Experimental Thermal and Fluid Science 36 (2012) 65–71

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Experimental Thermal and Fluid Science

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Viscosity of water based SWCNH and TiO2 nanofluids

Sergio Bobbo a,⇑, Laura Fedele a, Anna Benetti a, Laura Colla a, Monica Fabrizio b, Cesare Pagura b,Simona Barison b

a Consiglio Nazionale delle Ricerche, Istituto per le Tecnologie della Costruzione, Corso Stati Uniti, 4, 35127 Padova, Italyb Consiglio Nazionale delle Ricerche, Istituto per l’Energetica e le Interfasi, Corso Stati Uniti, 4, 35127 Padova, Italy

a r t i c l e i n f o

Article history:Received 13 January 2011Received in revised form 10 June 2011Accepted 19 August 2011Available online 12 September 2011

Keywords:NanofluidViscosityTitanium dioxide (TiO2)Single wall carbon nanohorn (SWCNH)

0894-1777/$ - see front matter � 2011 Elsevier Inc. Adoi:10.1016/j.expthermflusci.2011.08.004

⇑ Corresponding author. Tel.: +39 049 8295736; faxE-mail address: sergio.bobbo@itc.cnr.it (S. Bobbo).

a b s t r a c t

At present, literature data on viscosity of nanofluids are still scarce and controversial. The possible non-Newtonian behaviour of these fluids is frequently neglected and the problems related to the nanofluidsstability and the actual composition are often not considered. In this paper, viscosity data for nanofluidsformed by water, as base fluid, and solid nanoparticles of two different materials – single wall carbonnanohorn (SWCNH) or titanium dioxide (TiO2) – are presented. Viscosity was measured by using arheometer and obtained as a function of the nanoparticles mass fraction and the shear rate, thus allowingevaluation of the possible non-Newtonian behaviour for the nanofluid. Both the studied nanofluidsshowed a Newtonian behaviour.

The viscosity data were correlated by different equations and here an empirical correlation is proposed.� 2011 Elsevier Inc. All rights reserved.

1. Introduction

In the last years, huge attention has been given to technologiesaimed at increasing the energy efficiency of plants, machines andprocesses. A critical point in systems employing secondary fluidsfor the energy transportation (e.g. chillers for air conditioning,supermarket refrigeration plants, thermal solar plants) is the lowthermal conductivity of the traditionally used thermal vectors,i.e. water, oils and ethylene glycol. Nanotechnologies can givenew opportunities in solving this problem by means of nanofluids,a new family of fluids obtained by dispersing solid nanoparticles inliquids. The transport properties, mainly thermal conductivity andheat transfer coefficient, promise to be much higher when thesefluids are used instead of the base fluids, as suggested by the wideavailable literature. Among others, Keblinski et al. [1] comparedthe enhancement in thermal conductivity of water-based nanofl-uids taken from several works, finding improvement till 60%, whileYu et al. [2] observed various nanofluids that pointed out heattransfer enhancement in the range of 15–40%, with readily avail-able oxide nanoparticles in a variety of base fluids. However, thepresence of nanoparticles could increase the viscosity of nanofluidin respect to the base fluid, thus increasing the energy required topump the fluid in the circuits and reducing or nullifying the gain inheat transfer efficiency.

In order to understand the nanofluid effectiveness, the knowl-edge of thermal conductivity and viscosity is essential. In particu-lar, in the case of laminar flow, Timofeeva et al. [3] observed that a

ll rights reserved.

: +39 049 8295728.

nanofluid can be useful when the increase in viscosity is less thanfour times the increase in thermal conductivity.

Alternatively, in case of turbulent flow, the Mouromtseff num-

ber Mo ¼ kaqbcdp

le

� �was proposed as a figure of merit for the compar-

ison of the heat transfer capability of alternative thermal fluids [4].Here, k, q, cp and l are the thermal conductivity, density, specificheat and dynamic viscosity of the fluid, respectively. The expo-nents a, b, d, and e assume proper values corresponding to the heattransfer mode. Referring to fluids flowing in turbulent flow inside aspecified geometry at a given velocity, the highest heat transferrate is presented by the secondary fluid with the highest Mour-omtseff number [5].

