A model for temperature and particle volume fraction effect on nanofluid viscosity

Journal of Molecular Liquids 153 (2010) 139–145

Contents lists available at ScienceDirect

Journal of Molecular Liquids j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / m o l l i q

A model for temperature and particle volume fraction effect on nanofluid viscosity Marziehsadat Hosseini, Sattar Ghader ⁎ Department of Chemical Engineering, College of Engineering, Shahid Bahonar University of Kerman, Jomhoori blvd., Post Code 76175, Kerman, Iran

a r t i c l e

i n f o

Article history: Received 19 December 2009 Accepted 9 February 2010 Available online 14 February 2010 Keywords: Nanofluid Viscosity Local composition theory Eyring's theory

a b s t r a c t A theory based model is presented for viscosity of nanofluids and evaluated over the entire range of temperature and volume fraction of nanoparticles. The model is based on Eyring's viscosity model and the nonrandom two liquid (NRTL) model for describing deviations from ideality (Eyring-NRTL model). The equation for viscosity is composed of a contribution due to nonrandom mixing on the local level and another energetic section related to the strength of intercomponent interactions which inhibit components from being removed from their most favorable equilibrium position in the mixture. The experimental data were used to evaluate existing models which do not contain adjustable parameters and Eyring-NRTL model. The Eyring-NRTL model was found to agree well with the experimental data with the restriction that contains adjustable parameters which were interactions in the form of NRTL constants. However, the agreement was even better if temperature dependent interaction parameters were used. Comparisons of predicted and actual viscosity over the entire temperature and volume fraction range illustrate an improvement over the conventional nanofluid viscosity models with 2.91% AAD. © 2010 Elsevier B.V. All rights reserved.

1. Introduction An innovative technique has been studied extensively in recent years in which nanoparticles are dispersed in a base fluid (nanofluid) for enhancing physical properties. In spite of their promising features, there are only few published results on nanofluids. A review of relevant works on nanofluid's viscosity may be found in [1–3]. Viscosity is important in designing nanofluids for flow and heat transfer applications because the pressure drop and the resulting pumping power depend on the viscosity. Some of the experimental research for nanofluid's viscosity includes viscosity of carbon nanotubes [4] and graphite nanofluids [5], BaZrO3 suspensions [6], BaTiO3 suspensions [7], nickel-terpineol suspensions [8] and TiO2 nanoparticles in water [9–11]. Other investigations have focused on the rheology and viscosity of Al2O3 nanoparticles in water [12–14], copper oxide in EG at room temperature [15] and CuO nanoparticles in water and ethylene glycol mixture [16]. In spite of increasing interest for experimental report of nanofluids viscosity [4–16], researchers find that experimental results are larger than theoretical predictions of conventional models of nanofluids viscosity (shown in Table 1) especially at high nanoparticle volume fraction. It is very interesting to note that many of the existing formulas are derived based on the Einstein's pioneering work. Notice that all seven correlations in Table 1 are developed to relate viscosity to volume fraction only and there is no account of temperature dependence. Generally, fluids have higher viscosity near their freezing

⁎ Corresponding author. E-mail address: [email protected] (S. Ghader). 0167-7322/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.molliq.2010.02.003

point and fairly low viscosity near their boiling temperature, showing that viscosity is a strong function of the temperature. A correlation that relates viscosity of copper oxide nanoparticles suspended in water in temperature range of 5–50 °C is given in [16]:
 1 ln μ = A −B T

ð1Þ

where A and B are the functions of volume percentage. A formula has been proposed for calculating viscosity of nanofluids at particle concentrations of 1% and 4% by Nguyen et al. [14]: μnf = μbf ð1:1250−0:0007T Þ

ð2Þ

in which μnf and μ bf are viscosity of nano and base fluids. Unfortunately, for higher particle volume fractions, it was not possible for authors to provide any correlations that could take into consideration the combined effects of temperature and particle concentration [14]. In a recent work Abu-Nada [17] performed a two-dimensional regression on experimental data of Nguyen et al. [14] and developed the following relation including temperature T and volume fraction φ: 19:582 2094:47 + 0:794ϕ + T T2 ϕ 27463:863 2 −0:192ϕ −8:11 − + T T3 2 ϕ ϕ 3 + 2:1754 2 + 0:0127ϕ + 1:6044 T T

μAl2O3 = −0:155−

which had maximum error of 5%.

