prediction of thermal conductivity and convective heat transfer coefficient of nanofluids by local...

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M. S. Hosseini

A. Mohebbi1

e-mail: [email protected]: [email protected]

S. Ghader

Department of Chemical Engineering,College of Engineering,

Shahid Bahonar University of Kerman,76175-133, Kerman, Iran

Prediction of ThermalConductivity and Convective HeatTransfer Coefficient of Nanofluidsby Local Composition TheoryIn this study, a new method based on the local composition theory has been developed topredict thermal conductivity, convective heat transfer coefficient, and viscosity of nano-fluids. The nonrandom two liquid (NRTL) model is used for this purpose. The effects oftemperature and particle volume concentration on thermal conductivity, convective heattransfer coefficient, and viscosity are investigated. The adjustable parameters of theNRTL model were obtained by fitting with experimental data. The results of the localcomposition theory are compared with the experimental data of CuO/water, Al2O3/water,TiO2/water, Cu/water, Au/water, Ni/water, TiO2/ethylene glycol, and Al/ethylene glycol(EG) nanofluids and a good agreement between the theory and the experimental data isobserved. The absolute average deviation of the model for thermal conductivity was1.51% in comparison to 42% in conventional models. This parameter for viscosity andconvective heat transfer coefficient were 2.91% and 2.13%, respectively. Moreover, a newequation for calculating convective heat transfer coefficient of nanofluids is proposed andtested. �DOI: 10.1115/1.4003042�

Keywords: nanofluid, convective heat transfer coefficient, thermal conductivitycoefficient, viscosity, local composition theory, nonrandom two liquid model

IntroductionNanofluids, which are suspensions of nanoparticles in a base

uid, have been found to provide a considerable heat transfernhancement in comparison to conventional fluids such as waternd ethylene glycol. In spite of their promising feature, there arenly few published results on nanofluids. A review of relevantorks may be found in Ref. �1–4�.There are a few theoretical formulas for predicting the thermal

onductivity of nanofluids. The Maxwell model �Eq. �1�� is validor well-dispersed noninteracting spherical particles with negli-ible thermal resistance at the particle/fluid interface �5�.

k

kf=

kp + 2kf + 2��kp − kf�kp + 2kf − ��kp − kf�

�1�

o include the effects of particle geometry and finite interfacialesistance, Maxwell’s model was generalized to yield the follow-ng expression for the thermal conductivity ratio:

k

kf=

3 + ��2�11�1 − L11� + �33�1 − L33��3 − ��2�11L11 + �33L33�

�2�

here the parameters of the above equation are given in Ref. �5�.Classical molecular Monte Carlo �MC� and molecular dynam-

cs �MD� simulations also are useful in elucidating the fundamen-als of nanofluids. As reviews indicate, MC simulation has beenpplied in the study of nanofluids �6,7�. The earliest such work8,9� lead to the important observation that the force per unit areaetween two surfaces confining a simple fluid oscillates betweenttraction and repulsion in a decaying manner with a period equalo the size of the fluid molecule before reaching the bulk pressure

1Corresponding author.Contributed by the Heat Transfer Division of ASME for publication in the JOUR-

AL OF HEAT TRANSFER. Manuscript received August 29, 2009; final manuscript re-eived November 9, 2010; published online February 1, 2011. Assoc. Editor: Patrick
. Phelan.
ournal of Heat Transfer Copyright © 20

aded 01 Feb 2011 to 192.43.227.18. Redistribution subject to ASME

at 5–6 molecular diameters. This finding, which classical con-tinuum theories do not predict, was subsequently confirmed ex-perimentally a few years later by Horn and Israelachvili �10,11�using their now famous surface force apparatus �SFA�.

