Calculation of the thermal conductivity of nanofluids containing nanoparticles and carbon nanotubes

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1 High Temperatures ^ High Pressures, 2003/2007, volume 35/36, pages 611 ^ 620 DOI: /htjr142 Calculation of the thermal conductivity of nanofluids containing nanoparticles and carbon nanotubes Jurij Avsec, Maks Oblak Faculty of Mechanical Engineering, University of Maribor, Smetanova 17, 2000 Maribor, Slovenia; jurij.avsec@uni-mb.si Paper presented at the Seventeenth European Conference on Thermophysical Properties, 5 ^ 8 September 2005, Bratislava, Slovak Republic Abstract. A mathematical model for the calculation of thermal conductivity of nanofluids containing nanoparticles and nanotubes on the basis of statistical mechanics is presented. This appears to be the first such attempt published in the scientific literature. We focus on the influence of the liquid layer structure around nanoparticles as the reason for enhancement of the thermal conductivity of nanofluids. Analytical results of both the current analysis and statistical mechanics are compared with experimental data reported in the scientific literature, which show good agreement. Nomenclature A constant Greek symbols B constant a volume fraction C heat capacity g Gru«neisen parameter C v0 heat capacity at constant volume Z viscosity C el electronic heat capacity y D Debye temperature C har lattice heat capacity y E Einstein temperature C int reduced internal heat capacity m r relative dipole movement CLS Chung ^ Lee ^ Starling k correction factor for hydrogen d pt diameter of particle bonding effect f function l thermal conductivity h n thickness of nanolayer l 0 constant HC Hamilton ^ Crosser model l f thermal conductivity of reference k B Boltzmann constant fluid k e constant l k dilute gas thermal conductivity k r spatial average l p high-density thermal conductivity L temperature dependent constant l pt thermal conductivity of nanol el electron mean free path particle LJ Lennard ^ Jones r density M molecular mass r reduced density N number of molecules in system C influence ofpolyatomic molecules n d number of conduction electrons per volume C s sphericity R m universal gas constant t electron or phonon lifetime RD relative deviation o acentric factor RHC revised Hamilton ^ Crosser model O a atomic volume T temperature Superscripts and subscripts T 0 reduced temperature 0 dilute gas state T c critical temperature c critical condition v el electron speed el electron v g average phonon group velocity f fluid V volume ph phonon V c critical volume p influence of high densities X Xue model pt particle x, y, z coordinate axes

2 612 J Avsec, M Oblak 1 Introduction Accurate knowledge of nonequilibrium or transport properties of pure gases and liquids is essential for optimal design of the different components of chemical process plants, for determination of intermolecular potential energy functions, and for development of accurate theories of transport properties in dense fluids. Transport coefficients describe the process of relaxation to equilibrium from a state perturbed by application of temperature, pressure, density, velocity, or composition gradients. The theoretical description of these phenomena constitutes that part of nonequilibrium statistical mechanics known as kinetic theory. The term `nanofluid' is used here to describe a solid ^ liquid mixture which consists of nanoparticles suspended in a liquid, and this is one of the new challenges set by nanotechnology for the thermal sciences. Possible application areas for nanofluids include advanced cooling systems and micro/nano electromechanical systems. The investigation of the effective thermal conductivity of a liquid with nanoparticles is attracting a lot of interest experimentally and theoretically, as the effective thermal conductivity of a nanoparticle suspension can be much higher than that of a fluid without nanoparticles. 2 Calculation of the thermal conductivity of a pure fluid (1) We use the Chung ^ Lee ^ Starling model (CLS) ([9], [10], [32]) for calculation of the thermal conductivity of a pure fluid. The equations are based on kinetic gas theories and are correlated with the experimental data. The low-pressure transport properties are extended to fluids at high densities by introducing empirically correlated, densitydependent functions. These correlations use acentric factor o, dimensionless (relative) dipole moment m r and empirically determined association parameters to characterize the molecular structure effect of polyatomic molecules k, the polar effect and the hydrogen bonding effect. In this paper we are determining new constants for fluids. The dilute gas thermal conductivity for the CLS model is written as: l ˆ l k l p. (1) The first term on the right-hand side represents the high-density function: Z0 l k ˆ 3119:41 Cf m, o, k. (2) M We use the Taxman theory ([13]). Taxman solved the problem of influence of internal degrees of freedom on the basis of WCUB theory and the approximations given by Mason and Monschick ([8], [13], [23], [24]). The final expression for the influence of internal degrees of freedom is: 2 0:215 1:061b 0:2665 0:28288 C 3 int C ˆ 1 C Z coll Z coll 6 int b 0:6366 Z 1:061bC 7 4 5, (3) int coll Z coll where C int is the reduced internal heat capacity at constant volume, b is the diffusion term, and Z coll is the collision number. The heat capacities are calculated by the use of statistical thermodynamics. This features all important contributions (translation, rotation, internal rotation, vibration, intermolecular potential energy, and influence of electron and nuclei excitation). The residual part l p of the thermal conductivity can be represented by: l p ˆ f T, V, T c, V c, M, m r, o, k. (4) (1) Consult also references [2], [3], [4], [10], [11], [16], [19], [20], [21], [26], [27], [29].