ð3Þ

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Table 1 Conventional models of nanofluids viscosity. μe is ratio of nanofluid viscosity (μnf) to base fluid viscosity (μbf) and ϕ is particle volume fraction [14,16]. Model

Expression

Einstein Brinkman Bicerane et al.

μe = μe = μe =

μnf μbf μnf μbf μnf μbf

Frankel and Acrivos

μe =

μnf μbf

Remarks Spherical particles and low particle volume fractions, ϕ b 0.02 are considered Extended Einstein formula ν is the virial coefficient and kH is Higgins coefficient

= 1 + 2:5ϕ 1 = ð1−ϕÞ2:5 = 1 + νϕ + kH ϕ2 " # ðϕ= ϕm Þ1 = 3 = 98 1=3 1−ðϕ =ϕm Þ

ϕm is the maximum particle fraction that must be determined experimentally

Lundgren Batchelor

μe = μe =

μnf μbf μnf μbf

 3 2 = 1 + 2:5ϕ + 25 4 ϕ + f ϕ 2 = 1 + 2:5ϕ + 6:5ϕ 2

Graham

μe =

μnf μbf

6 = 1 + 2:5ϕ + 4:56 4


 c  2+ dp

3 1 c dp



1+

c dp

7 2 7 5

Investigation of the viscosity of compounds at various pressure and temperature conditions has indicated the requirement of including the structural and thermal effects for accurate viscosity prediction of dense fluids [18]. In this study, local composition theory is used to predict the viscosity of nanofluids. For this purpose Eyring's theory [19] for viscosity has been applied to nonideal mixtures. Then an equation is introduced for viscosity which is composed of a contribution due to nonrandom mixing on the local level and another energetic section related to the strength of intermolecular interactions which inhibit components from being removed from their most favorable equilibrium positions in the mixtures. The effects of temperature and particle volume concentration have been investigated. Also results of this theory have been compared with experimental data and conventional models.

2. Eyring-NRTL model for nanofluid viscosity 2.1. Local composition theory Local composition expressions are derived assuming the components of a mixture are of different size and therefore interact based on their size and concentration in a mixture. These relations account for the difference in intermolecular forces and molecule size and even molecular structure of the mixture. Looking back to solution thermodynamics, one of the major assumptions of regular solution theory was that the mixture interactions were independent of each other [20]. But in some cases, like completely different strengths of attraction, the mixture interaction can be strongly coupled to the mixture composition. One way of treating this prospect is to recognize the possibility that the “local compositions” in the mixture might deviate strongly from the bulk compositions. As an example, consider a lattice consisting primarily of type A atoms but with two B atoms right beside each other. Suppose all these atoms were the same size and that the coordination number was 10. Then the local compositions around a B atom are xAB = 9/10 and xBB = 1/10 (notation of subscripts is AB which means “A around B”). Specific interactions such as polarity and hydrogen bonding might lead to such effects, and thus, the basis of the hypothesis is that energetic differences lead to the nonrandomness in the mixture. For “random mixing”, the probability of any “i–j interaction” is the same and goes as the product of the “i–j concentrations”. For example, in a binary mixture there are three types of interactions for molecules 1 and 2. First, a molecule can interact with itself (1–1 or 2–2 interactions), or it can interact with a molecule of the other type (a 1–2 interaction). In a random fluid, the probability of finding a 1 molecule is the fraction of 1 atoms, x1. The probability of a 1–1 interaction is a conditional probability. A conditional probability is the probability of finding a second interacting molecule of a certain type given the first is a certain type. For independent events, a conditional probability is calculated by the product of the individual probabilities. Therefore, the probability of a 1–1 interaction is x21. By similar arguments, the probability of a 2–2

