STUDY OF THE EFFECTIVE THERMAL CONDUCTIVITY OF NANOFLUIDS

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1 Proceedings of IMECE25 25 ASME International Mechanical Engineering Congress and Exposition Noveber 5-, 25, Orlando, Florida, USA IMECE5-828 STUDY OF THE EFFECTIVE THERMAL CONDUCTIVITY OF NANOFLUIDS Ratnesh K. Shukla and Vijay K. Dhir Mechanical and Aerospace Engineering Departent Henry Saueli School of Engineering and Applied Science University of California, Los Angeles Los Angeles, CA 995 Ph: (3) , Fax: (3) Eail: ABSTRACT Nanofluids, that is liquids containing nanoeter sized etallic or non-etallic solid nanoparticles, show an increase in theral conductivity copared to that of the base liquid. In this paper a odel for theral conductivity of nanofluids based on the theory of Brownian otion of particles in a hoogeneous liquid cobined with the acroscopic Hailton-Crosser odel is presented. The odel is shown to predict a teperature and particle size dependent theral conductivity. Coparison between the predicted and experiental results show that the odel is able to accurately predict the teperature and volue fraction dependence of the theral conductivity of water based aluina and gold nanofluids. INTRODUCTION Nanofluids are fluids which contain nanoeter sized etallic or non-etallic particles suspended in the. Copared to ordinary fluids containing icro-sized particles, nanofluids are ore stable, do not clog flow channels and also show a considerable increase in theral conductivity for very sall volue fraction of solid particles [-8]. Due to their iproved heat transfer characteristics nanofluids have potential application in any heat transfer areas. Conventional theories eploying continuu odels underpredict the relative increase in theral conductivity obtained by addition of nanoparticles. This failure has been attributed to the lack of particle size as a paraeter in the continuu odels and as a consequence other possible echaniss naely, Brownian otion, clustering of nanoparticles, liquid layering close to nanoparticles and nature of heat transport in nanoparticles have been suggested [9]. However there is still a lack of proper understanding of which echanis contributes significantly to an increase in theral conductivity of the nanofluid. In [9] Molecular Dynaics siulations of a single particle of size 2n oving in a liquid showed that inside the nanoparticle heat flows in a ballistic rather than diffusive anner. Xue et al [] conducted olecular dynaics siulations for a solid-liquid interface using Lennard-Jones potential to odel the interactions between various atos assuing that the ass of solid and the liquid atos was the sae. They concluded that for such a syste in which the difference between the nuber density of solid and the liquid is only about 2%, layering of liquid atos at the solid-liquid interface does not have any significant effect on the theral conductivity and that a thin fil of liquid confined between two solid walls would show the sae theral conductivity as the bulk fluid. Jang and Choi [] analyzed Brownian otion of nanoparticles as a possible echanis for explanation of high theral conductivity of nanofluids and developed a odel based on the assuption that ordered layers of liquid atos close to the nanoparticle surface act as a hydrodynaic boundary layer. Unlike conventional Hailton Crosser odel [2] their odel was able to predict a particle-size and teperature dependent theral conductivity of nanofluids. The values of theral conductivity enhanceent predicted by their odel copared well with the experiental data for theral conductivity of aluina/water and copper/water nanofluids. In [2] Copyright c 25 by ASME

