Is classical energy equation adequate for convective heat transfer in nanofluids? Citation Advances In Mechanical Engineering, 2010, v.

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1 Title Is lassial energy equation adequate for onvetive heat transfer in nanofluids? Authors Wang, L; Fan, J Citation Advanes In Mehanial Engineering, 200, v. 200 Issued Date 200 URL Rights This work is liensed under a Creative Commons Attribution- NonCommerial-NoDerivatives 4.0 International Liense.

2 Advanes in Mehanial Engineering Volume 200, Artile ID 79406, 5 pages doi:0.55/200/79406 Researh Artile Is Classial Energy Equation Adequate for Convetive Heat Transfer in Nanofluids? Jing Fan and Liqiu Wang Department of Mehanial Engineering, The University of Hong Kong, Pokfulam Road, Hong Kong Correspondene should be addressed to Liqiu Wang, lqwang@hku.hk Reeived 20 August 2009; Aepted 7 September 2009 Aademi Editor: Oronzio Mana Copyright 200 J. Fan and L. Wang. This is an open aess artile distributed under the Creative Commons Attribution Liense, whih permits unrestrited use, distribution, and reprodution in any medium, provided the original work is properly ited. To address whether the heat transfer in nanofluids still satisfies the lassial energy equation, we theoretially examine the marosale manifestation of the mirosale physis in nanofluids. The mirosale interation between nanopartiles and base fluids manifests itself as thermal waves/resonane at the marosale. The energy equation that governs the heat transfer in nanofluids is of a dual-phase-lagging type instead of the postulated and ommonly-used lassial energy equation. The interplays among diffusion, onvetion, and thermal waves/resonane enrih the heat transfer in nanofluids onsiderably.. Introdution Choi oined the term nanofluids for the fluids with nanoelements nanopartiles, nanotubes, or nanofibers suspended in them ]. Reent experiments on nanofluids have shown, for example, twofold inreases in thermal ondutivity 2], strong temperature dependene of thermal ondutivity 3], substantial inreases in onvetive heat transfer oeffiient 4, 5], and threefold inreases in ritial heat flux CHF in boiling heat transfer 2, 3, 6]. Stateof-the-art expositions of major advanes on the synthesis, haraterization, and appliation of nanofluids are available, for example, in 2, 3, 6 2]. These harateristis make them very attrative for a large number of industries suh as transportation, eletronis, defense, spae, nulear systems ooling, and biomediine. The study of nanofluids is still in its infany 2, 3, 6 2]. The preise nature and mehanism of the signifiant improvement of thermal performane are still not known. There is also a lak of agreement between experimental results and between theoretial models. The fat that the enhanement in thermal properties omes from the presene of nanopartiles has direted researh efforts nearly exlusively towards thermal transport at nanosale. The lassial onservation equations inluding the energy equation have been postulated as the marosale model of nanofluid onvetive heat transfer but without adequate justifiation. Thermal ondutivity and onvetive heat transfer oeffiient are a marosale phenomenologial haraterization of heat transfer and their measurements are not performed at the nanosale, but rather at the marosale. Therefore, interest should fous not only on what happens at the nanosale but also on how the presene of nanoelements affets the heat transport at marosale. We attempt to examine whether the lassial energy equation is adequate for desribing onvetive heat transfer in nanofluids at marosale based on a marosale heat transfer model in nanofluids, whih is rigorously developed by saling-up the mirosale model for the heat transfer in the nanopartiles and in the base fluids. The approah for saling-up is the volume averaging 3 5] with help of multisale theorems 5]. 2. Marosale Energy Equation The mirosale model for heat transfer in nanofluids is well known. It onsists of the field equation and the onstitutive equation. The field equation omes from the onservation laws of mass, momentum, and energy. The ommonly-used