Taylor series for ϕ – dp is particle radius and c is inter-particle spacing

interaction is x22. The probability of a 1–2 interaction is x1x2 and the probability of a 2–1 interaction is also x1x2. After 1964, researchers [21] began to realize that the distributions of molecules around each other might not be entirely random, as implied by the simple form of the conditional probabilities assumed above. Instead, nonrandom distributions might give rise to local compositions which are different from the bulk compositions. Recognizing the significance of local composition theory, it should not be surprising that many researchers have studied it [22–25]. In essence, our effort in this paper attempts to apply the same reasoning for nanofluids that was so successful for mixtures in the form of local composition theory. The resulting expression contains parameters which are intended to characterize the component interactions within the context of the theory. The values of these parameters must then be regressed from experimental data and the utility of the theory is judged by how accurately the experimental data are correlated. The problem with mixtures is that there are many different kinds of interactions occurring simultaneously, e.g., size asymmetry and disperse attractions. As a result, many specific terms must be invoked to describe these many specific interactions. If all the interactions were to be treated in their entirety, the resulting theory would be too heavy to be of any practical use. Therefore, we have made approximation and used local composition theory for nanofluid viscosity. The perspective we will adopt is that these are empirical equations which can usually fit the data, and that extrapolations beyond the available experimental data must be performed at some risk. The assumptions of local composition theory are: (a) the average energy of an i–j interaction is independent of temperature, density and other species present, (b) the coordination number of a specie in a mixture is the same as that of the pure specie and (c) the temperature dependent part of the energy is given by exponential form. In this theory it is assumed that the local compositions are given by some weighting factor, Ωij, relative to the overall compositions (if we denote x21 as mole fraction of “2's” around “1” and x11 as mole fraction of “1's” around “1”) x21 x = 2 Ω21 x11 x1

ð4Þ

x12 x = 1 Ω12 : x22 x2

ð5Þ

Therefore, if Ω12 = Ω21 = 1, the solution is random. We can also write local mole balances for cells 1 and 2 as x11 + x21 = 1

ð6Þ

x12 + x22 = 1:

ð7Þ

This approach seems to better reflect real dispersions containing interacting nanoparticles with irregular spatial arrangement as opposed to the ideal homogeneously dispersed systems.

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141

2.2. Two fluid theory

and local mole fractions are calculated from the relation

Two fluid theory provides a useful point of departure for deriving semi-empirical equations to represent thermodynamic excess functions [26]. In a binary mixture as shown in Fig. 1 each molecule is closely surrounded by other molecules; we refer to the immediate region around any central molecule as that molecule's cell. In a binary mixture of components 1 and 2, we have two types of cells: one type contains molecule 1 at its center and the other contains molecule 2 at its center. This theory can be extended to apply for multicomponent mixtures for an n-component mixture, there will be n cells so that properties of this mixture, J, can be expressed mathematically as

xij x = i Gij xjj xj

n

J = ∑ zi J

ðiÞ

ð8Þ

i=1

where z is an arbitrary composition variable like volume fraction, and J(i) represents the properties of cell i containing central molecule i. J(i) can be expressed in terms of local rather than overall compositions. On the other hand, deviations from a composition average of pure component properties are therefore attributable to unlike interactions, hence the nonrandom mixing nature of the components. The local composition of molecule i surrounding a central molecule of type j is represented by zij. For an n-component system, the local compositions must be conserved n

∑ zij = 1:

ð9Þ

i=1

Wilson's local composition assumption [21] is one of the efforts to estimate local compositions   xi exp −λij = RT xij   = xjj xj exp −λjj = RT

Renon and Prausnitz [27] applied Wilson's local composition assumption to two fluid theory and developed the nonrandom two liquid model (NRTL), which can successfully correlate excess Gibbs free energies [26]. In the NRTL model, the Gibbs free energy is ð1Þ

+ x2 g

ð2Þ

:

ð11Þ

The cell 1 and 2 properties, g(1) and g(2), can be related to the local compositions by g g