For a nanofluid, it is well known that q and cp can be calculatedby means of standard linear equations, while k and l have to bemeasured.

For this reason, extensive experimental measurements on ther-mal conductivity and viscosity are necessary.

At present, viscosity data for nanofluids are still scarce in liter-ature and frequently discordant and controversial. Various papershave detected significant differences from the base fluids only fornanofluids at nanoparticle concentration higher than 1% by mass[6,7]. A systematic investigation on nanofluids viscosity turnedout to be fundamental in order to understand the main parametersinfluencing nanofluids rheological behaviour and optimise themalso in relation to the thermal properties.

Considering the high potentiality of nanofluids as energy vec-tors in different application, as solar panels [8], air conditioningmachines and refrigeration plants [9,10], a research project ontheir thermal and rheological properties has begun in our institute.

Nomenclature

a prime nanoparticles radiusaa aggregates nanoparticles radiusD fractal indexT temperature (K)/ particle volume fractionl dynamic viscosity (Pa s)x particle mass fraction

Subscriptsexp experimentalf base fluidnf nanofluidSWCNH related to single wall carbon nanohornsTiO2 related to titanium oxide

66 S. Bobbo et al. / Experimental Thermal and Fluid Science 36 (2012) 65–71

In this paper, two different water-based nanofluids are studied.They were prepared dispersing two different kinds of nanoparti-cles, specifically single wall carbon nanohorns (SWCNH) and tita-nium dioxide (TiO2), in water. TiO2 is a stable, inexpensive anddurable oxide, that already showed some enhancements in thethermal conductivity of water-based fluids [2]. The SWCNHs areroughly spherical aggregates of nanohorns, consisting in a singlelayer of a graphene sheet wrapped into an irregular tubule witha variable diameter of generally 2–5 nm and a length of 30–50 nm, with their tips cone-shaped. The SWCNHs are mainly ofthree types: dahlias, buds and seeds [11,12]. The critical point thatdifferentiates SWCNHs from carbon nanotubes (CNTs), thatshowed important thermal conductivity increase [12], is theirmuch lower toxicity [13], due to both the lack of fibril-like struc-ture and the absence of any metal nanoparticles used to catalysenanotube growth during their production. Moreover, their hetero-geneous surface structure favours their dispersion in water.

The use of dispersants was necessary to stabilize the nanoparti-cle dispersions. After careful analysis of the average size distribu-tion of the nanoparticles in solution, long time, by means of ananosizer apparatus [14], sodium n-dodecyl sulphate (SDS) andpolyethylene glycol (PEG) were used as dispersants for the nanofl-uids based on SWCNH and TiO2, respectively.

The viscosity for these fluids was measured at ambient pressureand in the temperature range between 283.2 K and 353.2 K bymeans of a rheometer, in order to evaluate their Newtonian behav-iour, too. The data were regressed by viscosity correlations.

2. Experimental section

2.1. Materials

Deionised water (Millipore, Billerica MA, USA, 18.2 X) was usedas base fluid.

The TiO2 nanoparticles used for the dispersions were purchasedfrom Degussa (TiO2, P25). They had a spherical shape with a de-clared 21 nm diameter. The SWCNHs used in this work were pro-duced and provided by Carbonium Srl (info@carbonium.it). Themorphological characterisation of nanoparticles was performedby field emission scanning electron microscopy (FE-SEM) with aSIGMA Zeiss instrument (Carl Zeiss SMT Ltd., UK). SEM picturesof TiO2 and SWCNHs are shown in Fig. 1, where the actual dimen-sions of nanoparticles can be deduced to be 20–30 nm and 60 nmfor TiO2 and SWCNH, respectively.

As dispersant, sodium n-dodecyl sulphate (SDS, 99%, Alfa Aesar)and polyethylene glycol 600 (PEG, Alfa Aesar) were used.