Experimental research for nanofluids includes investigation ofthermal conductivity coefficient, viscosity, and convective heattransfer coefficient of some nanofluids such as CuO-water suspen-sion �12–14�, Al2O3-EG suspension �15,16�, Cu nanoparticles dis-persed in EG �17,18�, TiO2 nanoparticles dispersed in water �19�,Al2O3-water nanofluid �20�, Al–EG, and TiO2-EG nanofluids�21�. Table 1 shows conventional models of thermal conductivityof solid/liquid suspensions �22�.

Most of the nanofluid studies reported in the literature haveconcluded or assumed that nanofluids provide heat transfer en-hancement with respect to their respective base fluids �12,23�.Nonetheless, the assessment of what constitutes an enhancementhas not been determined on the same basis. An increased heattransfer coefficient may simply reflect the changes in the thermalphysical properties of the nanofluid being tested while the modelsand correlations developed for simple fluids still apply. As anexample, a recent study measured the convective heat transfer andpressure loss behavior TiO2–water nanofluids in fully developedlaminar and turbulent flows. The results indicated that the turbu-lent heat transfer and pressure loss can be predicted by the tradi-tional correlations and models, as long as the measuredtemperature- and loading-dependent nanofluid properties are usedin calculating the dimensionless numbers �19�.

In the present study, local composition theory �LCT� is used topredict the thermal conductivity coefficient, viscosity, and convec-tive heat transfer coefficient of nanofluids. The effects of tempera-ture and particle volume concentration have been investigated.Results of this theory have also been compared with experimentaldata and conventional models. Finally, a new formula for calcu-lating convective heat transfer coefficient of nanofluids is intro-
duced.
MAY 2011, Vol. 133 / 052401-111 by ASME

license or copyright; see http://www.asme.org/terms/Terms_Use.cfm

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Local Composition Theory for Nanofluids PropertiesNanofluid is a suspension of nanoparticles in a base fluid. Be-

ause of the difference in intermolecular forces, nanofluids can beonsidered as a nonrandom mixture, in which the fluid propertyan be expressed in terms of local rather than overall composition.herefore, the local composition theory can be used for nano-uids. Transport properties of nanofluids, such as thermal conduc-

ivity, convective heat transfer coefficient, and viscosity, appertaino interaction of particles. Thus, these properties of nanofluids cane calculated by the local composition theory, which is based ononrandom mixture and intermolecular forces.

2.1 Two Liquid Theory. The two liquid theory uses two flu-ds as references. This theory provides a useful point of departureor deriving semi-empirical equations to represent, for example,hermodynamic functions for mixtures �24�. In a binary mixture ashown in Fig. 1, each molecule is closely surrounded by otherolecules, which is referred to the immediate region around any

entral molecule as that molecule’s cell. In a binary mixture ofomponents 1 and 2, there are two types of cells: One type con-ains molecule 1 at its center and the other contains molecule 2 atts center.

This theory can be extended to apply for multicomponent mix-ures for an n-component mixture, there will be n hypotheticalure fluids so that properties of this mixture J can be expressedathematically as

J = �i=1

n

ziJ�i� �3�

here zi is an arbitrary composition variable, and J�i� representshe properties of hypothetical fluid i containing central molecule i.�i� can be expressed in terms of local rather than overall compo-itions. The local composition of molecule i surrounding a central

Table 1 Conventional models of thermal

odels Expression

axwellkeff =

kp + 2k1 + 2�kp − k1��kp + 2k1 − �kp − k1��

k1

amiltonkeff =

kp + �n − 1�k1 − �n − 1��kp − k1��kp + �n − 1�k1 + �kp − k1��

k1

ang et al.

keff =

�1 − �� + 3�0� kc1�r�n�r�

kc1�r� + 2k1dr

�1 − �� + 3�0� kc1n�r�

kc1�r� + 2k1dr

k1

ukeff =

kp + 2k1 + 2�kp − k1��1 + ��3�

kp + 2k1 − �kp − k1��1 + ��3�k1

angkeff = k1�1 − �� + kp� + 3C

d1

dpk1 Redp

2 Pr �

ig. 1 Two types of hypothetical pure fluids of binary mixtures

52401-2 / Vol. 133, MAY 2011


molecule of type j is represented by zij. For an n-component sys-tem, the local compositions must be conserved,

�i=1

n

zij = 1 �4�

2.2 Thermal Conductivity. Equations �3� and �4� and NRTLmodel �see Appendix� can be used for liquid mixture thermal con-ductivity. In these equations, J is replaced by thermal conductivityof liquid mixture �k� and volume fraction ��’s are adopted in-stead of mole fraction.

