Viscosity and thermal conductivity of nanofluids containing multi-walled carbon nanotubes stabilized by chitosan

Viscosity and thermal conductivity of nanofluids containing multi-walled carbon nanotubes stabilized by chitosan

International Journal of Thermal Sciences 50 (2011) 12e18 Contents lists available at ScienceDirect International Journal of Thermal Sciences journa...

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International Journal of Thermal Sciences 50 (2011) 12e18

Contents lists available at ScienceDirect

International Journal of Thermal Sciences journal homepage: www.elsevier.com/locate/ijts

Viscosity and thermal conductivity of nanofluids containing multi-walled carbon nanotubes stabilized by chitosan Tran X. Phuoc a, *, Mehrdad Massoudi a, Ruey-Hung Chen b a b

National Energy Technology Laboratory, Department of Energy, P. O. Box 10940, MS 84-340, Pittsburgh, PA 15236, USA Department of Mechanical, Materials and Aerospace Engineering, University of Central Florida, Orlando, Fl 32816-2450, USA

a r t i c l e i n f o

a b s t r a c t

Article history: Received 20 April 2010 Received in revised form 3 September 2010 Accepted 6 September 2010 Available online 12 October 2010

Thermal conductivity, viscosity, and stability of nanofluids containing multi-walled carbon nanotubes (MWCNTs) stabilized by cationic chitosan were studied. Chitosan with weight fraction of 0.1%, 0.2 wt%, and 0.5 wt% was used to disperse stably MWCNTs in water. The measured thermal conductivity showed an enhancement from 2.3% to 13% for nanofluids that contained from 0.5 wt% to 3 wt% MWCNTs (0.24 to 1.43 vol %). These values are significantly higher than those predicted using the Maxwell’s theory. We also observed that the enhancements were independent of the base fluid viscosity. Thus, use of microconvection effect to explain the anomalous thermal conductivity enhancement should be reconsidered. MWCNTs can be used either to enhance or reduce the fluid base viscosity depending on the weight fractions. In the viscosity-reduction case, a reduction up to 20% was measured by this work. In the viscosity-enhancement case, the fluid behaved as a non-Newtonian shear-thinning fluid. By assuming that MWCNT nanofluids behave as a generalized second grade fluid where the viscosity coefficient depends upon the rate of deformation, a theoretical model has been developed. The model was found to describe the fluid behavior very well. Published by Elsevier Masson SAS.

Keywords: MWCNT nanofluids Thermal conductivity Viscosity Stability

1. Introduction Carbon nanotubes (CNTs) are relatively new materials that possess some unique properties including high moduli of elasticity, high aspect ratios, and high thermal conductivity. Thus, by dispersing CNTs into a liquid phase such as water, ethylene glycol, or engine oil, its thermal and transport properties could be enhanced. This work, therefore, will focus on nanofluids prepared by dispersing multiwalled carbon nanotubes (MWCNTs) in deionized water (DW) and we will look at their potential, in terms of their thermal conductivity, stability, and viscosity in heat transfer applications. Stability study is motivated by the fact that CNTs can bundle together easily because of their high van der Waals interaction forces [1,2], non-reactive surface properties, very large specific surface areas and aspect ratios [3]. Therefore, to prepare a stable nanofluid containing CNTs, their surface properties need to be modified either physically or chemically. Physical methods such as ultrasonication and high shear mixing have been used [4,5]. However, such approaches are not very effective and can fragment nanotubes causing a decrease in their aspect ratios. Chemical