2 Brownian dynaics siulation was used to calculate the theral conductivity of nanofluids. Two unknown paraeters in the potential function between two nanoparticles were chosen to reproduce two experiental data points for theral conductivity of a given nanofluid syste. Using this epirical potential good agreeent was found between calculated and experiental values of theral conductivity. Xuan et al. [3] developed a theoretical odel for predicting theral conductivity of nanofluids by taking into account physical properties of both fluid and solid nanoparticle and the structure of nanoparticle clusters. Xue [4] developed a theoretical odel based on Maxwell and average polarization theory for predicting theral conductivities of nanofluids by taking in to account presence of a thin liquid layer around the nanoparticle whose theral conductivity is higher than theral conductivity of the bulk liquid. Good agreeent with the experiental data was reported for nanotube/oil and aluina/water nanofluids, however, the choice of thickness and theral conductivity of the liquid layer reained epirical. Wang et al. [5] developed a theoretical odel for estiating theral conductivity of nanofluids based on effective ediu approxiation and fractal theory for a description of a nanoparticle cluster and its radial distribution. Our objective in this paper is to develop a icroscopic odel for accurate prediction of theral conductivity of a nanofluid by concentrating on Brownian otion of nanoparticles as priary echanis for the enhanced theral conductivity of nanofluids. BROWNIAN MOTION BASED MODEL In this section a odel for predicting theral conductivity of nanofluids based on the Brownian otion of particles in a liquid at equilibriu is developed. It is assued that the Brownian otion of solid nanoparticles can be described using Langevin dynaics. The contribution of nanoparticles to the net theral conductivity of the nanofluid is then related to the flux autocorrelation function by applying Green-Kubo relation [7]. Brownian otion Brownian otion of particles suspended in a liquid is described using an equation of otion for the particle representing a balance between rando fluctuating force exerted by the fluid particles and the particle inertia and fluid resistance. Consider a syste consisting of N Brownian particles suspended in a hoogeneous liquid. The coupled Langevin equations of otion for these N identical particles are [9] du dt = Z.u + F(t), () with u = u(),x = x at t = describing the initial state of the syste. u = {u i } and F = {F i },i =,2,... N represent the velocities and rando forces acting on particle i respectively, being its ass. Also, Z = { ζ ij } represents eleents of the generalized friction tensor which deterines the force on ith particle due to velocity of the jth. It is assued that the net force acting on the particle can be split into two parts, Z.u representing the friction force exerted on the particle by surrounding fluid and F(t) representing fluctuating force exerted on the particle by surrounding fluid which is a characteristic of Brownian otion. For the fluctuating part, following assuptions are ade: F(t) is independent of u. F(t) varies extreely rapidly copared to the variations of u. The effects of the fluctuating force can then be suarized by F(t) =, F(t)F T (t ) = F δ(t t ). (2) Application of equipartition theore gives uu T eq = k BT I at theral equilibriu. This syste can be solved exactly and it can be shown that [9] F = 2k B T Z, (3) which is the Fluctuation-Dissipation Theore. Also, the velocity autocorrelation function is given by the following expression u(t)u T (t + τ) = k BT e τ Z. (4) Theral Conductivity In order to estiate the contribution of N spherical Brownian particles suspended in a syste, occupying volue V and at an equilibriu teperature T, to the theral conductivity λ we use the Green-Kubo relation [7] λ = V Z k B T 2 J x (τ)j x () dτ, (5) where J x represents the heat current vector in the syste in the x direction. For our syste consisting of N Brownian particles the icroscopic definition of heat current, assuing that potential energy of interparticle interaction is negligible, is J x = N i= 2 (u2 i + v 2 i + w 2 i )u i (6) where u i,v i,w i are the coponents of the velocity vector of the ith particle in the x, y and z directions respectively. The expres- 2 Copyright c 25 by ASME

3 sion for flux autocorrelation function then siplifies to J x (t)j x (t + τ) = 2 4 { N i=(u 3i + u iv 2i + u iw 2i ) } { N i=(u 3i + u iv 2i + u iw 2i ) } t+τ t (7) where the subscripts denote that corresponding expressions are evaluated at ties t and t + τ respectively. In order to proceed further with the calculation of flux autocorrelation we assue that the interparticle contributions to the flux autocorrelation are negligible. Physically this corresponds to a situation in which the Brownian spheres are well separated so that the friction tensor takes a siple for given by ζ ij = 6πµaδ i j (8) where µ is the fluid viscosity and a is the particle radius. The flux autocorrelation is now given by J x (t)j x (t +τ) = 2 4 N i= (u 3 i + u i v 2 i + u i w 2 i ) t (u 3 i + u i v 2 i + u i w 2 i ) t+τ (9) For calculation of averages showing up in the above equation one needs to relate higher oents to lower ones. Two saple calculations based on relations in [8] are shown below, u 3 i (t)u3 i (t + τ) = 2 u i(t)u i (t) u i (t)u 3 i (t + τ) + 3 u i (t)u i (t + τ) u 2 i (t)u 2 i (t + τ) = 9 u i (t)u i (t) u i (t + τ)u i (t) u i (t + τ)u i (t + τ) + 6 u i (t)u i (t + τ) 3 u 3 i (t)u i(t + τ)v 2 i (t + τ) = 2 u i(t)u i (t) u i (t)u i (t + τ)v 2 i (t + τ) + u i (t)u i (t + τ) u 2 i (t)v 2 i (t + τ) = 2 u 2 i (t) u i(t)u i (t + τ) v 2 i (t + τ) + u i (t)u i (t + τ) u 2 i (t) v 2 i (t + τ) () where u i (t)v i (t + τ) = has been used. Other averages can be calculated in a siilar way and are oitted here for brevity. Also since velocity autocorrelation function is given by u i (t)u i (t + τ) = k BT e ζτ/,ζ = 6πµa () Eq () reduces to u 3 i (t)u3 i (t + τ) = { kb T u 3 i (t)u i(t + τ)v 2 i (t + τ) =3 { kb T } 3 { } 9e ζτ + 6e 3ζτ } 3 { } e ζτ (2) The Green-Kubo relation iplies that the diffusion coefficient is given by area under the velocity autocorrelation curve whereas the theral conductivity is given by area under the flux autocorrelation curve. Thus the diffusivity of the particle can be obtained as Z D = u(t)u T () dt = k BT I, (3) ζ which is the Einstein s relation for diffusivity. For contribution of particles to the net theral conductivity of the liquid-particle nanofluid syste we have λ particle = Z Vk B T 2 = N2 4Vk B T 2 = 85 NkB 2T 2 Vζ Z J x (τ)j x () dτ } {25e ζτ + e 3ζτ dτ (4) For nanoparticles with radius a on substitution of Stokes expression for drag i.e. ζ = 6πaµ and φ = N( 4 3 πa3 )/V for volue fraction above expression yields λ particle = 85 96π 2 φk 2 B T a 4 µ. (5) This relation agrees qualitatively with the experiental results in that contribution of nanoparticles to the increase in theral conductivity rises with an increase in particle volue fraction and fluid teperature. In addition, saller nanoparticles cause a higher increase in theral conductivity of the base fluid. The advantage of using an approach, as described above, is that an explicit expression for theral conductivity contribution due to nanoparticles, which follow Langevin dynaics, can be obtained. Besides, using a proper definition of the generalized friction tensor Z theral conductivity enhanceent in a ore coplex nanofluid syste, exaple carbon nanotube oil suspension, can be predicted. In the next section a coparsion between theoretical predictions obtained using a odel based on above result and the experiental data for aluina/water and gold/water nanofluids is given. 3 Copyright c 25 by ASME