The local volume fraction is defined as

�ij = Vixij��i=1

n

Vixij� �5�

The local volume fractions are computed from overall volumefractions by the above equation. Combining Eq. �5� with the fol-lowing equation �see Appendix�:

xij

xjj=

xi

xjGij �6�

yields

�ij = �iGij��i=1

n

�iGij� �7�

where Gij =exp�−�Aij /RT� and Aij =gij −gjj. gij’s are characteris-tic interactions contributed by molecule i to the cell property g�i�.Moreover, Gij =1 whenever i= j and �i=xiVi /� j=1

n xjVj.Thus

�21 =�2G21

�1 + �2G21�8�

�12 =�1G12

�2 + �1G12�9�

kE is defined by:

kE = k − kid �10�

And kid is defined a volume fraction average of k of two purecomponents,

kid = �1k11 + �2k22 or kid = �1k1 + �2k2 �11�

In the nonrandom two liquid model, deviation from a volumefraction average of the pure component, k values are attributed tononrandom mixing. Applying the two fluid theory from Eq. �3� for

ductivity of solid/liquid suspensions †22‡

Remarks

Spherical particles and low particle volume fractions, ��0.02, are considered

Extended Einstein formula

�m is the maximum particle and � is the ratio of theordered layer thickness to the nanoparticle radius

Taylor series for �, where C is a constant related to theKapitza resistance

con

property of k, one can obtain the following equation:

Transactions of the ASME


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k = �1k�1� + �2k�2� �12�

here � is volume fraction and k�i� represents the thermal con-uctivity of hypothetical fluid i containing central molecule i.

From the NRTL model, the hypothetical fluid properties k�1�

nd k�2� can be related to the local compositions by

k�1� = �11k11 + �21k21 �13a�

nd

k�2� = �12k12 + �22k22 �13b�

hus

k = �1��11k11 + �21k21� + �2��12k12 + �22k22� �14�

ombining above equations yield

kE = �1�21�k21 − k11� + �2�12�k12 − k22� �15�

y defining the following parameters:

A21 = k21 − k11 A12 = k12 − k22 �16�

here k12 and k21 are binary thermal conductivity parameters; so,q. �15� becomes

kE = �1�21A21 + �2�12A12 �17�

ith Eqs. �8�–�10� and �17�, the thermal conductivity for the bi-ary liquid mixture becomes

k = �1�2� A21G21

�1 + �2G21+

A12G12

�2 + �1G12� + �1k1 + �2k2 �18�

here k1 and k2 are the pure component value for thermal con-uctivity.

2.3 Shear Viscosity Model. A predictive local compositionheory for nanofluid shear viscosity has been developed based onhe Eyring and NRTL local composition models. For this purpose,yring’s theory for shear viscosity has been applied to nonidealixtures. Then an equation is introduced for shear viscosity,hich is composed of a contribution due to nonrandom mixing on

he local level and another energetic portion related to the strengthf intermolecular forces inhibiting molecules from removal fromheir most favorable equilibrium positions in the mixtures. Theormer portion was labeled as ��Vloc� and computed by the NRTLquation.

Eyring’s theory �25� for shear viscosity is as follows:

� =Nh�

Vexp��G�/RT� �19�

here �G� and V are activation free energy and molar volume,espectively. For nonideal mixtures,

�G� = �GID� + GE �20�

here �GID� represents the activation energy for an ideal mixture.