ð1Þ

ð2Þ

E

= x11 g11 + x21 g21

ð12Þ

= x12 g12 + x22 g22

ð13Þ




g = x1 x2

A21 G21 A12 G12 + x1 + x2 G21 x2 + x1 G12

 ð15Þ

the Aij's are treated as adjustable parameters which can be obtained by fitting experimental data. Therefore, the NRTL model allows predicting effect of composition and temperature on the excess Gibbs free energy of mixture. 2.4. Eyring's theory Considering liquid flow a rate process, Eyring's theory [19] for shear viscosity μ=

2.3. NRTL model

g = x1 g

where Gij = exp(−αAij/RT) and Aij = gij − gjj. gij's are characteristic interactions contributed by molecule i to the cell property g(i). In NRTL, as a modification of Wilson's equation, the expression for the local composition ratio also uses a Boltzmann distribution, but introduces another parameter (α12 = α21 = α) to “account for the nonrandomness of the liquid solution”. α is an empirical constant independent of temperature. When this parameter is zero, the solution is said to be completely random and local mole fractions reduce to overall mole fractions. The value of α is usually 0.2–0.4. All NRTL calculations in this study used α = 0.2 as many other studies [22,23,25,28]. The explanation why α is usually taken as 0.2 can be found in [26,27]. Using Eqs. (11)–(13) and local mole balances (Eqs. (6) and (7)), the excess Gibbs energy for the binary liquid mixture relative to ideal solution – gE = g − (x1g11 + x2g22) where g11 and g22 are Gibbs energy of components 1 and 2 – becomes

ð10Þ

where xij is the local mole fraction of i around j and λij is an energy parameter characterizing the interaction of component i with component j.

ð14Þ

Nh exp ðΔG = RT Þ V

ð16Þ

involves the activation free energy, ΔG, required to remove molecules within the fluids from their most energetically favored state to the activated state. In this equation μ is shear viscosity of the mixture, h is Planck's constant, N is Avogadro's number, V molar volume, R gas constant and T absolute temperature. 2.5. Eyring-NRTL viscosity model Extension of the concepts of classical thermodynamics to the viscous flow behavior of liquid mixtures can be made by assuming equivalence between the Gibbs energy of activation for flow and the equilibrium Gibbs energy of mixing. ΔG for mixture can be related to excess Gibbs free energy GE by E

G = ΔG−ΔGID

ð17Þ

where ΔGID represents the activation energy for ideal mixture (in the thermodynamic sense) of the components. Substitution of Eq. (17) into Eq. (16) and regrouping of GE into terms of SE and HE(GE = HE − TSE) yields     E E μV = ðμV ÞID exp −S = R exp H = RT

ð18Þ

where (μV)ID represents the volume–viscosity product for ideal mixture of the constituent components and has the following definition: ðμV ÞID = Nh exp ðΔGID = RT Þ

Fig. 1. Two types of cells in binary mixture in accordance to two fluid theory.

ð19Þ

A nonzero excess entropy term is indicative of nonrandom mixing by the fluid components and can be grouped with μV to yield a local

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Fig. 2. Comparison of calculated viscosity with experimental data and conventional models for Al2O3/H2O nanofluid at T = 25 °C.

viscosity including nonrandom mixing effects. It is this term for which the local composition model is applicable. The viscosity is therefore composed of one contribution due to nonrandom mixing on the local level and another energetic portion related to the strength of intermolecular interactions which inhibit molecules from being removed from their most favorable equilibrium positions in the mixtures. We label the former part as (μV)loc thus   E μV = ðμV Þloc exp H = RT

ð20Þ

Fig. 4. Comparison of calculated viscosity with experimental data of CuO nanoparticles dispersed in a 60% by weight ethylene glycol and 40% water.