* Corresponding author. E-mail address: [email protected] (T.X. Phuoc). 1290-0729/$ e see front matter Published by Elsevier Masson SAS. doi:10.1016/j.ijthermalsci.2010.09.008

methods use either hydrophilic functional groups or surfactants to attach onto CNTs to stabilize them. For this approach, hydrophilic functional groups such as nitric/sulfuric acid mixture, potassium hydroxide group [6,7] and a wide range of surfactants such as sodium dodecylbenzene sulphonate (SDBS), sodium dodecyl sulfate (SDS), gum Arabic (GA) [4,8] have been used. For the chemical methods to be effective either aggressive chemical functionalization or high concentrations of surfactant need to be used. For example, Jiang et al. [8] reported that the optimum amounts of SDS used to obtain a stable homogeneous suspension of 0.5 wt% CNTs was 2 wt%. Aggressive chemical functionalization can cause defects on CNTs altering their thermal and physical properties. High surfactant concentrations can significantly increase the base fluid viscosity and the interface thermal resistance between the carbon nanotubes, thus, limiting the thermal transport in the nanotubematrix [9]. In this work, we explore the use of chitosan to stabilize MWCNTS dispersed in deionized water. Chitosan is a cationic surfactant having a positive charge on the polar portion of a solution. Cationic surfactants such as hexadecyltrimethylammonium bromide (CTAB), gemini surfactant [10], and cationic-onionic mixed surfactant [11] have been found to be effective in stabilizing CNTs and various metal particles [12] with low surfactant concentrations. We use chitosan because it is biocompatible and is a natural polymer isolable from crustacean cell.

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Reports on thermal conductivity of nanofluids containing CNTs have not been consistent. This might be due to the differences in the experimental conditions such as carbon nanotube aspect ratios, dispersants used and the approaches used for preparing the experimental nanofluids. Garg et al. [5] reported an increase from 3 to 5% for nanofluids containing 1 wt% MWCNT measured at 25  C and the ultrasonication from 20 to 80 min, respectively. Chen et al. [6] reported an enhancement of 17.5% at volume fraction of 0.01 for an ethylene glycol based nanofluid containing multi-walled carbon nanotubes without using surfactant and prepared by a wetmechanical reaction. Chen and Xie [9] used the wet-mechanochemical reaction method to functionalize single and double-walled carbon nanotubes. The enhancement was 15.6%, 14.2%, and 12.1% for SWCNT, DWCNT, and MWCNT, respectively at volume fraction of 0.002 (0.2%.). Choi et al. [13] reported significant enhancements for multi-walled carbon nanotubes (mean diameter of 25 nm and a length of 50 mm) in oil suspension at room temperature. At 1% volume fraction, the thermal conductivity increased more than 160% compared to that of the base fluid. A modest (20%) enhancement, however, was reported by Xie et al. [14] for dispersing the nanotubes in distilled water, ethylene glycol and decene. Assael et al. [15] observed a thermal conductivity enhancement of 38% for 0.6 vol% CNTs in water stabilized by SDS and CTAB. Ding et al. [4] reported a maximum enhancement of 79% at 1 wt% MWCNT dispersed in water with SDBS as surfactant. Despite of such a discrepancy, all of these experimental results are found to be higher than those predicted by either the classical theory of Maxwell [16] or its subsequent modifications [17e19]. Many mechanisms such as the ballistic nature of heat transport in nanoparticles [20], the effects of nanoparticle clustering, the Brownian motion of the particles [20e22], and the layering of liquid molecules at the particle-liquid interface [20,23e25] have been suggested to be responsible for such anomalous enhancement. The effect of microconvection due to particle motion has been also considered [21e29]. Kumar et al. [21], Prasher et al. [26] and Jang and Choi [27] argued that the contribution of the microconvection is due to the energy transport by the particles which absorb heat from the liquid at one location and transfer it to an another location. Li and Peterson [28], Koo and Kleinstreuer [29] argued that, due to the Brownian motion, a portion of the surrounding fluid is affected leading to micro-scale mixing and heat transfer. When the particles are moving in the direction of the bulk heat transfer, this kind of mixing action will then enhance the mass transport in the direction of the temperature gradient and enhance the bulk heat transport. Recently, Eapen et al. [30] decided to verify the microconvection theory by testing to see if lighter nanoparticles will increase the nanofluid thermal conductivity. They used well-dispersed silica and Teflon particles, both of which are lighter than commonly used alumina and copper oxide nanoparticles. They observed that the thermal conductivities agreed with the predictions given by the mean-field theory of Maxwell and were far lower than what the microconvection model developed by Prasher, et al. [26] would predict. It is clear that the mechanisms mentioned above may be responsible for the enhancement but none of these can fully explain such enhancement. Since the microconvection theory predicts the heat transport being proportional to the particle velocity, the base fluid viscosity must play a significant role. Our goal, therefore, is to investigate if the base fluid viscosity would affect the nanofluid thermal conductivity enhancement. Although viscosity is as critical as thermal conductivity in establishing adequate pumping power as well as the heat transfer coefficient in engineering systems that employ fluid flow, only limited studies on CNT nanofluid viscosity have been reported [4e7,11]. These studies reveal that aqueous CNT nanofluids behave as non-Newtonian shear-thinning fluids but no attempts have been