The relation between excess functions is

GE = HE − TSE �21�

ubstitution of Eq. �21� into Eq. �20� and regrouping yields:

�V = ��V�ID exp�− SE/R�exp�HE/RT� �22�

here ��V�ID represents the volume-viscosity product for a hypo-hetical ideal mixture of the constituent components and has theollowing definition:

��V�ID = Nh� exp��GID� /RT� �23�

nd ��Vloc� can be defined as

��V�loc = ��V�ID exp�− SE/R� �24�

hus

ournal of Heat Transfer


�V = ��V�loc exp�HE/RT� �25�

Or a property � is defined as

� ln��V� �26�

� = �loc + HE/RT �27�

For nanofluids, the heat for dissolving nanoparticles in the basefluid is small, so that

HE = 0 �28�

Then

� = �loc �29�

The NRTL model has been used to compute �loc with the follow-ing parameters defined as

A21 = �21 − �11 A12 = �12 − �22 �30�

where �12 and �21 are binary viscosity parameters, and with theNRTL model and Eq. �30�, the viscosity for the binary liquidmixture becomes

� = �1�2� A21G21

�1 + �2G21+

A12G12

�2 + �1G12� + �1�1 + �2�2 �31�

where �1 is a hypothetical value for nanoparticle viscosity, whichcan be determined by extrapolation up to 100% for nanoparticlevolume fraction beyond available experimental data. Also, �2 isthe viscosity of base fluid.

Then mixture viscosity of mixture can be calculated from Eq.�26� as follows:

� = exp��/V �32�

where V is the molar volume of the mixture at the mixture at thecomposition in question.

In Eqs. �18� and �31�, Gij =exp�−�Aij /RT�, in which Aij’s areassumed a linear function of temperature, i.e.,

Aij = Bij�0� + Bij

�1�T �33�

where Bij�0� and Bij

�1� are calculated for each nanofluid by the re-gression method.

2.4 Convective Heat Transfer Coefficient Model. Equations�3� and �4� and the NRTL model �see Appendix� can be used forthe liquid mixture convective heat transfer coefficient. In theseequations, J is replaced by convective heat transfer coefficient ofliquid mixture �h�, volume fraction ��’s are adopted instead ofmole fraction, and convective heat transfer coefficient for a two-component mixture can be expressed as

h = �1�2� A21G21

�1 + �2G21+

A12G12

�2 + �1G12� + �1h1 + �2h2 �34�

In this equation, Gij =exp�−�Aij /RT� and Aij’s are calculated foreach nanofluid by regression method.

In Eq. �34�, the convective heat transfer coefficient of nanofluidis calculated by the local composition theory directly. As anothermethod for calculating the convective heat transfer coefficient,first, the thermal conductivity and viscosity of the nanofluid arecalculated by the local composition theory. With having two men-tioned properties, the Reynolds number �Re� and Prandtl number�Pr� are calculated. Then the Nusselt number is obtained. Tradi-tionally, the Nusselt number is related to the Reynolds numberdefined as Re=fuD /�f and the Prandtl number defined as Pr=� /�. In general, the Nusselt number Nu of a nanofluid may be
expressed as follows �18�:
MAY 2011, Vol. 133 / 052401-3


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Nunf

= f�Re,Pr,kp

kf,�Cp�p

�Cp� f, �, dimensions and shape of particles�

�35�

n this study, the following formula is proposed to correlate thexperimental data for the nanofluid:

Nunf = c1�1 + c2�m1Pepm2�Renf

m3 Prnf0.4 �36�

ompared with the heat transfer correlation for conventionalingle-phase flow, the volume fraction �� of suspended nanopar-icles and the Peclet number are involved in the above expression.he Peclet number �Pe� describes the effect of thermal dispersion,hich is caused by microconvective and microdiffusion of the

uspended nanoparticles. The case c2=0 refers to zero thermalispersion, which corresponds to the pure base fluid. The particleeclet number Pep in Eq. �36� is defined as

