

1.5

Fig. 9. Effects of the length-to-diameter ratio on the normalized thermal conductivity of nanofluids (unit-cell model analysis).

b 1.6

1

1

ð8Þ

Here, K is the shape factor and K ¼ 2/0:2 v ðlp =d p Þ for the cylindrical particles, and lp and dp are the length and diameter of the cylindrical particles. In the present measurements, the average length and diameter of the CNTs are 10 lm and 150 nm, respectively. Based on the unit-cell model, Figs. 9 and 10 further show the effects of the ratio of the length-to-diameter and the thermal conductivity of cylindrical particles on the normalized effective thermal conductivity of CNT/water nanofluids. As shown in Fig. 9, the normalized effective thermal conductivity of nanofluids increases greatly as the lengthto-diameter of cylindrical particles increases for a given ratio of the thermal conductivities. This can help us to understand the reason why the present effective thermal conductivities of CNT/water nanofluids are much higher than those of the nanofluids dispersed with the spherical nanoparticles. As shown in Fig. 10, the normalized effective thermal conductivity of nanofluids also increases greatly

1

φv=0.3 %

0

200

400

λp / λ f

600

800

1000

Fig. 10. Effects of the particle thermal conductivity on the normalized thermal conductivity of nanofluids (unit-cell model analysis).

with the ratio of the thermal conductivities at the beginning, then becomes almost independent of the thermal conductivity of particles when the particle thermal conductivity goes up to a critical value. The results mentioned above, together with the consideration of the present measurement uncertainty, clearly indicate that the dispersion of nanoparticles does not cause anomalous enhancements for the thermal conductivity and thermal diffusivity as those reported by some other authors [6,11]. Furthermore, the magnitude of the enhancements can be predicted using the existing models. Although it is very difficult to point out clearly the problems encountered in the previous experiments [6,11], it is found that the electrical insulation of the hot wire probe is very important when the hot wire method is applied to measurements of nanofluids. Even in our measurements, an insufficient insulation of the wire has resulted in clustering of nanoparticles and CNFs on the wire surface, and then in much different values of the thermal conductivity and thermal diffusivity of nanofluids. Furthermore, proper mixing and stabilization of the particles are also very important. To check the stabilization of the particles, we have measured the volume fraction of the particles after the experiments as well as observed the time variation of the thermal conductivity and thermal diffusivity.


4. Conclusions The effective thermal conductivity and thermal diffusivity of Au/toluene, Al2O3/water, TiO2/water, CuO/water and CNT/water nanofluids have been measured accurately for various volume fractions and temperatures. The present results show that the effective thermal conductivity and thermal diffusivity increase with an increase in the particle concentration, in the particle thermal conductivity and in the ratio of the length-to-diameter of CNFs. The effective thermal conductivities of nanofluids have not shown any anomalous enhancements. All of the measured values at lower volume fractions agree well with those predicted by the H–C equation for the spherical particles, and by the unit-cell model equation of Yamada and Ota for the CNFs. Acknowledgements The authors acknowledge a lot of assistance and useful discussion given by Dr. H. Q. Xie from Shanghai Institute of Ceramics, Chinese Academy of Sciences, China and Associate Professor H. Ago and Dr. W.M. Qiao from the Institute for Materials Chemistry and Engineering, Kyushu University, Japan. References [1] R.L. Hamilton, O.K. Crosser, Thermal conductivity of heterogeneous two component systems, Ind. Eng. Chem. Fundam. 1 (1962) 187–191. [2] T. Kumada, Thermal conductivity of suspensions – measurement and discussions of shaped effects of dispersions, Trans. Jpn. Soc. Mech. Engrs. 41 (1975) 1209–1218. [3] E. Yamada, T. Ota, Effective thermal conductivity of dispersed materials, Wa¨rme- und Stoffu¨bertragung 13 (1980) 27–37. [4] H. Sasaki, H. Masuda, Control of thermal conductivities of liquids by dispersing ultra-fine particles of metals, in: Proc. 3rd Asian Thermophys. Prop. Conf., 1992, pp. 425–431. [5] H. Masuda, A. Ebata, K. Teramae, N. Hishinuma, Alteration of thermal conductivity and viscosity of liquid by dispersing ultra-fine

[6]

[7] [8]

[9]

[10] [11]

[12] [13]

[14]

[15]

[16]

[17] [18] [19]

[20]

[21]

599

particles (Dispersion of Al2O3, SiO2 and TiO2 ultra-fine particles), Jpn. J. Thermophys. Prop. 7 (4) (1993) 227–233. S.U.S. Choi, Enhancing thermal conductivity of fluids with nanoparticlesDevelopments and Applications of Non-Newtonian Flows, FED-231/MD 66, ASME, New York, 1995, pp. 99–103. X. Wang, X. Xu, S.U.S. Choi, Thermal conductivity of nanoparticlefluid mixture, J. Thermophys. Heat Transfer 13 (1999) 474–480. S. Lee, S.U.S. Choi, S. Li, J.A. Eastman, Measuring thermal conductivity of fluids containing oxide nanoparticles, J. Heat Transfer 121 (1999) 280–289. H.Q. Xie, J. Wang, T. Xi, Y. Liu, F. Ai, Thermal conductivity enhancement of suspensions containing nanosized alumina particles, J. Appl. Phys. 91 (2002) 4568–4572. Y.M. Xuan, Q. Li, W. Hu, Aggregation structure and thermal conductivity of nanofluids, AIChE J. 49 (2003) 1038–1043. D.H. Kumar, H.E. Patel, V.R.R. Kumar, T. Sundararajan, T. Pradeep, S.K. Das, Model for heat conduction in nanofluids, Phys. Rev. Lett. 93 (2004) 144301. P. Keblinski, J.A. Eastman, D.G. Cahill, Nanofluids for thermal transport, Mater. Today 8 (2005) 36–44. X. Zhang, T. Tomimura, M. Fujii, The simultaneous measurements of thermal conductivity and diffusivity with a short hot wire, in: Proc. 14th Japan Symp. Thermophys. Prop., 1993, pp. 23–26. M. Fujii, X. Zhang, N. Imaishi, S. Fujiwara, T. Sakamoto, Simultaneous measurements of thermal conductivity and thermal diffusivity of liquids under microgravity conditions, Int. J. Thermophys. 18 (2) (1997) 327–339. X. Zhang, M. Fujii, Simultaneous measurements of the thermal conductivity and thermal diffusivity of molten salts with a transient short-hot-wire method, Int. J. Thermophys. 21 (1) (2000) 71–84. C.A. Nieto de Castro, S.F.Y. Li, A. Nagashima, R.D. Trengove, W.A. Wakeham, Standard reference data for the thermal conductivity of liquids, J. Phys. Chem. Ref. Data 15 (3) (1986) 1073–1086. JSME Data Book: Heat Transfer, fourth ed., 1986. J.C. Maxwell, A Treatise on Electricity and Magnetism, vol. 1, third ed., Dover, New York, 1954. S.K. Das, N. Putra, P. Thiesen, W. Roetzel, Temperature dependence of thermal conductivity enhancement for nanofluids, J. Heat Transfer 125 (4) (2003) 567–574. M.J. Assael, C.-F. Chen, I. Metaxa, W.A. Wakeham, Thermal conductivity of suspensions of carbon nanotubes in water, Int. J. Thermophys. 25 (4) (2004) 971–985. H.Q. Xie, H. Lee, W. Youn, M. Choi, Nanofluids containing multiwalled carbon nanotubes and their enhanced thermal conductivities, J. Appl. Phys. 94 (8) (2003) 4967–4971.