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made to interpret the measured data using theoretical models. For many fluids such as polymers, slurries and suspensions, some generalizations have been made to model their shear dependent viscosities. These fluids are known as the Power-Law or the generalized Newtonian fluid models; these widely used models are deficient in many ways, for example, they cannot predict the normal stress differences or yield stresses and they cannot capture the memory or history effects [31,32]. In an effort to obtain a model that does exhibit both normal stress effects and shear-thinning/ thickening, Man and Sun [33], and Man [34] modified the constitutive equation developed by Rivlin and Ericksen [35], (See also Truesdell and Noll [36], Dunn and Fosdick [37] and Dunn and Rajagopal [38]) for a second grade fluid by allowing the viscosity coefficient to depend on the rate of deformation, that is, meff ¼ mPm=2 , where P is the second invariant of the symmetric part of the velocity gradient, and m is a material parameter. When m < 0, the fluid is shear-thinning, and if m > 0, the fluid is shearthickening. Our goal here is to develop a theoretical model that can correlate both the shear stress and the viscosity of a CNT nanofluid to the shear rate. 2. Experiments Multi-walled carbon nanotubes (MWCNT) with >95 wt% purity were purchased from Cheap Tubes Inc. According to the manufacturer’s specification, the tubes have an outside diameter of 20e30 nm, inside diameter of 5e10 nm and length of 10e30 mm. The average density is 2.1 g/cm3. Low molecular chitosan (>75% deacetylation) was purchased from SigmaeAldrich. The base fluid was prepared by mixing an appropriate amount of chitosan into deionized water having 0.5 vol% acetic acid then was stirred for 24 hours using a magnetic stirrer. Three different weight percents (0.1 wt%, 0.2 wt%, and 0.5 wt%) of chitosan and four different weight percents (0.5 wt%, 1wt%, 2 wt%, and 3 wt%) of MWCNTS were used for the present study. All the weights were measured using a micro balance (VP64CN, OHAUS Corporation). For all samples, the nanofluids were prepared by dispersing an appropriate amount of MWCNTs into the previous prepared base fluid (200 mL). The mixture was then ultrasonicated for 10 min at 100% amplitude using a 130 W, 20 kHz ultrasonic processor (VCX 130, Sonics & Materials Inc.) and it was followed by 20 min stirring using a magnetic stirrer. The process was repeated until the total mixing time was 1 h. The prepared samples were set at rest for 24 h before conducting any viscosity and thermal conductivity measurements. Viscosity measurements were performed using a Brookfield R/S Coaxial Cylinder Rheometer (RS115LS, Brookfield Engineering). The device consists of a 40 mm diameter spindle (CC3-40), a cooling jacket with temperature control (FTKY3) and a sample cup that can hold a sample volume of 60 mL for measurements. The device can provide rotational steady state controlled shear stress or controlled shear rate measurements. The rheometer drives the spindle immersed into the sample cup containing the test fluid sample with shear rates up to 12,000 1/s. It measures viscosity by measuring the viscous drag of the fluid against the spindle when it rotates. The cooling jacket is connected to a refrigerated circulating bath (RE204 Bath/Circulator) that controls the temperature from 20 to 150  C. The sample temperature is monitored by a temperature sensor embedded into the cooling jacket. Thermal conductivity was measured using a KD 2 Pro thermal properties analyzer. The instrument has a probe of 60 mm long and 1.3 mm diameter. The instrument measures the thermal conductivity based on the transient hot wire technique. It has a heating element, a thermo-resistor and a microprocessor to control and measure the conduction in the probe. It provides a specified