Pe =udp

�nf�37�

here �nf is defined as

�nf =knf

�Cp�nf=

knf

�1 − ��Cp� f + ��Cp�p�38�

ith calculating the Nusselt number by Eq. �36�, the convectiveeat transfer coefficient of nanofluid is obtained by the followingquation:

Nu�x� = h�x�D/kf �39�

here D is the tube inner diameter. In the above equations, theensity of nanofluid is calculated by the following equations:

nf = �1 − �� f + �p �40�

Results and DiscussionThe local composition based model was applied to calculate the

hermal conductivity, viscosity, and convective heat transfer coef-cient of nanofluids. Experimental data for these properties foruspensions of Al2O3, CuO, TiO2, Au, Ni, Cu, and SiO2 nanopar-icles in water base fluid and suspensions of TiO2 and Al nano-articles in ethylene glycol base fluid are given in Refs.19–21,26–31�. To quantify the adjustable parameters of theRTL model �i.e., A12, A21, B12

�0�, B12�1�, B21

�0�, and B21�1��, experimental

ata must be fitted to corresponding equations, namely, Eqs. �18�,31�, and �34� for thermal conductivity, viscosity, and convectiveeat transfer coefficient, respectively. The nonlinear regressionethod was used for fitting numerical values and experimental

ata and finding the NRTL parameters. An objective function isefined as

I = �� Jexp − Jcal

Jexp�2

�41�

ere Jexp and Jcal are arbitrary nanofluids property, for example,hermal conductivity, viscosity, or convective heat transfer coeffi-ient from experimental data and numerical calculation, respec-ively. Numerical result is fitted to experimental data by changingRTL parameters on the numerical calculation and obtaining the

east value of I. The Microsoft Office Excel software is used toptimize parameters.

3.1 Nanofluid Thermal Conductivity. Figures 2 and 3 com-are calculated thermal conductivity using LCT with experimentalata �21,26,27� for Al2O3/water, CuO/water, TiO2/water,iO2/EG, and Al/EG nanofluids at room temperature. It can be
een that there is a good agreement between the results of the
52401-4 / Vol. 133, MAY 2011


local composition theory and the experimental data. The averageabsolute deviation is 1.51%. The mentioned parameter is definedas

AAD =1

n�i=1

n �� Jcal − Jexp

Jexp��

i

�42�

where J is an arbitrary nanofluid property, for example, thermalconductivity.

Moreover, the results of the local composition theory have beencompared with Maxwell’s model in Fig. 2. This figure shows thatMaxwell’s model underestimates the nanofluid thermal conductiv-ity and the comparison of Maxwell’s model with the experimentaldata yields an average absolute deviation �AAD� of 42%. Figures2 and 3 also show that the thermal conductivity of nanofluid in-creases with increasing nanoparticle percentage.

Figure 4 compares calculated thermal conductivity versus tem-perature using LCT with experimental data �26,27� for Ni/waterand Au/water nanofluids. It can be seen that there is a good agree-ment between the results of the local composition theory and theexperimental data. These figures �i.e., Figs. 2–4� also show thatthe nanofluid’s thermal conductivity considerably increases withparticle volume fraction in the same temperature and increaseswith temperature in the same particle volume fraction.