terms of volume than mole fractions and better fit to experimental was obtained by volume fraction. In addition, local compositions of an ideal binary mixture should be equal to the bulk compositions only when the molar volumes of the pure components are the same [22]. However, local compositions based on volume fraction satisfy this requirement [29]. Therefore, in NRTL model, deviations from a volume fraction average of the pure component ξ values are attributed to nonrandom mixing and NRTL equation was based on volume fraction. Defining local volume fraction by

or defining a property, ξ, as ξ = ln ðμV Þ


ð21Þ

E

ξ = ξloc + H = RT

E

ð23Þ

 ð25Þ

k=1

allows local volume fractions to be computed from overall volume fractions and the equilibrium thermodynamic parameters Gij by recalling Eq. (14)


then ξ = ξloc

n

∑ Vk xkj

ð22Þ

for nanofluids, it is reasonable to assume H =0

φij = Vi xij =

φij = φi Gij =

n



∑ φk Gkj

ð26Þ

k=1

ð24Þ n

The NRTL model has been used to compute ξloc. While many correlations have been based on mole fractions, the same correlations have also been successfully used with volume fractions as the correlating composition variable. While it is not clear which variable should be used, volume fraction was chosen here because most of the experimental data are reported by volume fraction and because additive equations give slightly better results in

Fig. 3. Comparison of calculated and experimental viscosity of Al2O3/H2O and CuO/H2O nanofluids at T = 25 °C.

where Gij = 1 whenever i = j and ϕi = xi Vi = ∑ xj Vj . Attributing j=1

deviation from a volume fraction average of the pure component ξ values to nonrandom mixing and applying the two fluid theory yields n

ðjÞ

ξloc = ∑ φj ξ j=1

ð27Þ

Fig. 5. Comparison of calculated viscosity of Al2O3/H2O nanofluid at different temperatures and volume fractions with experimental data.

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143

Table 3 NRTL parameters of binary systems. For complete set of model results and parameters see Supplementary data. System

ϕ (%)

T (°C)

a12

b12

a21

b21

Al2O3(1)/H2O(2) CuO(1)/H2O(2)

1–9.4 1–9

20–70 20–65

25.699 41.999

12.169 16.120

4.999 30.479

− 9.175 − 11.158

Eqs. (31) and (32) are the final relations for calculating viscosity which becomes


A21 G21 A12 G12 + φ1 + φ2 G21 φ2 + φ1 G12

ξ = φ1 φ 2

 + φ1 ξ1 + φ2 ξ2

ð33Þ

for two component nanofluid, and Fig. 6. Comparison of calculated CuO/H2O viscosity at different temperatures and volume fractions with experimental data.

with ðjÞ

ξ

n

= ∑ φij ξij

ð28Þ

i=1

Here, ξij is a parameter that characterizes local interaction of component i in ξ property of cell j. Then n

n

j=1

i=1

ξloc = ∑ φj ∑ φij ξij

ð29Þ


n  n n ξloc = ∑ φj ∑ φi Gij ξij = ∑ φk Gkj i=1

ð30Þ

k=1

in the pure component limit Eq. (30) requires that ξ = ξjj = ξ0j where ξ0j is the pure component j value for ξ. After deriving excess ξ relative to ideal solution (ξE = ξ − ∑xjξ0j ), the final result is n

n

0

n

ξ = ξloc = ∑ φi ξi + ∑ i=1

i=1

 φ2 G21 φ3 G31 A21 + A31 φ1 + φ2 G21 + φ3 G31 φ1 + φ2 G21 + φ3 G31
 φ1 G12 φ3 G32 + φ2 A12 + A32 φ2 + φ1 G12 + φ3 G32 φ2 + φ1 G12 + φ3 G32
 ð34Þ φ1 G13 φ2 G23 + φ3 A + A φ3 + φ1 G13 + φ2 G23 13 φ3 + φ1 G13 + φ2 G23 23

+ φ1 ξ1 + φ2 ξ2 + φ3 ξ3

for three component nanofluid. 2.6. Interaction parameters determination The model thus obtained has Aij parameters which are obtained by minimizing the difference between model and experimental viscosity:

combining Eqs. (26) and (29) yields

j=1




ξ = φ1

∑ φj Aji Gji

j=1 φi n

ð31Þ

∑ φk Gki

k=1

where Gij = exp(− αAij/RT) and Aij = ξij − ξjj. The mixture viscosity can be calculated from μ = exp ðξÞ = V

mod

ε=∑

ξ

exp

−ξ

ξexp

!2 :