Fig. 1. SEM (scanning electron microscope) images of (a) SWCNH and (b) TiO2

nanoparticles.

2.2. Nanofluids preparation

The nanofluids were prepared by dispersing the nanoparticlesin water by a two-step method. Different preparation methods(ultrasonic agitation, ball milling and homogenisation) and

different dispersants were proven [14]. The high pressure homog-enisation method turned out to be the best process to improve thesuspension stability and then it was used to prepare both SWCNHand TiO2 nanofluids. The nanoparticles were mechanically dis-persed in water at different concentrations, i.e. 0.01%, 0.1%, 1% bymass. Then, a high pressure homogenizer (up to 1000 bar) was em-ployed to optimise the dispersion. Different dispersants wereadded to stabilize the solutions. After some trials, SDS and PEGwere identified as the best dispersants for the nanofluids basedon SWCNH and TiO2, respectively.

For the nanofluids based on SWCNH at concentrations of 0.1%and 1% by mass, the ratio between nanoparticles and dispersantmass was 1:1. For the lowest concentration (0.01% by mass), theratio was 1:3.

For the water–TiO2 nanofluid, the ratio between nanoparticlesand dispersant mass was 1:2 for each concentration.

2.3. Nanofluids stability characterisation

A Zetasizer Nano ZS (Malvern) was used to analyse the averagedimension of the nanoparticles in solution. The Zetasizer works

0

2

4

6

8

10

12

14

16

0.1 1 10 100 1000 10000

Inte

nsity

%

Size / nm

(a)

0

2

4

6

8

10

12

14

16

0.1 1 10 100 1000 10000

Inte

nsity

%

Size / nm

(b)

Fig. 2. Particle diameter size distribution, according to the intensity, for (a) thewater–SWCNH and (b) the water–TiO2 at 0.1% nanofluids (with dispersants), (d)just after preparation and (s) after 18 days.

S. Bobbo et al. / Experimental Thermal and Fluid Science 36 (2012) 65–71 67

measuring the Brownian motion of the particles in the sample bymeans of the Dynamic Light Scattering (DLS) and then calculatingthe size from this, basing on theory. Fig. 2 shows the particle sizedistribution, according to the intensity detected by the Zetasizer,for the water–SWCNH and the water–TiO2 nanofluids, respectively,just after preparation and after 18 days.

Fig. 3. Dynamic viscosity of water at (s) 283.2, (d) 293.2, (h) 303.2, (j) 313.2, (D)323.2, (N) 333.2, (e) 343.2 and (�) 353.2 K compared to (—) Refprop 8.0.

2.4. Viscosity measurements

The dynamic viscosity data were measured at ambient pressureand in a temperature range between 283.2 K and 353.2 K by meansof an AR-G2 rheometer (TA Instruments). It is a rotational rheom-eter, with a plate-cone geometry. A 1� cone, with a diameter of40 mm, was employed. The rotational speed range is between 0and 300 rad s�1. In order to stabilize the measurement tempera-ture at ±0.1 K, an upper heated plate (UHP) was used. Temperaturewas measured by a Pt100 X thermo-resistance inside the Peltierplate of the rheometer, with a resolution of 0.01 K and a declaredaccuracy of 0.1 K. A critical point in this measurement is the sam-ple loading. After some trials with water, a constant quantity ofabout 0.33 ml was considered optimal for the analysis. The samplewas deposited using a pipette, taking care no air bubbles were in-side. Before the measurements, the rheometer was carefully cali-brated at each temperature, i.e. the non-zero moment of inertiaof the rheometer spindle, the non-zero moment of inertia of themeasurement geometry and the instrument friction were cali-brated. Then, due to thermal expansion, zero reference point atthe experimental temperature had to be found. Finally, the rota-tional mapping of the instrument allowed finding the small varia-

tions in behaviour around one revolution of the shaft, monitoringthe torque required to maintain this speed through a full 360� ofrotation.