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accuracy of 5%. About 60 mL of the sample was used and the temperature was kept constant at 25  C. A number of measurements were taken for each sample and only measurements resulted with the mean correlation coefficient r2 > 0.9999 were considered. 3. Results and discussions Fig. 1 shows a photograph of aqueous nanofluids containing 1 wt% MWCNTs with and without chitosan as a stabilizer. The weight fraction of chitosan for this case was 0.2 wt%. As shown in Fig. 1a, the nanofluid prepared without chitosan was not stable at all, the solid precipitated quickly and settled down at the bottom of the vial about 30 min after preparation. The fluid in Fig. 1b was stabilized by 0.2 wt% chitosan and it was stable for months. We have tried to tilt the vial gently and we observed no solid particles accumulated on the vial bottom. Various procedures such as zeta potential, UVeVIS absorption, etc., can be used for nanofluid stability study. Zeta potential values reveal surface charges of solids in aqueous solutions, which can be used to deduce the mechanism by which the solids are stabilized. Such information, however, does not reveal if the solids are well dispersed or stabilized. The UVeVIS absorption is difficult and unreliable because the fluid needs to be very dilute for the probe beam to be able to propagate through. Therefore, in this work the fluid densities at two locations of 6 cm vertically apart are measured to determine if the CNTs are well dispersed and stabilized. To do so, for each location, a fluid sample of about 3 cm3 was gently pipetted out and its density and sound speed were measured using a density and sound analyzer (DSA 5000, Anton Paar). The measurement conditions and results are tabulated in Table 1. It is clear that the densities and sound speeds measured at these locations were the same and they did not change even for 45 days after the fluid was prepared. The chitosan weight percent used by this work was as low as 0.1% yet we could disperse 3 wt% MWCNTs stably. Chen et al. [9] used 0.6 wt% cationic gemini to obtain stable 0.5 wt% MWCNT dispersion, while Jiang et al. [8] used 2.0 wt% SDS to disperse 0.5 wt% CNTs. Thus, comparing with these studies, chitosan can be considered as an effective stabilizer for dispersing CNTs in water. The results for thermal conductivities of various aqueous suspensions containing MWCNTs stabilized by low molecular weight chitosan are tabulated in Table 2. These samples contained from 0.5 wt% to 3 wt% MWCNTs. Since the average density of the MWCNTs is 2.1 g/cm3 as provided by the vendor, the volume fractions of the solid are ranging from 0.24 to 1.43 vol%, respectively.

Table 1 Density and sound speed of nanofluid containing 1 wt% MWCNTs dispersed in deionized water and stabilized by 0.2 wt% chitosan. Location 1 is at 1 cm from the surface, location 2 is at 1 cm from the container bottom. These locations were vertically 6 cm apart.

Deionized water Base fluid (DW þ 0.2wt% Chitosan) MWCNT nanofluid Days (after preparation) 1 15 45

Sound speed (ms)

0.99825 1.000214

1482.69 1486.37

Location 1 1.002625 1.002624 1.002626

Location 2 1.002624 1.002625 1.002627

Location 1 1489.03 1488.99 1488.96

Location 2 1489.01 1488.98 1489.01

Taking the thermal conductivity of MWCNTs of 6000 W/m-K [9], the suspension thermal conductivities are calculated using the Maxwell’s theory [16] given by Eq. (1). The results are also included for easy comparison.