3 Thermal conductivity of nanofluids Calculation of the thermal conductivity of pure solids (2) 3.1 Electronic contribution to the thermal conductivity calculated using the Eliashberg transport coupling function The expression for the electronic contribution l el of the thermal conductivity can be calculated on the basis of the theory of thermal conductivity of a classical gas: l el ˆ 1 3 nc el v el l el (5) where C el is the electronic heat capacity (per electron), n is the number of conduction electrons per volume, v el is the electron velocity and l el is the electron mean free path. In equation (5) it is assumed that in a temperature gradient electrons travel the same average distance l before transferring their excess thermal energy to the atoms by collisions. We can also describe the electronic part of the thermal conductivity by: l el ˆ nk B T m b * + 2 e k E F t e, k k B T, (6) where m b represents electron band mass and t is electron lifetime that depends both on the direction of the wave vector k and on the energy distance e. The brackets h::i describe an average over all electron states. The lifetime for the scattering of electrons by phonons contains quantum-mechanical quantum matrix elements for the electron ^phonon interaction and statistical Bose ^ Einstein and Fermi ^ Dirac factors for the population of phonon and electron states. A very useful term in this context is the Eliashberg transport coupling function a 2 tr F(o). A detailed theoretical expression can be found in the work of Grimvall ([15]). The Eliashberg coupling function allows us to write thermal conductivity as follows: 1 ˆ 4p 2 omax ho=k B T ho a 2 l el L 0 Topl 2 0 exp ho=k B T 1 Š 1 exp ho=k B T Š 2p 2 tr F o k B T 3 2 ho a 2 2p 2 tr F o do, (7) k B T from which: " 1 ˆ k l E C har T=y E A 2 el B ye 1 3 A # T 2p 2 B. (8) In equation (8) k E represents a constant, y E is the Einstein temperature and C har represents the lattice heat capacity in the Einstein model. Motakabbir and Grimvall ([25]) discussed equation (8) with A=B as a free parameter, with the assumption that A=B Phonon contribution to thermal conductivity It is more difficult to determine the thermal conductivity when there are non-free electrons. We shall call solids which obey this rule non-metallic crystals. Because the atoms in a solid are closely coupled together, an increase in temperature will be transmitted to other parts. In modern theory, heat is considered as being transmitted by phonons, which are the quanta of energy in each mode of vibration. We can again use the expression: l ph ˆ 1 Cvl. (9) 3 (2) Consult also references [1], [6], [7], [14], [17], [24], [25], [28], [31].