All the measurements were performed at constant temperatureand variable shear rate, starting from 200 s�1 to 1600 s�1and viceversa, at constant step of about 150 s�1 (except for temperatureshigher than 333.2 K, at which faster measurements had to be per-formed, due to water evaporation). A conditioning step of 10 s wascarried out and a pre-shear rate at 200 s�1 was applied before themeasurements to remove any possible fluid ‘‘memory’’, due to thesample preparation, storage and loading. Each experimental pointis the average of three values of viscosity, sampled under constantshear rate.

2.5. Viscosity measurement validation

To evaluate the rheometer uncertainty, a well known fluid aswater was analysed at each temperature and the viscosity datawere compared with Refprop 8.0 database [15]. As shown byFig. 3 (and in Table 2), all the measured data for water are quiteclose to the literature data in the shear rate range between400 s�1 and 1400 s�1, being the percentage absolute average devi-ation (AAD%) about 1.5%. The deviations at low shear rates shouldbe due to difficulties in the torque control by the rheometer, whileat high shear rate to changes in the fluid laminar flow.

The estimated uncertainty in the viscosity measurements is lessthan 2%.

3. Results and discussion

3.1. Experimental data

The selections of the most suitable preparation method and dis-persant type and concentration have been done trying to optimisethe stability of the nanofluids by analysing the average size distri-bution of the particles along the time. In fact, nanoparticles canaggregate and settle down after dispersing in the base fluid, yield-ing to a loss in the stability of the nanofluid and thus hinderingtheir application.

In particular, the dispersants can act in different ways, beingdifferent the interactions between these fluids and the nanoparti-cles. The SDS, an anionic surfactant, gives a negative charge tothe SWCNH through the sulphate group, leading to electrostaticrepulsions between the particles and stabilizing the suspension[16]. On the contrary, the PEG molecules are adsorbed by the sur-

Fig. 4. Dynamic viscosity at 283.2 K of (d) water, water and SDS at (D) 0.03%, (e)0.1% and (h) 1% in mass, (N) water–0.03% SDS and 0.01% SWCNH, (�) water–0.1%SDS and 0.1% SWCNH and (j) water–1% SDS and 1% SWCNH; (—) water calculatedby Refprop 8.0.

Fig. 5. Dynamic viscosity at 283.2 K of (d) water, water and PEG 600 at (D) 0.02%,(e) 0.2% and (h) 2% in mass, (N) water–0.02% PEG 600 and 0.01% TiO2, (�) water–0.2% PEG 600 and 0.1% TiO2 and (j) water–2% PEG 600 and 1% TiO2. (—) watercalculated by Refprop 8.0.

68 S. Bobbo et al. / Experimental Thermal and Fluid Science 36 (2012) 65–71

face of TiO2 nanoparticles, forming a dense layer around the parti-cles, producing stabilization by steric effects and leading to the for-mation of more compact aggregates [17].

The nanofluids formed by water, SDS and SWCNH are very sta-ble even after several days. The measured nanoparticle averagediameter was around 140 nm, 188 nm and 120 nm for the 0.01%,0.1% and 1% mass concentrations, respectively.

Solutions of water, PEG and TiO2 tend to be less stable, since, inrest condition, part of nanoparticles slowly sediments under grav-ity. However, these solutions can easily recover the initial size dis-tribution by simple mechanical agitation. Considering the possibleuse of these fluids in plants with forced circulation and then a con-tinuous mixing condition, nanoparticles settling effects are practi-cally negligible. All the measurements provide much higher valuesthan the 21 nm correspondent to the nominal diameter of thenanoparticles: at 0.01%, 0.1% and 1% mass concentrations the mea-sured average diameter is actually around 180 nm, 121 nm and132 nm, respectively, indicating a tendency of titania particles torearrange in liquid media forming aggregates, but maintainingdimensions that can still be defined as nanometric.