3f lp =lf  1 leff   ¼ 1þ lf lp =lf þ 2  f lp =lf  1

(1)

where leff is the thermal conductivity of the nanofluid, lf is the thermal conductivity of the base fluid, lp is the thermal conductivity of the solid, and 4 is the solid volume fraction. From Table 2, three distinct results are observed: firstly, the thermal conductivity enhancement increased from 2.3% to 13% for all suspensions that contained from 0.5 wt% to 3 wt% MWCNTs (0.24 to 1.43 vol %). These results are generally similar to those reported by Garg et al. [5], Chen et al. [6] and Chen and Xie [9] but significantly lower than those reported by Choi et al. [13], Xie et al. [14], and Assael et al. [15]. Secondly, the measured thermal conductivity enhancements were the same for all fluids having the same solid weight fraction even though their viscosities were different (5.7 cp, 7.2 cp, and 17.7 cp for the base fluids prepared with 0.1 wt%, 0.2 wt%, and 0.5 wt% chitosan, respectively). Thus, the enhancement is independent of the base fluid viscosity. And thirdly, by comparing with the calculated values, the enhancements were significantly higher than those predicted using the Maxwell’s theory. The reason for such an anomalous enhancement has been explained using the “microconvection” theory and many models have been developed which Table 2 Thermal conductivity of aqueous nanofluids containing MWCNTs stabilized by chitosan. Thermal conductivity of Deionized water is about 0.589. Caculated values obtained using the Maxwell’s equation. Eq. (1). Nanofluids

Fig. 1. Sample nanofluid prepared by dispersing MWCNTs in deioinzed water: (a) without chitosan; (b) with 0.2 wt% chitosan.

r (g/cm3)

Vol %

lnf (W/mK)

%Increase ¼ 100 (lfelf)/lf Measured

Calculated

0.581 0.593 0.606 0.620 0.652

e 2.4 4.3 6.7 12.7

e 0.7 1.435 2.88 4.346

0.00 0.24 0.47 0.95 1.40

0.580 0.594 0.606 0.621 0.656

e 2.3 4.3 6.8 13.0

e 0.7 1.435 2.88 4.34

0.00 0.24 0.47

0.580 0.590 0.606

e 2.58 4.30

e 0.7 1.435

DW-0.1 wt% Chitosan þ0.0WT% þ0.5 WT% CNTs þ1.0WT% CNTs þ2.0WT% CNTs þ3.0WT% CNTs

0.00 0.24 0.47 0.95 1.40

DW-0.2 wt% Chitosan þ0.0 WT% þ0.5 WT% þ1.0 WT% þ2.0 WT% þ3.0 WT% DW-0.5 wt% Chitosan þ0.0 WT% þ0.5 WT% þ1.0 WT%