3.2 Nanofluid Viscosity. Figure 5 compares the calculatedviscosity from the LCT and conventional models with experimen-

Fig. 2 The comparison of calculated thermal conductivity byLCT with experimental data and Maxwell’s model forAl2O3/water, CuO/water, and TiO2/water nanofluids at roomtemperature

Fig. 3 The comparison of calculated thermal conductivity bylocal composition theory with experimental data for TiO2/EG
and Al/EG nanofluids at room temperature


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al data �21,29� for Al2O3/water and TiO2/water at room condi-ions. It can be seen that there is a good agreement between theesults of the local composition theory and the experimental data.he average absolute deviation is 2.91%. This figure also shows

hat Einstein’s, Brinkman’s, and Lundgren’s formulas underesti-ate nanofluid viscosity.Figure 6 shows the dependence of the calculated viscosity on

anofluid temperature for the Al2O3/water nanofluid. As one canee, there is a good agreement between the local compositionheory and the experimental data �29�.

The nanofluid’s viscosity considerably increases with particleolume fraction and decreases with increasing in temperature.ore particles in solution have a direct effect on fluid shear stresshile the temperature effect is obviously due to a decreasing in-

erparticle and intermolecular attraction forces. Generally, the re-ults are indicative that interaction terms are essential in the pre-iction of viscous behavior. Nguyen et al. �28� in theirxperimental study also reported relative viscosity as a function ofemperature for Al2O3/water and CuO/water nanofluids. It is alsonteresting to note that for all nanofluids studied, viscosity gradi-nt is particularly more for high particle fraction. This result sug-ests that temperature effects on particle suspension propertiesay be very different for high particle fraction than lower ones.Einstein’s formula, and all other models originating from it,

as obtained based on the theoretical assumption of a linear fluidurrounding isolated particles. Such a model may well representhe situation of a liquid that contains a small number of dispersed

ig. 4 The comparison of calculated thermal conductivity byocal composition theory with experimental data for Ni/waternd Au/water nanofluids

ig. 5 The comparison of calculated viscosity by local com-osition theory with experimental data for Al2O3/water and
iO2/water nanofluids at room temperature


particles. However, for higher particle concentrations, the linearfluid theory is no longer appropriate to represent nanofluid reality.A possible explanation of this is the fact that particle suspensionproperties can affect the interaction energies. Meanwhile, experi-mental results indicate that the presence of more particles insidethe base fluid would have more pronounced effect on nanofluidviscosity. In other words, more interaction energy between par-ticles requires more activation energy to move molecules withinthe base fluid from their most energetically favored state. In otherwords, it can be concluded that interaction energies are stronglydependent on both particle concentration and temperature �32�.

3.3 Nanofluid Convective Heat Transfer Coefficient. Figure7 compares calculated convective heat transfer coefficient fromthe local composition theory with the experimental data �19� ofTiO2/water at room temperature. It can be seen that there is a goodagreement between the results of this theory and the experimentaldata. The average absolute deviation is 2.13%. NRTL parametersfor calculating thermal conductivity coefficient, viscosity, andconvective heat transfer coefficient, which are obtained by fittingwith experimental data, are given in Tables 2–4, respectively.

In the next step, a relatively general formula as Eq. �36� forcalculating the Nusselt number of nanofluid is proposed. The co-efficients c1 and c2 as well as the exponents m1, m2, and m3 in Eq.�36� were determined by a proper data-reduction procedure. For agiven flow velocity, a set of all these coefficients and exponents isfitted for predicting convective heat transfer coefficient of nano-fluids with different volume fractions of suspended nanoparticles.For TiO2/water, SiO2/water, and Cu/water nanofluids inside atube, the coefficient c1 and c2 as well as the exponent m1, m2, andm3 in Eq. �36� are given in Table 5. The flow regimes are alsogiven in Table 5.

Figures 8 and 9 show the theoretical predictions of convectiveheat transfer coefficients of nanofluids from Eq. �36�. Obviously,there exists a good coincidence between the results calculatedfrom this equation and the experimental data �19�. It reveals thatEq. �36� correctly incorporates the main factors of affecting theconvective heat transfer coefficient of the nanofluid in a simpleform.