The Zeta potential of nanofluids was also measured by ZetasizerNano and in Table 1 the values for SWCNH and TiO2 nanofluidswith dispersants are shown. All the measured nanofluids show aZeta potential higher than |30| mV.

Dynamic viscosity data of the water-dispersants mixtures andwater based nanofluids were measured from 283.2 K to 353.2 Kby increments of 10 K per step.

The investigated fluids, apart from bidistilled water, were

– water + SDS at 0.03%, 0.1% and 1% by mass;– water + PEG 600 at 0.02%, 0.2% and 2% by mass;– water + SWCNH at 0.01%, 0.1%, 1% by mass + SDS at 0.03%, 0.1%

and 1% by mass, respectively;– water + TiO2 at 0.01%, 0.1%, 1% by mass + PEG 600 at 0.02%, 0.2%

and 2% by mass, respectively.

The nanofluid compositions are indicated both in mass and vol-ume percentage in Table 2.

In Figs. 4 and 5, viscosity data of the measured fluids at 283.2 Kare represented.

As shown in Fig. 4, base fluids formed by water and SDS, both atthe 0.03% and 0.1% by mass, have viscosities very similar to water.SDS shows its influence at concentration of 1% by mass, with a vis-cosity enhancement of about 7%. Even viscosities of nanofluids

Table 1Zeta potential for SWCNH and TiO2 nanofluids with dispersants.

Nanofluid Zeta potential/|mV|

SWCNH 1 wt.%, SDS 1 wt.% 56SWCNH 0.1 wt.%, SDS 0.1 wt.% 57SWCNH 0.01 wt.%, SDS 0.03 wt.% 50TiO2 1 wt.%, PEG 2 wt.% 40TiO2 0.1 wt.%, PEG 0.2 wt.% 43TiO2 0.01 wt.%, PEG 0.02 wt.% 37

Table 2Nanofluids particle concentrations.

Nanofluid wt.% vol.%

Water–SWCNH 0.01 0.004760.1 0.04761 0.4787

Water–TiO2 0.01 0.002560.1 0.02571 0.2583

with SWCNH at 0.01% and 0.1% are similar or lower than those ofwater. On the contrary, the viscosity of water–SDS–SWCNH at 1%nanofluid increases of about 13%.

In Fig. 5, it is shown that base fluids of water and PEG at 0.02%and 0.2% have viscosity slightly lower than water. Larger differ-ences are given by solution of water and PEG at 2%, with viscosity5% higher than water. Also the nanofluids with TiO2 at 0.01% and at0.1% have viscosities similar to the base fluids, while the solution ofwater, PEG and TiO2 at 1% shows a viscosity higher than 7% in re-spect to water. These behaviours are analogous at eachtemperature.

In Fig. 6, the relative viscosity at 293.2 K is given as a function ofnanofluid volume composition. The viscosity enhancement is evi-dent only at the highest composition for both fluids.

This behaviour could be due also to nanoparticles aggregation,together with concentration, since a high relative viscosity couldbe observed when agglomerations are formed inside the nanofluid,as already suggested by [18,19]. However, this phenomenon is ob-served only at the highest composition.

Table 3 summarises the viscosity measurements for all theSWCNH–nanofluids and the TiO2–nanofluids at the different com-positions at constant shear rate (about 800 s�1). It should be noted

Fig. 6. Relative viscosity as a function of volumetric composition at 293.2 K for (�)SWCNH and (s) TiO2 water-based nanofluids.

S. Bobbo et al. / Experimental Thermal and Fluid Science 36 (2012) 65–71 69

that measurements at 353.2 K are difficult to perform, since waterbegins to vapourize and nanoparticles, especially TiO2, begin toaggregate.

In Figs. 7 and 8, the trend of the shear stress as a function of theshear rate is shown for the two studied nanofluids, at each compo-sition, at 283.2 K, evidencing a Newtonian behaviour of bothnanofluids.