4 614 J Avsec, M Oblak For an isotropic solid we can express the thermal conductivity as an integral over o containing the phonon density of states F(o) ([2]): l ph ˆ N omax 3V v 2 g t o C o F o do, (10) 0 where v g is some average phonon group velocity, C is the heat capacity of a single phonon mode, and the ratio N=V is the number of atoms per volume. Relaxation time can be expressed as the ratio of mean free path to velocity, so that the thermal conductivity can be expressed as: l ph ˆ N omax 3V v g l o C o F o do. (11) 0 The crucial point in equation (11) is the determination of the relaxation time. For the low-temperature region (where the temperature is lower than Debye temperature y D )we have used the expression: l ph ˆ l 0 exp y D, (12) T where l 0 is a constant. For the high-temperature region (T 4 y D ) the solution of equation (11) gives: l ph ˆ B MO 1=3 a k 3 B y 3 D, (13) 2p 3 h 3 g 2 T where B is a dimensionless constant, O a is atomic volume, and g is the Gru«neisen constant. 4 Calculation of thermal conductivity of nanoparticles (3) In nanoparticle ^ fluid mixtures, other effects such as microscopic motion of particles, particle structures and surface properties may cause additional heat transfer in nanofluids. Nanofluids also exhibit superior heat-transfer characteristics to conventional fluids. One of the main reasons is that the suspended particles appreciably increase the thermal conductivity of nanofluids. The thermal conductivity of a nanofluid is strongly dependent on the nanoparticle volume fraction. Until now there has been no sophisticated theory allowing us to predict the thermal conductivity of nanofluids. Here we are making an attempt to calculate the thermal conductivity of a nanofluid analytically. Hamilton and Crosser developed a model for the effective thermal conductivity of two-component mixtures as a function of the conductivity of the pure materials, and the composition and shape of dispersed particles. The thermal conductivity can be calculated with the expression ([12], [18], [22], [34], [35], [36]): lpt n 1 l f n 1 a l f l pt l ˆ l f, (14) l pt n 1 l f a l f l pt where l is the thermal conductivity of the mixture, l f is the thermal conductivity of the liquid, l pt is the thermal conductivity of solid particles, a is the volume fraction, and n is the empirical shape factor given by n ˆ 3, (15) C s where C s is sphericity, defined as the ratio of the surface area of a sphere (with a volume equal to that of a particle) to the area of the particle. The volume fraction a of the (3) Consult also references [30] and [33].

5 Thermal conductivity of nanofluids 615 particles is defined as: V pt a ˆ ˆ n p V f V pt 6 d 3 pt, (16) where n is the number of particles per unit volume, and d p is the average particle diameter. An alternative expression for calculating the effective thermal conductivity of solid ^ liquid mixtures was introduced by Wasp ([5]): lpt 2l f 2a l f l pt l ˆ l f. (17) l pt 2l f a l f l pt Comparison between equations (14) and (17) shows that Wasp's model is a special case of the Hamilton ^ Crosser (HC) model with a sphericity of 1.0. From the literature ([12], [22], [35], [36]) we can find other models (Maxwell, Jeffrey, Davis, Lu-Lin) with almost identical analytical results. The HC model gives very good results for particles larger than 13 nm. For smaller particles, the theory gives wrong results with a deviation greater than 100% from the experimental results. The theoretical models for the calculation of thermal conductivity of nanofluids are dependent only on the thermal conductivity of the solid and the liquid and their relative volume fraction, but not on particle size and the interface between the particles and the fluid. In the literature we can find many reasons for thermal conductivity enhancement in nanofluids. In the present paper we have made the assumption that the thermal conductivity enhancement in nanofluids is due to formation of liquid layers around nanoparticles. For the calculation of effective thermal conductivity we have used the Xue theory [36], based on Maxwell's theory and average polarization theory. Because interfacial shells exist between the nanoparticles and the liquid matrix, we can regard the interfacial shell together with the nanoparticle as a complex nanoparticle. The nanofluid system should therefore be regarded as consisting of complex nanoparticles dispersed in a fluid. We assume that l is the effective thermal conductivity of the nanofluid, and l c and l f are the thermal conductivities of the complex nanoparticles and fluid, respectively. The Xue model yields the following equations: 9 1 a k r l lf 2l l f a l r l l cx, l B 2x l cx l e 4 l l cy, ˆ 0,(18) 2l 1 B 2 x l cy l,,,, 1 B 2j, l 1 B 2j, l 2 1 B 2j, k r l 2 l 1 l cj, ˆ l 1. (19) 1 B 2j l 1 B 2j l 2 B 2j k r l 2 l 1,,, We assume that the complex nanoparticle is composed of an elliptical nanoparticle with thermal conductivity l 2 with half-radii a, b, c, and an elliptical shell of thermal conductivity l 1 with a thickness t. In equations (18) and (19) k r represents the spatial average of the heat flux component. For simplicity we assume that all fluid particles are balls and all the nanoparticles are the same rotational ellipsoids. Many studies have focused on the effect of the liquid layer; we considered nanoparticlein-liquid suspension with mono-sized spherical particles of radius r and particle volume concentrations a. We have used the model of Yu and Choi ([37]) which assumes that a nanolayer could be combined with the particle to form an `equivalent' particle, and that the particle volume concentration is so low that there is no overlap of those equivalent particles. On this basis we can express the effective volume fraction: a e ˆ a 1 2h 3 n, (20) d p