ð35Þ

Generalized reduced gradient (GRG2) algorithm was used for minimizing objective function ε. This algorithm is developed by Leon Lasdon, of the University of Texas at Austin, and Allan Waren, of Cleveland State University. Linear and integer problems use the simplex method with bounds on the variables and the branch and bound method, implemented by John Watson and Dan Fylstra, of Frontline Systems, Inc. Because a pure trial and error approach would take an extremely long time especially for problems involving many variables and constraints, first derivatives play a crucial role in this iterative method. The first derivative (or gradient) of the objective function measures its rate of change with respect to each of the variables. When all of the partial derivatives of the objective function are zero (that is, the

ð32Þ

where V is the molar volume of the mixture at composition in question. Aij's are assumed to be temperature dependent as Aij = aij + bijT. The reason for temperature dependency of Aij is described in correlation of temperature dependent viscosity section. Table 2 Percentage of absolute average deviation (AAD) of different models over the entire range of temperature and nanoparticle volume fraction. For detail of model results and model parameters see Supplementary data. ϕ (%) Al2O3/H2O CuO/H2O

1–9.4 1–9

T (°C)

Eyring-NRTL model (Eq. (31))

22–70 2.34 20–65 2.91   cal μ −μ exp   × 100: %AAD = 1n ∑  μ exp

Einstein

Brinkman

Lundgren

38.276 49.86

37.61 49.45

37.42 49.33

Fig. 7. Comparison of experimental viscosity of Al2O3/H2O nanofluid at different temperatures and volume fractions with temperature independent model (bij = 0 in Eq. (36)).

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Fig. 8. Comparison of experimental viscosity of CuO/H2O nanofluid at different temperatures and volume fractions with temperature independent model (bij = 0 in Eq. (36)).

gradient is the zero vector), the first-order conditions for optimality have been satisfied (some additional second order conditions must be checked as well) and we have found the lowest possible minimum value for the objective function [30]. 3. Results and discussions The local composition based model was applied to correlate viscosity of Al2O3/water, CuO/water and CuO/(water, ethylene glycol) nanofluids. Experimental data were taken from [13,14,16]. 3.1. Correlation of nanofluid viscosities at ambient temperature Figs. 2 and 3 compare the calculated viscosity of Eyring-NRTL model with experimental data of Al2O3/water and CuO/water systems at 25 °C [14]. The particle volume fraction changes from 0 to 13%. It can be seen that Eyring-NRTL model predicts experimental data accurately. Moreover, the results of Eyring-NRTL model have been compared with three conventional models in Fig. 2 and high difference between Brinkman, Einstein and Lundgren models and experimental data is observed. Fig. 4 compares the calculated viscosity with experimental data [16] of copper oxide nanoparticles dispersed in ethylene glycol and water mixture. Good agreement exists between the results of EyringNRTL model and experimental data. 3.2. Correlation of temperature dependent viscosity Figs. 5 and 6 compares calculated viscosity with experimental data of Al2O3/water and CuO/water nanofluids at different temperatures. As it can be seen, there is good agreement between the results of Eyring-NRTL model and experimental data [14]. Viscosity is also calculated by conventional models and absolute average deviation (AAD) between different models and experimental data are given in Table 2. The maximum AAD between Eyring-NRTL model and experimental data is 2.91%. Complete set of model predictions at different temperatures and volume fractions and model parameters can be found in Supplementary data. Eyring-NRTL and conventional models are also compared to experimental data in Supplementary data and AAD for each model is given. Nanofluid's viscosity considerably increases with particle volume fraction and decreases with temperature. More particles in solution have direct effect on fluid shear stress while the temperature effect is obviously due to a decreasing inter-particle and intermolecular attraction forces. Generally, the results are indicative that interaction terms are essential in prediction of viscous behavior. Nguyen et al.