3.2. Literature analysis

Up today, no literature data are available for the same nanofl-uids here considered, i.e. fluids formed by the same nanoparticles,base fluids, dispersants, at the same compositions and with thesame preparation methods. However, Tseng and Lin [20], Chenet al. [21] and Alphonse et al. [17] published viscosity data of dif-ferent TiO2-based nanofluids.

In the first paper, TiO2 suspensions in water, without any dis-persant and prepared by ball milling, were considered at composi-tions ranging between 5% and 12% by volume. They foundpseudoplastic flow behaviour, deducing the presence of aggregatesin the suspension. At shear rate 100 s�1, they found viscosity from100% to 1200% higher than that of water for composition from 5%to 12%, respectively.

Chen et al. studied a water–TiO2 nanofluid (without any disper-sant) prepared by ultrasonication, followed by high shear homog-enisation, with compositions roughly ranging between 0.1% and1.2% by volume. They used a rheometer at ambient temperature,finding viscosity enhancement till roughly 10%, for the highestcomposition. Their nanofluids were stable for at least 1 monthand showed a Newtonian behaviour. These results can be consid-ered in good agreement with ours, since we also found a Newto-

Table 3Experimental viscosity data for water-based nanofluids with SWCNH and TiO2 and water,

T/K SWCNH (0.01 wt.%/mPa s)

SWCNH (0.1 wt.%/mPa s)

SWCNH (1 wt.%/mPa s)

TiO2 (0.01 wtmPa s)

283.2 1.29 1.31 1.48 1.28293.2 1.04 1.00 1.19 1.02303.2 0.80 0.76 0.92 0.76313.2 0.65 0.64 0.76 0.64323.2 0.55 0.55 0.65 0.53333.2 0.47 0.49 0.53 0.48343.2 0.41 0.40 0.46 0.43353.2 0.36 0.32 0.43 0.37

nian behaviour for the TiO2-based nanofluids, with viscosityenhancement till 7% for composition of 0.3% by volume (around1% by mass).

Finally, Alphonse et al. studied rheology and stability of TiO2–water nanofluids at compositions ranging between 3.4% and12.5% by mass at 293.2 K, using a rheometer with shear rate vary-ing over the range of 1–1000 s�1. They found a Newtonian behav-iour in the shear rate range of 1–100 s�1 and a shear thinningbehaviour for higher shear rate. After the addition of low amountof PEG 2000 (less than 20 g L�1), they observed a marked decreaseof viscosity with respect to the nanofluids without dispersant, untila minimum of about 50–60%.

3.3. Theoretical models

In literature, several theoretical models have been proposed tocorrelate viscosity data of nanofluids and few of them were appliedto these experimental data.

In general, they derived from the Einstein model [22],

lnf ¼ lf ð1þ 2:5/Þ ð1Þ

based on the assumption of a viscous fluid containing spherical par-ticles. Here, / is the particle volume fraction and lnf and lf are thedynamic viscosity of the nanofluid and the based fluid, respectively.In general, this formula is applicable when / is lower than 1% andthere are not nanoparticle interactions.

Starting from the Einstein’s formula, Brinkman suggested anequation applicable to moderate particle volume concentration,roughly 4% [23], in the form

lnf ¼ lf1

ð1� /Þ2:5ð2Þ

In [24], Batchelor considered the nanoparticle Brownian motionand their interaction, proposing the formula

lnf ¼ lf ð1þ 2:5/þ 6:5/2Þ ð3Þ

All these equations base on the assumptions that the viscosityof the nanofluid is only a function of the base fluid viscosity andthe particle concentration and that the nanoparticles can be mod-elled as rigid spherical particles.

As shown in Fig. 9, Eqs. (1)–(3) are able to estimate nanofluidsviscosity for the lowest compositions, but overestimate the sus-pensions at 1 wt.%. These results are in contrast with literature,e.g. [25], where these equations underestimated nanofluids viscos-ity for concentrations higher than 1 vol.%. It could be due to differ-ent employed preparation methods, dispersants and nanoparticledimensions.