6 616 J Avsec, M Oblak where h n represents the liquid layer thickness. We have also assumed that the thermal conductivity of the equivalent particles has the same value as the thermal conductivity of the particle. On the basis of all the presented assumptions we have derived the following model (RHC) for the thermal conductivity for nanofluids: lpt n 1 l f n 1 a e l f l pt l ˆ l f. (21) l pt n 1 l f a e l f l pt 5 Results and comparison with experimental data We have made analytical computations for the mixture of nanoparticles or nanotubes with a reference fluid. The copper and aluminium oxide nanoparticles dispersed in a fluid are very interesting for industrial application because of their very high thermal conductivity in comparison with copper or aluminium oxides. We have used experimental results from the literature ([12]) with copper average nanoparticle diameter smaller than 10 nm. For Al 2 O 3 nanoparticles Eastman et al reported an average diameter of 35 nm. The carbon nanotubes have attracted much attention owing to their unique structure and remarkable mechanical and electrical properties ([33]). Nanofluids comprising copper nanoparticles directly dispersed in ethylene glycol exhibit significantly greater thermal conductivity enhancements compared to nanofluids containing oxide nanoparticles ([36]). Figure 1 shows the results for copper for analytical computation of Eliashberg coupling function (E) and Eishah models ([1]). The Eishah model is obtained on the basis of comparison between experimental data and empirical equation with optimisation procedure with fitting parameters. The agreement between models is satisfactory with the maximum relative deviation between the two models of around 9%. CF-1 is based on the Eishah model using equation (8) and CF-2 is based on the Eishah model using equation (9) ([1]). For ethylene glycol (figure 2) we have compared our analytical results with the simulations obtained on the web onlinetools/airprop/airprop.html). The comparison shows very good agreement. Figure 3 shows the comparison between analytical calculation and experimental data for aluminium oxide. Figures 4 and 5 show the analytical calculation for a mixture of ethylene glycol and copper and aluminium oxide (Al 2 O 3 ) nanoparticles of the thermal conductivity ratio. The results for thermal conductivity obtained by Xue and RHC models show very good agreement with experimental results. Thermal conductivity predicted by the HC model gave much lower values than the experimental results. Figure 6 shows the thermal conductivity ratio enhancement for the mixture between carbon nanotubes (CNT) and decene (DE) ([33]). In figure 6 there is satisfactory agreement between analytical data obtained by the Xue (X) model and the experimental data. For the Thermal conductivity=w m 1 K Temperature=K Figure 1. Thermal conductivity of copper. E CF-1 CF-2

7 Thermal conductivity of nanofluids Relative deviation RD 0.02 Temperature=K Figure 2. Thermal conductivity of ethylene glycol. 30 Thermal conductivity=w m 1 K E CRC Temperature=K Figure 3. Thermal conductivity of aluminium oxide, Al 2 O Thermal conductivity ratio HC X experimental RHC Volume fraction of nanoparticles Figure 4. Thermal conductivity of a mixture of copper nanoparticles ethylene glycol of various compositions.

8 618 J Avsec, M Oblak 1.2 Thermal conductivity ratio 1.1 HC experimental RHC Volume fraction of nanotubes=% Figure 5. Thermal conductivity of aluminium oxide nanoparticles ethylene glycol. 1.6 Thermal conductivity ratio X experimental Volume fraction of nanotubes=% Figure 6. Thermal conductivity ratio of a mixture between decene (DE) and carbon nanotubes (CNT). 3 Thermal conductivity ratio Volume fraction of nanotubes=% Figure 7. Analytical prediction of the thermal conductivity ratio with the X model.