[14] in their experimental study also reported relative viscosity as a function of temperature for Al2O3/water and CuO/water nanofluids (not reported here). The relative viscosity was almost constant for all nanofluids independent of temperature for particle volume fraction lower than 4%. However, the difference became more for higher particle fraction, e.g. for 7% and 9%, where temperature and particle volume fraction dependence can be observed. It is also interesting to note that for all nanofluids studied, viscosity gradient is particularly more for high particle fraction. This result suggests that temperature effects on particle suspension properties may be very different for high particle fraction than lower ones. Einstein's formula, and all other models originating from it, were obtained based on the theoretical assumption of a linear fluid surrounding isolated particles. Such a model may well represent the situation of a liquid that contains a small number of dispersed particles. However, for higher particle concentrations linear fluid theory is no longer appropriate to represent nanofluid reality. A possible explanation of this is the fact that particle suspension properties can affect the interaction energies. Meanwhile, experimental results indicate that the presence of more particles inside the base fluid would have more pronounced effect on nanofluid viscosity. In other words, more interaction energy between particles requires more activation energy to move molecules within the base fluid from their most energetically favored state. In other words, it can be concluded that interaction energies are strongly dependent on both particle concentration and temperature. In the resulting model (Eq. (31)) each set of two components is associated with two interaction parameters: Aij and Aji. With the arguments given above, it is more reasonable to take into account temperature effect into interaction energy of pair i and j, Aij, because the presence of more particles inside the base fluid would have different behavior at different temperatures. Therefore, linear temperature dependency was used for Aij: Aij = aij + bij T

ð36Þ

and in calculations of Eqs. (31), (36) was used for Aij. NRTL parameters, which obtained by fitting experimental data are given in Table 3. These parameters have been fitted to viscosity data, at or above room temperature. Even though the temperature effect is taken into consideration by Eq. (36), this may not represent a drawback considering the low temperature of the process because better representation of experimental viscosity data is obtained. To illustrate this point, we present calculated nanofluid viscosity at low and high volume fractions for Al2O3/H2O and CuO/H2O system for the case in which temperature dependency of interaction energies are not taken into account, i.e., bij = 0 in Eq. (36) (Figs. 7 and 8). Deviations are much more than previous case especially at high volume fraction. More troublesome, however, is the trend. It is clear that volume fraction extrapolations to 9% and more will give bad predictions. It is clear that model may give bad predictions if temperature dependency of interaction parameters is not used. Many empirical or semi-empirical equations can correlate viscosity data using multiple adjustable parameters. Our model also suffers from this drawback. In addition to the obvious disadvantage of requiring viscosity data for parameter estimation, correlations may not be immediately extendable to multicomponent nanofluids as they may require more parameters (such as three- and four-body interaction terms) for nanofluids containing more than two components. The equation is based entirely on interaction parameters which are themselves obtained from experimental data; meanwhile, accurate viscosity data can be obtained by the model at low and high temperatures and volume fraction. An interesting feature of model is easy extending of viscosity prediction to high pressure by reflecting pressure effect on NRTL parameters like refs. [31,32] which

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was not performed in this paper due to lack of access to experimental viscosity data of nanofluids at high pressure. Generally, the model has been indicative that interaction terms are essential in prediction of viscosity. Unfortunately, determination of the interaction term has been by data fitting and so the shortcomings of these models are similar to the correlations except that at least the adjustable parameters have some physical interpretation. They are, however, still entirely system specific. Nguyen et al. [14] also reported the presence of particle agglomerates on the inner surface of the measuring chamber at high temperatures (beyond temperatures given in Figs. 5 and 6). This suggests nanoparticles agglomeration in solution will be important at high temperatures (nanoparticle solution is not stable in temperatures higher than what concerned in this paper) which was not also considered in this study. The tendency to form agglomerates results in observed drastic increase of the nanofluid viscosity [14]. 4. Conclusions In this study, a model incorporating the Eyring and local composition theory has been developed, which allows reasonably prediction of nanofluids viscosity. The model predictions are better than conventional models like Einstein's, Brinkman's and Lundgren's formulas for nanofluid viscosity. According to good predictions the model is reliable for correlating viscosity of nanofluids at low and high temperatures and volume fractions. The model suffers from the shortcoming of relying on the adjustable parameters which are system specific. The results are indicative that interaction terms are essential in prediction of viscous behavior and better agreement between model and experimental data is obtained for case of temperature dependent parameters. Appendix A. Supplementary data Supplementary data associated with this article can be found, in the online version, at doi:10.1016/j.molliq.2010.02.003.

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