Figures 10 and 11 show the theoretical predictions of the Nus-selt number of nanofluids from Eq. �36�. As it can be seen, there isa good agreement between the calculated results from this corre-lation and the experimental data �30�. All the coefficients andexponents in Eq. �36� have been determined by correlating experi-

Fig. 6 Comparison of experimental data of viscosity for vari-ous volume concentrations of Al2O3 nanoparticles dispersed inwater with respect to temperature and those predicted by localcomposition theory

mental data of heat transfer for nanofluids; therefore, these formu-

MAY 2011, Vol. 133 / 052401-5


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as �see Table 5� can be used to predict the convective heat trans-er coefficient for such suspensions with different volumeractions of nanoparticles.

ConclusionsIn this study, to the best of our knowledge for the first time, the

ocal composition theory has been used to predict thermal conduc-ivity, viscosity, and convective heat transfer coefficient of nano-uids. The results of this theory for calculating the mentionedroperties of Al2O3/water, CuO/water, Cu/water, SiO2/water, Ni/ater, Au/water, TiO2/water, TiO2/EG, and Al/EG nanofluids are

n good agreement with the experimental data. The results showhat Maxwell’s formula underestimates nanofluid thermal conduc-ivity. The comparison of results of the local composition theoryith the experimental data yields an AAD of 1.51% for thermal

onductivity, while the comparison of the conventional modelsith the experimental data yields an AAD of 42%. The AAD for

onvective heat transfer was 2.13%. A relatively general formulaor calculating the convective heat transfer coefficient of nanofluids introduced. The results of this new formula are in good agree-

ent with experimental data. Finally, the agreement between theesults of this work with experimental data approves the local

Fig. 7 The comparison of calculated cocomposition theory with experimental da

Table 2 NRTL parameters for calculating thermal cond

ystemT

�°C��

�%� A12 A

l2O3 /H2O �26� 21 0–4 1.26uO /H2O �dp=29 nm� �27� 21 0–4 0.81iO2 /H2O �27� 21 0–5 1.30i /H2O �27� 0–100 5 48.18 4u /H2O �26� 25–60 0.26 16,601.30 81,2iO2/EG �21� 21 0–5 0.24l/EG �21� 21 0–5 0.32

Table 3 NRTL parameters for calculating vi

ystemT

�°C��

�%� A12

l2O3 /H2O �dp=47 nm� �29� 20–70 0–9.4 21,809uO /H2O �dp=29 nm� �28� 20–70 0–9 2.86l2O3 /H2O �dp=36 nm� �29� 21 0–1.3 419.09iO2/water �15 nm� �21� 21 0–5 402.48

52401-6 / Vol. 133, MAY 2011


composition theory as a new useful method for calculating ther-mal conductivity, viscosity, and convective heat transfer coeffi-cient of nanofluids in comparison to other models.

NomenclatureAAD � absolute average deviation

Aij � NRTL parametera � thermal diffusivity

Cp � specific heatD � test tube inner diameterdp � particle average diameter

Gij � NRTL nonrandomness factorGE � excess free energy

�G� � free energy of activationg � molar Gibbs energy

gij � binary free energy parameterg�i� � free energy of hypothetical fluid i

H � enthalpyHE � excess enthalpy

h � convective heat transfer coefficienth� � Planck’s constant

ective heat transfer coefficient by localor TiO2/water „dp=95 nm… nanofluid

ivity coefficient and AAD of LCT and experimental data

B12�0� B12

�1� B21�0� B21

�1�AAD�%�

26 0 0.35 0.26 77.99 0.164452 657,902 2366 4.91 3011.8 0.534302 0.22 65.45 17.04 5111 8.438092 400.18 131.33 29.03 107.87 0.723328 14,225.46 118.62 924.08 148.38 0.025656 0.22 65.43 17.04 5111 0.338307 0.22 65.43 17.04 5111 0.3823