9 Thermal conductivity of nanofluids 619 analytical calculation by the X model we have used the assumption that the average nanotube is a ˆ nm long (b ˆ c ˆ 12:5 nm). Figure 7 shows the analytical prediction of the mixture obtained by the X model between ethylene glycol and carbon nanotubes (a ˆ nm, b ˆ c ˆ 12:5 nm). 6 Conclusion and summary The paper presents the mathematical model for computation of transport properties for nanofluids containing nanoparticles or nanotubes. In the present paper we have focused on the influence of liquid layer as the reason of thermal conductivity enhancement of nanofluids. With the presented theory it is possible to predict thermal conductivity of nanofluids in dependence of volume concentration of nanoparticles or nanotubes. The analytical results are compared with the experimental data and they show relatively good agreement. References [1] Abu-Eishah S I, 2001, ``Correlations for the thermal conductivity of metals as a function of temperature'' Int. J. Thermophys ^ 1868 [2] Assael M J, Dalaouti N K, Dymond J H, Feleki E P, 2000, ``Correlation for viscosity and thermal conductivity of liquid halogenated ethane refrigerants'' Int. J. Thermophys ^ 376 [3] Assael M J, Dalaouti N K, Gialou K E, 2000, ``Viscosity and thermal conductivity of methane, ethane and propane halogenated refrigerants'' Int. J. Thermophys ^ 371 [4] Assael M J, Dymond J H, Papadaki M, Patterson P M, 1992, ``Correlation and prediction of dense fluid transport coefficients. II, Simple molecular fluids'' Fluid Phase Equilib ^255 [5] Avsec J, Oblak M, 2005, ``The calculation of thermal conductivity for nanofluids on the basis of statistical nano-mechanics V'', paper presented at the 43rd AIAA Aerospace Sciences Meeting and Exhibit, 10 ^ 13 January 2005, Reno, Nevada, AIAA 2005 ^ 759 [6] Barman R, 1976 Thermal Conductivity in Solids (Oxford: Clarendon Press) [7] Bodzenta J, 1999, ``Influence of order ^ disorder transition on thermal conductivity of solids'' Chaos Solitons and Fractals ^ 2098 [8] Chapman S, Cowling T G, 1970 The Mathematical Theory of Non-Uniform Gases third edition (Cambridge: Cambridge University Press) [9] Chung T-H, Ajlan M, Lee L L, Starling K E, 1984, ``Generalized multiparameter correlation for nonpolar and polar fluid transport properties'' Ind. Eng. Chem. Fundam. 23 8^13 [10] Chung T-H, Lee L L, Starling K E, 1988, `Àpplications of kinetic gas theories and multiparameter correlation for prediction of dilute gas viscosity and thermal conductivity'' Ind. Eng. Chem. Res ^679 [11] Dymond J H, 2000, ``Second virial coefficients and liquid transport properties at saturated vapour pressure of haloalkanes'' Fluid Phase Equilib ^ 32 [12] Eastman J A, Choi S U S, Li S, Yu W, Thompson L J, 2001, `Ànomalously increased effective thermal conductivities of ethylene glycol-based nanofluids containing copper nanoparticles'' Appl. Phys. Lett ^720 [13] Ferziger J H, Kaper H G, 1972 Mathematical Theory of Transport Processes in Gases (London: North-Holland) [14] Grimvall G, 1980 The Electron ^ Phonon Interaction in Metals (Amsterdam: North-Holland) [15] Grimvall G, 1999 Thermophysical Properties of Materials (Amsterdam: Elsevier) [16] Holland P M, Hanley H J M, 1979, ``Correlation of the viscosity and thermal-conductivity data of gaseous and liquid propane'' J. Phys. Chem. Ref. Data 8 559^575 [17] Incropera F P, Dewitt D P, 2001 Heat and Mass Transfer fifth edition (New York: John Wiley) [18] Kebliski P, Phillpot S R, Chou S U S, Eastman J A, 2002, ``Mechanisms of heat flow in suspensions of nano-sized particles'' Int. J. Heat Mass Transf ^ 863 [19] Kihara T, 1976 Intermolecular Forces (Chichester/New York: John Wiley) [20] Kiselev S B, Huber M L, 1998, ``Transport properties of carbon dioxide ethane and methane ethane mixtures in the extended critical region'' Fluid Phase Equilib ^ 280 [21] Kiselev S B, Perkins R A, Huber M L, 1999, ``Transport properties of refrigerants R32, R125, R143a, and R125 R32 mixtures in and beyond the critical region'' Int. J. Refrig ^ 520 [22] Lee S, Choi S U S, Li S, Eastman J A, 1999, ``Measuring thermal conductivity of fluids containing oxide nanoparticles'' J. Heat Transf., Trans. ASME ^289

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