4 .5.4 Model Prediction ( % vol. fraction) Model Prediction (4 % vol. fraction) Experiental data [6] ( % vol. fraction) Experiental data [6] (4 % vol. fraction) Hailton-Crosser Prediction [2] ( % vol. fraction) Hailton-Crosser Prediction [2] (4 % vol. fraction).5 Model Prediction (.26 % vol. fraction) Model Prediction (.3 % vol. fraction) Experiental data [7] (.26 % vol. fraction) Experiental data [7] (.3 % vol. fraction).25. eff f λ / λ.3.2 eff f λ / λ Teperature (K) Teperature (K) Figure. Coparison of theoretical predictions with the experiental data for aluina(38.4 n)/water nanofluid. Figure 2. Coparison of theoretical predictions with the experiental data for gold(7 n)/water nanofluid. ANALYSIS AND DISCUSSION Based on the result given by Eq (5) in the previous section we propose the following for of effective theral conductivity of a nanofluid { } λp + 2λ f + 2φ(λ p λ f ) λ e f f = λ f + C φ(t T ) λ p + 2λ f φ(λ p λ f ) µa 4 (6) where λ f and λ p are the theral conductivities of fluid and nanoparticle respectively and C and T are constants. The first ter represents contribution due to acroscopic Hailton- Crosser odel [2] whereas the second ter represents contribution due to Brownian otion of nanoparticles. Thus the contribution of Brownian otion to the theral conductivity increases with rising teperature and is expected to be ore for saller nanoparticles. Note that T represents a reference teperature below which contribution of Brownian otion to effective theral conductivity is negligible and the acroscopic Hailton Crosser odel is sufficient to accurately predict the theral conductivity. This reference teperature is expected to vary for different solid/liquid cobination, for exaple there will not be any Brownian otion once the liquid is below freezing point; thus freezing point provides a lower liit for T. Also, since at a sae given teperature saller nanoparticles show ore Brownian otion (diffusion coefficient D = k BT 6πµa ) than the ones with larger radius this value T is expected to be a function of particle radius as well. A coparison between theoretical predictions and experiental data for the theral conductivity enhanceent ratio (effective theral conductivity of nanofluid to theral conductivity of the base fluid) with teperature for 38.4 n aluina/water nanofluid [6] is presented in Fig.. The values of constants C and T are set equal to and 2 C respectively. Reasonable agreeent is found between predicted values and experiental data for particle volue fractions of % and 4%. Teperature dependence of theral conductivity and viscosity of water is taken in to account for calculation of effective theral conductivity enhanceent. It is also noted that the acroscopic Hailton Crosser odel predicts no change in conductivity enhanceent with teperature since it does not account for the effect of Brownian otion as shown in Fig.. Figure 2 shows a coparison between theoretical predictions and experiental data for 7 n gold/water nanofluid. Good agreeent is found between experiental data and predicted values for a choice of constants C=.5 33,T = C. For this nanofluid acroscopic Hailton Crosser odel predicts negligible increase in effective theral conductivity due to very sall volue fraction of gold nanoparticles in water. CONCLUSIONS In this paper a icroscopic odel, which takes into account the dependence of size of nanoparticles and teperature on the effective theral conductivity of nanofluids along with acroscopic Hailton Crosser odel, is proposed based on the theory of Brownian otion of particles in a fluid. The odel predicts a linear dependence of the increase in theral conductivity of nanofluid with the volue fraction of solid nanoparticles. Satisfactory coparison of the theoretical predictions with the experiental data is observed for variation of effective theral con- 4 Copyright c 25 by ASME

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