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predict that the thermal conductivity enhancement is proportional to the particle velocity [21e29]. Since the elementary treatment of the diffusive motion leads to the diffusion constant Do being proportional to lv; where l is the diffusion length and v is the mean particle velocity. Thus with Do ¼ ðkB T=3pmdp Þ [39] (where kB is the Boltzmann constant, T is temperature, dp is the particle diameter, and m is the fluid viscosity) then v ¼ Do =l ¼ kB T=3pmldp . Thus, a change in the base fluid viscosity would lead to a change in the particle velocity and hence the thermal conductivity enhancement should vary as predicted by the microconvection theory. Our results, however, show the contrary, that is, the enhancement is indeed independent of the base fluid viscosity. Similar observation was made recently by Eapen et al. [30] in an effort to verify the microconvection theory by testing if lighter nanoparticles would increase the nanofluid thermal conductivity enhancement. They used well-dispersed silica and Teflon particles, both of which are lighter than the commonly used alumina and copper oxide nanoparticles. They observed no microconvection involved. In fact, their measured thermal conductivities agreed with the prediction given by the mean-field theory of Maxwell and were far lower than what the microconvection theory model developed by Prasher et al. [26] would predict. Thus, the use of microconvection theory to explain the anomalous thermal conductivity enhancement should be questioned. The viscosity and the corresponding shear stress measured for aqueous nanofluids containing MWCNTs stabilized by different chitosan weight fractions are shown in Figs. 2 and 3. Also included in the figures are the calculated results that were obtained using our present theoretical approach which will be discussed below. In general, we observed that dispersing chitosan into deionized water (DW) increases its viscosity significantly. If the concentration of the added chitosan is high, the resultant solution will behave as a nonNewtonian fluid and its viscosity decreases as the shear rate increases. For example, for the range of the shear rates up to 200 1/ s, the viscosity of DW with 0.1 wt% and 0.2 wt% chitosan remained at about 5.7 cp and 7.2 cp, respectively, while those measured for DW with 0.5 wt% chitosan decreased slightly from about 17.70 cp to 16.60 cp. A nanofluid that is prepared by dispersing MWCNTs into a base fluid, depending on the weight percent of the added solid, can be either a viscous-enhancement or viscous-reduction nanofluid due to the lubricating effect of the CNTs. Such a lubricating effect was also reported by Chen et al. [6]. The solid weight percent at which the fluid could switch from viscous-reduction to viscousenhancement is seen to depend also on the weight percent of the stabilizer. For example, when the base fluid contained 0.1 wt% chitosan, the lubricating effect occurred when the added CNTs was about 0.5 wt% or less while for 0.2 wt% chitosan it prevailed when the added solid was about 1 wt%. All the nanofluids prepared under these conditions behaved similarly to the base fluid, their viscosities remained independent of the shear rate and the shear stresses increased linearly as the shear rate increased. The viscous enhancement case was seen at higher solid weight percents. In this case, the nanofluids became non-Newtonian. Their viscosities were significantly higher than that of the base fluids and they decreased rapidly while the shear stresses increased nonlinearly as the shear rate increased. We define a drag reduction index as



Dindex

s  sf ¼ 100 sf

 (2)

where s is the shear stress and sf is the shear stress of the base fluid. The drag reduction can be calculated to be about 20% for the present experimental conditions. Thus, it can be said that, CNTs can be used either as a viscosity enhancer or drag-reducer agent.

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Fig. 2. Viscosity and shear stress as a function of shear rate showing the effect of the MWCNT weight percent. The base fluid is DW þ 0.1 wt% chitosan. The measured values are shown by symbols while the calculated values are shown by the solid and dotted lines. The calculated viscosity values were obtained using Eq. (16) and the shear stresses were calculated using Eq. (15) with m ¼ 0.547, 0.65, and 0.647 and ¼ 0.134; 0.331; and 0.523 for MWCNT weight percent increased from 1%, 2%, and to 3%, respectively.

In many studies, the behavior of the viscosity and the shear stress of nanofluids such as those presented in these figures has been interpreted using the widely used empirical model developed by Casson [40]

s1=2 ¼ s1=2 m1=2 g_ 1=2 O þ N

(3)

In this equation so is the yield stress, mN is the suspension viscosity at infinite shear rate and g_ is the shear rate. Phuoc and Massoudi [41] used this equation and obtained mN being 0.1225 cp and 0.0225 cp for Fe2O3 e Deionized water nanofluids with PEO and PVP as a dispersant, respectively. These values are about two orders of magnitude lower than the viscosity of the base fluid (the liquid phase prepared with PVP as a dispersant (DW-0.2% PVP) had a viscosity similar to that of water, while the viscosity of water with PEO as a dispersant, (DW-0.2% PEO), was about 12.5 cp). Choi et al. [42] used this equation and calculated the intrinsic viscosities of CrO2 e ethylene glycol, g-Fe2O3, a-Fe2O3 e EG and Ba-ferrite-EG nanofluids at infinite shear rate and reported a decrease of the viscosity with an increase in the particle volume fraction. This could be problematic since the intrinsic viscosity should reach the viscosity of the base fluid in case of dilute suspensions or increase as the particle volume fraction increases if the suspension is dense.