sity and AAD of LCT and experimental data

A21 B12�0� B12

�1� B21�0� B21

�1�AAD�%�

574.3 162.78 108.61 7.96 598.66 3.9203.17 3.34 1000.3 3.77 1000.3 10

64.15 0.29 84.22 0.03 11.68 5.6087310.7 0.05 8.97 1.04 302.88 2.5884

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J � fluid propertyJij � binary parameterJ�i� � property of hypothetical fluid i

k � thermal conductivityL � tube length

LCT � local composition theoryN � Avogadro number

Pe � Peclet numberPr � Prandtl numberq � heat fluxR � universal gas constant

Re � Reynolds numberS � entropyT � absolute temperatureu � mean velocityV � molar volumexi � mole fraction of component

xij � local mole fraction of component i around cen-tral molecule j

zi � arbitrary composition variable

reek Symbols� � NRTL nonrandomness factor� � shear viscosity� � ln��V�

� volume fractionij � local volume fraction of i around component j

� density

uperscriptsi , j � hypothetical fluid i , j� � activation propertyE � excess° � pure component

ubscriptsave � average

bf � base fluideff � effective

f � fluidnf � nanofluid

loc � local compositionID � ideal1,2 � nanoparticle and base fluid

able 4 NRTL parameters for calculating convective heatransfer coefficient of TiO2/water nanofluid †19‡ and AAD of LCTnd experimental data

Re x/D A12 A21

AAD�%�

49 18,707 54,210 3.6709900 151 23,486 74,638 2.8795

200 21,480 76,993 2.68549 20,097 55,992 1.6238151 24,997 82,528 1.2687

1500 200 24,654 81,152 2.071250 7,715 7,123 0.7082

Table 5 The coefficients c1 and c2 as well

System

TiO2/water �laminar flow� �19� 1.9TiO2/water �turbulent flow� �19� 0.4CuO/water and SiO2/water �turbulent flow� �30� 0.0Cu/water �turbulent flow� �30� 0.0



Appendix: NRTL ModelIn 1968, Renon and Prausnits �24� applied local composition

assumption to two liquid theory and developed the NRTL model,which can successfully correlate excess Gibbs free energies. In theNRTL model, the Gibbs energy is,

g = x1g�1� + x2g�2� �A1�

The hypothetical fluid properties, g�1� and g�2� can be related to thelocal compositions by

g�1� = x11g11 + x21g21 �A2�

g�2� = x12g12 + x22g22 �A3�

where local mole fractions are calculated from the following re-lation:

xij

xjj=

xi

xjGij �A4�

where Gij =exp�−�Aij /RT�, and Aij =gij −gjj, gij’s are characteris-tic interactions contributed by molecule i to the hypothetical fluidproperty g�i� and � is a nonrandomness factor, which when equalto zero, reduces the local mole fractions to the overall mole frac-tions indicating totally random mixing.

The value of � is usually 0.2–0.4. All NRTL calculations in thisstudy used �=0.2 as many other studies �33–35�. The explanationwhy � is usually taken as 0.2 can be found in Refs. �24,36�.

Using Eqs. �A1�–�A4� and conservation of mass, the excessGibbs energy for the binary liquid mixture becomes

gE = x1x2� A21G21

x1 + x2G21+

A12G12

x2 + x1G12� �A5�

The Aij’s are treated as adjustable parameters, which can be ob-tained by fitting with experimental data. Therefore, the NRTLmodel allows the predicting effect of composition and temperatureon the excess Gibbs free energy of mixture.

the exponents m1, m2, and m3 in Eq. „36…

c2 m1 m2 m3

0.2103 0.2409 0.379 0.049831.4342 0.3406 1.0247 0.4083

76 14.22155 2.037843 0.520494 0.86047949 10 5.683711 0.1955 0.875408

Fig. 8 Comparison between the experimental data †19‡ and thecalculated values from Eq. „36… for fully developed laminar andflow for TiO2/water nanofluid

as

c1

535551108092

MAY 2011, Vol. 133 / 052401-7


R

Fcn

0

Downlo

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