Recent studies suggested correlations between the nanofluidshigh viscosity and the nanoparticles aggregation [18,26–28]. Dif-ferent models have been proposed taking into account this phe-nomenon, as the Krieger–Dougherty equation [25]

at constant shear rate (about 800 s�1).

.%/ TiO2 (0.1 wt.%/mPa s)

TiO2 (1 wt.%/mPa s)

Waterexp (mPa s) Watercalc11 (mPa s)

1.32 1.40 1.31 1.311.01 1.13 1.02 1.000.80 0.84 0.81 0.800.65 0.72 0.66 0.650.56 0.58 0.55 0.550.50 0.52 0.47 0.470.41 0.48 0.41 0.400.35 0.45 0.34 0.35

Fig. 9. Viscosity as a function of temperature for water–SWCNH–SDS nanofluid. (�)Experimental data; (—) Eq. (1); (- -) Eq. (2); (� � �) Eq. (3); (-e-) Eq. (4); (-s-) Eq. (6);(-h-) Eq. (7); (-�-) Eq. (8).

Fig. 10. Viscosity as a function of temperature for water–TiO2–PEG nanofluid. (�)Experimental data; (—) Eq. (1); (- -) Eq. (2); (� � �) Eq. (3); (-e-) Eq. (4), (-s-) Eq. (6), (-h-) Eq. (7), (-�-) Eq. (8).

Table 4Regressed parameters of Eq. (8).

Nanofluid a b

Water–SWCNH �0.50437 1.74486Water–TiO2 0.36838 0.25271

Fig. 7. Shear stress as a function of shear rate for water–SWCNH–SDS nanofluid at283.2 K. (D) 0.01% SWCNH, (�) 0.1% SWCNH and (h) 1% SWCNH.

Fig. 8. Shear stress as a function of shear rate for water–TiO2–PEG nanofluid at283.2 K. (D) 0.01% TiO2, (�) 0.1% TiO2 and (h) 1% TiO2.

70 S. Bobbo et al. / Experimental Thermal and Fluid Science 36 (2012) 65–71

lnf ¼ lf 1� /a

/m

� ��½g�/m

ð4Þ

where /m is the maximum concentration at which nanofluid canflow, /a the effective aggregates volume fraction (and here it is con-

sidered as /) and [g] is the intrinsic viscosity (for non-interacting,rigid spherical particles, 2.5).

Afterwards, Chen et al. [18] assumed that the aggregates den-sity change with the radial position and then it is not uniform inthe nanofluid, by means of the equation

/a ¼ /aa

a

� �3�D

ð5Þ

where aa and a are the aggregates and prime nanoparticles radii,respectively. D is the fractal index, that is 1.8 for nanoparticles[27–29]. So, Eq. (4) becomes

lnf ¼ lf 1� //m

aa

a

� �1:2� ��½g�/m

ð6Þ

Then, a simplified equation was proposed [30] as

lnf ¼ lf 1� //m

� ��2

ð7Þ

In order to apply Eqs. (4)–(7), /m should be calculated. Althoughthe present experimental data are only at three differentconcentrations, restricting the validity range of the models, /m

was calculated, basing on [31], on all the experimental data, being6.85% and 5.00% for SWCNH and TiO2 nanofluids, respectively.

The correlation results are added in Figs. 9 and 10. In Table 5, allthe deviations of the viscosity data from the regressed models aresummarised at three different temperatures. It is evident that alsoEqs. (4) and (6) can estimate nanofluids viscosity only for the low-est compositions, overestimating the suspensions at 1 wt.% Eq. (8)better represents the experimental data, showing higher devia-tions, although not always acceptable.

Nevertheless, the applicability of theoretical models to nanofl-uids is a still unsolved problem. Here, a simple equation, with sim-ilar form to Eq. (3), is proposed to correlate these experimentaldata

lnf ¼ lf 1þ a/þ b/2� �ð8Þ

Table 5Percentage deviation of the viscosity experimental data (from Table 3) from Eqs. ((1)–(4), (6), (7), (and) (8)) at 283.2 K, 313.2 K and 353.2 K.