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where d/dt is the total time derivative, given by ðdð$Þ=dtÞ ¼ ðvð$Þ=vtÞ þ ½gradð$Þu; where u is the velocity vector. The thermodynamics and stability of fluids of second grade have been studied in detail by Dunn and Fosdick [37]. They show that if the fluid is to be thermodynamically consistent in the sense that all motions of the fluid meet the ClausiuseDuhem inequality and that the specific Helmholtz free energy of the fluid be a minimum in equilibrium, then

m  0; a1 S0; a1 þ a2 ¼ 0:

(8)

By allowing the viscosity coefficient to depend on the rate of deformation, Man and Sun [33], and Man [34] modified the constitutive equation, Eq. (4), and proposed the following equation

T ¼ p1 þ mPm=2 A1 þ a1 A2 þ a2 A21

(9)

where

1 2

P ¼ tr A21

(10)

is the second invariant of the symmetric part of the velocity gradient, and m is a material parameter. When m < 0, the fluid is shear-thinning, and if m > 0, the fluid is shear-thickening. A subclass of models given by Eq. (9) is the generalized power-law model, which can be obtained by setting a1 ¼ a2 ¼ 0 in Eq. (9) (see Gupta and Massoudi [43], Massoudi and Phuoc [44], Massoudi and Vaidya [45], and Man and Massoudi [46] for further discussions of this model). Notice that if the normal stress parameters a1 and a2 are zero, then

T ¼ p1 þ mPm=2 A1 Fig. 3. Viscosity and shear stress as a function of shear rate showing the effect of the MWCNT weight percent. The base fluid is DW þ 0.2 wt% chitosan. The measured values are shown by symbols while the calculated values are shown by the solid and dotted lines. The calculated viscosity values were obtained using Eq. (16) and the shear stresses were calculated using Eq. (15) with m ¼ 0.584 and 0.678 and ¼ 0.354 and 0.641 for 2 wt% and 3 wt% CNTs, respectively.

In this paper, we will use this simplified form, which can also be considered as a generalized power-law fluid model. Using the cylindrical coordinate system for our present measurements and assuming u ¼ wðrÞez where ez denotes a unit vector along the z direction yields the following calculations

2

To the best of our knowledge, it has not been reported whether nanofluids, in general, exhibit the normal stress effects, a non-linear phenomenon related to the stresses that are developed orthogonal to the planes of shear. Therefore, we propose to use a general model that can exhibit both the normal stress effects and the shearthinning/thickening effects as shown by our present results. To do so, we assume that nanofluids, such as the one studied in the present work, behave as generalized second grade fluids. For a second grade fluid the Cauchy stress tensor is given by [35e38]

T ¼ p1 þ mA1 þ a1 A2 þ a2 A21

(4)

0 A1 ¼ 4 0

vw vr

2 6 A21 ¼ 4

0 0 0

 vw 2 vr 0 0

vw vr

3

05 0

0 0 0

(12)

3 0 7 0 5 vw 2

And

"   # 1 vw 2 P ¼ 2 2 vr The z-component of the stress tensor becomes:

A1 ¼ L þ LT ;

where shear-dependent viscosity can be defined as

A2 ¼

dA1 þ A1 L þ LT A1 ; dt

L ¼ grad u:

(13)

vr

where p is the indeterminate part of the stress due to the constraint of incompressibility, m is the coefficient of viscosity, a1 and a2 are material moduli which are commonly referred to as the normal stress coefficients. The kinematical tensors A1 and A2 are defined through

(5)

(11)

Trz ¼ m

"  #m=2     vw 2 vw ¼ mg_ m g_ vr vr

"

 #m=2 vw 2 ¼ mg_ m vr

(14)

(15)

(6)

meff ¼ m

(7)

In order to use Eqs. (15) and (16) to interpret the viscosity and the shear stress data, we need to determine m and m from the measured

(16)

T.X. Phuoc et al. / International Journal of Thermal Sciences 50 (2011) 12e18

viscosity data. To do so, from the experimental data we obtained m ¼ 0.547, 0.65, 0.647 when CNTs weight percent increased from 1 to 3 stabilized by 0.1 wt% chitosan. The values of m were more affected by the CNT weight percent and they increased from 0.134 to 0.331 to 0.523 for the same range of the CNTs weight percent. While using 0.2 wt% chitosan, it was found that m ¼ 0.584 and 0.678 and m ¼ 0.354 and 0.641 for 2 wt% and 3 wt % CNTs, respectively. It is clear that using the generalized powerlaw model, Eq. (11), as a subclass of the generalized second grade fluid models, the present measured values were very well predicted. For a given weight percent of the stabilizer, increasing the CNTs weight percent has a strong effect on the value of m. For a given value of CNTs weight percent, increasing the weight percent of the stabilizer increases both m and m. This is obvious because m is related to the fluid viscosity and m indicates the degree of the nonNewtonian nature of the fluid. The main factors that alter the fluid behavior and its properties are the weight percents of both the stabilizer and the CNTs. 4. Conclusions We have conducted a preliminary study on the stability, thermal conductivity, and viscosity of MWCNT nanofluids stabilized by cationic chitosan. Even with only 0.1 wt% chitosan, we could disperse and stabilize 3 wt% MWCNTs in water. The thermal conductivity enhancements obtained were significantly higher than the predictions made using the classical Maxwell’s theory. We also observed that the thermal conductivity enhancements were independent of the base fluid viscosity indicating that the particle velocity does not have a significant effect on the thermal conductivity enhancement. Thus, the use of the microconvection theory, which predicts the enhancement being proportional to the particle velocity, should be reconsidered. By assuming that MWCNT nanofluids, in general, behave as generalized second grade fluids whose viscosity coefficient depends on the rate of deformation, a theoretical model has been developed. The model was found to be suitable for describing the fluid behavior. Acknowledgements This work was supported by the DOE-NETL under the EPact Complementary program. Disclaimer: Reference in this paper to any specific commercial product is to facilitate understanding and does not necessarily imply its endorsement by the United States Department of Energy. References [1] L.A. Girifalco, M. Hodak, R.S. Lee, Carbon nanotubes, buckyballs, ropes, and a universal graphic potential, Phys. Rev. B 62 (2000) 13104e13110. [2] T. Lin, V. Bajpai, T. Ji, L. Dai, Chemistry of carbon nanotubes, Aust. J. Vhem. 56 (2003) 635e651. [3] C. Park, Z. Ounaies, K.A. Watson, R.E. Crooks, J.J. Smith, S.E. Lowther, Dispersion of single wall carbon nanotubes by in situ polymerization under sonication, Chem. Phys. Lett. 364 (2002) 303e308. [4] Yulong Ding, Hajar Alias, Dongsheng Wen, Richard A. Williams, Heat transfer of aqueous suspensions of carbon nanotubes (CNT nanofluids), Int. J. Heat Mass Transfer 49 (2006) 240e250. [5] Paritosh Garg, Jorge L. Alvarado, Charles Marsh, Thomas A. Carlson, David A. Kessler, Kalyan Annamalai, An experimental study on the effect of ultrasonication on viscosity and heat transfer performance of multi-wall carbon nanotube-based aqueous nanofluids, Int. J. Heat Mass Transfer 52 (2009) 5090e5101. [6] Lifei Chen, Huaqing Xie, Yang Li, Wei Yu, Nanofluids containing carbon nanotubes treated by mechanochemical reaction, Thermochim. Acta 477 (2008) 21e24. [7] Gwon Hyun Ko, Kyoungyoon Heo, Kyoungjun Lee, Dae Seong Kim, Chongyoup Kim, Yangsoo Sohn, Mansoo Choi, An experimental study on the pressure drop of nanofluids containing carbon nanotubes in a horizontal tube, Int. J. Heat Mass Transfer 50 (2007) 4749e4753.

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