SWCNH (0.01 wt.%) SWCNH (0.1 wt.%) SWCNH (1 wt.%) TiO2 (0.01 wt.%) TiO2 (0.1 wt.%) TiO2 (1 wt.%)

283.2 KEq. (1) 2.8 11.9 94.4 2.7 5.6 54.2Eq. (2) 2.8 13.0 351.1 2.7 5.9 97.8Eq. (3) 2.8 13.4 226.3 2.7 6.0 94.9Eq. (4) 1.6 0.1 �10.4 2.0 �0.7 �5.4Eq. (6) 1.6 0.3 �8.2 2.1 �0.1 9.9Eq. (7) 1.7 1.4 2.3 2.1 0.3 4.2Eq. (8) 1.3 �2.0 2.5 2.1 0.2 4.2

313.2 KEq. (1) 2.7 15.4 90.8 3.8 8.1 50.4Eq. (2) 2.8 16.5 342.6 3.8 8.4 92.9Eq. (3) 2.8 16.9 220.1 3.8 8.5 90.0Eq. (4) 0.0 1.7 �13.4 1.6 0.1 �9.2Eq. (6) 0.0 1.8 �11.3 1.6 0.6 5.6Eq. (7) 0.1 3.0 �1.1 1.7 1.0 0.1Eq. (8) 1.3 1.1 0.6 3.2 2.5 1.6

353.2 KEq. (1) �4.4 18.9 73.7 �7.5 3.4 25.7Eq. (2) �4.4 20.0 303.0 �7.5 3.7 61.3Eq. (3) �4.4 20.5 191.5 �7.5 3.8 58.9Eq. (4) �2.8 9.5 �17.6 �5.4 0.1 �20.6Eq. (6) �2.7 9.7 �15.6 �5.3 0.6 �7.7Eq. (7) �2.6 10.9 �5.9 �5.3 1.0 �12.6Eq. (8) �5.8 4.1 �8.4 �8.0 �1.9 �15.0

S. Bobbo et al. / Experimental Thermal and Fluid Science 36 (2012) 65–71 71

For the same base fluid and nanoparticle, this equation was re-gressed on the viscosity data at different temperatures (taken intoaccount by means of the base fluid viscosity at that temperature)and nanofluid concentrations. The regressed parameters are sum-marised in Table 4. As shown in Figs. 9 and 10, and in Table 5, thisequation well represents the experimental data.

4. Conclusions

The knowledge of viscosity is important for its influence on boththe heat transfer and the energy required to pump the nanofluid inthe circuits where they are used as secondary fluids.

In this paper, viscosity data of two nanofluids based on waterand SWCNH and TiO2, respectively, are presented. These two fluids,stabilized by addition of proper dispersants, were selected aftercareful analysis of the average nanoparticle dimensions in solution[14]. The experimental measurements were performed at atmo-spheric pressure and temperatures ranging between 283.2 K and353.2 K. Both fluids showed a Newtonian behaviour at each com-position. Negligible variations on the viscosity of the nanofluidsin relation to water are observed at nanoparticles concentrationsup to 0.1% in mass fraction. On the contrary, a significant increaseis measured for nanoparticles concentration of 1% in mass fraction.Part of this increment is due to the addition of the dispersants.

Few theoretical models were applied to regress the experimen-tal data, but they were found able to represent only nanofluidswith nanoparticle concentrations lower than 1 wt.%. Then, a newcorrelation was proposed to represent the experimental data forthe SWCNH/water and TiO2/water nanofluids.

However, the opportunity to use these nanofluids in practicalapplications is linked to their thermal and flow properties. In thenear future, measurements of these properties will be performedto assess the overall gain in energy efficiency achievable by usingthese nanofluids instead of simple water as secondary fluid.

Acknowledgement

The authors thank Mauro Scattolini for his fundamental help.

This work has been performed under the ‘‘Industria 2015’’ fund-ing of the Italian Ministry of Economic Development.

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