3 2 Advanes in Mehanial Engineering onstitutive equation inludes the Newton law of visosity and the Fourier law of heat ondution 6]. For transport in nanofluids, the marosale is a phenomenologial sale that is muh larger than the mirosale and muh smaller than the system length sale. Interest in the marosale rather than the mirosale omes from the fat that a predition at the mirosale is ompliated due to the omplex mirosale struture of nanofluids, and also beause we are usually more interested in large sales of transport for pratial appliations. Existene of suh a marosale desription equivalent to the mirosale behavior requires a good separation of length sales and has been well disussed in 5, 7]. To develop a marosale model of heat transfer in nanofluids, the method of volume averaging starts with a mirosale desription. Both onservation and onstitutive equations are introdued at the mirosale. The resulting mirosale field equations are then averaged over a representative elementary volume REV, the smallest differential volume resulting in statistially meaningful loal averaging properties, to obtain the marosale field equations. In the proess of averaging, the multisale theorems 5] are used to onvert integrals of gradient, divergene, url, and partial time derivatives of a funtion into some ombination of gradient, divergene, url, and partial time derivatives of integrals of the funtion and integrals over the boundary of the REV. Consider heat transfer in nanofluids with the ontinuous base fluid and the dispersed nanopartile denoted by - and d-phases, respetively. Note that the dispersed nanopartiles an be liquid droplets for general nanofluids 0, 2]. As a two-omponent mixture base fluid + nanopartiles, its mirosale model an be written as 8] v = 0, ρ v + ρ v v in the -phase, = p + ρ g + μ 2 v, T = k T, v d = 0, ρ d v d + ρ p v T + ρ d v d v d in the -phase, in the d-phase, = p d + ρ d g + μ d 2 v d, d T d = k d T d, v = v d, at A d, T = T d, at A d, + ρ p d v d T d in the d-phase, n d k T = n d k d T d, at A d. in the -phase, in the d-phase, d-phase n d -phase V REV Figure : Representative elementary volume REV. Here v and T are the veloity and the temperature, respetively. ρ, p, k, p, μ, and g are the density, speifi heat, thermal ondutivity, pressure, visosity, and gravitational aeleration, respetively. Subsripts and d refer to the - and d-phases, respetively. A d represents the area of the -d interfae ontained in the REV, and n d is the outward-direted surfae normal from the d-phase toward the -phase, Figure. To be thorough, we must also speify the initial onditions and the boundary onditions at the entranes and exits of the REV; however, we need not do so for our disussion. Applying the volume averaging and multisale theorems to sale-up the mirosale model yields a marosale model 2, 9], where the energy equation reads ε T u d T d d + ε v T u T = K T + K d T d d ε d a v h T T d d, d T d d u dd T d d + ε d = Kd T + K dd T d d a v h T d d T. d v d d T d d u d T Here ε i is the volume fration of the i-phase the index i an take or d, with V i and V REV as the volume of the i-phase in REV, and the volume of the REV respetively: ε i = 2 V i V REV. 3

4 Advanes in Mehanial Engineering 3 The intrinsi average is defined by Ψ i i = V i V i Ψ i dv. 4 K, K d, K d, K dd, u, u d, u d, u dd,anda v h represent the effets of mirosale physis on the marosale heat transfer. The readers are referred to2, 9, 20] for their governing equations and numerial omputations. 3. Results and Disussion Rewrite 2 in their operator form: A B T = 0. 5 C D T d d where A = γ /+γ v u K +a v h, B = u d K d a v h, C = u d K d a v h,andd = γ d /+γ d v d d u dd K dd. We then obtain an unoupled form by evaluating the operator determinant suh that γ + γ v u K + a v h γ d + γ d v d d u dd K dd + a v h u d + K d + a v h ] u d + K d + a v h T i i = 0, where the index i an take or d. γ = ε ρ and γ d = ε d ρ d are the -phase and d-phase effetive thermal apaities, respetively. Its expliit form reads, after dividing by a v hγ + γ d, T i i γ γ d + a v h 2 T i i γ + γ d 2 + γ + γ d 6 + a v h γ + γ d { { Kd ] K d ] K ] K dd ] } T i i + γ udd T i i + γ d u T i i] { u K dd ] + u dd K ] u d K d ] u d K d ] γ v u dd + K dd ] γ d v d d u + K ] } T i i u u dd u d u d +γ γ d v } v d d ] T i i. 7 When the system is isotropi and the physial properties of the two phases are onstant, it redues to where T i i + τ q 2 T i i 2 + v d d T i i = α Δ T i i + ατ T Δ Ti i + α k Fr, t + τ q Fr, t ], 8 = γ v + γ d v d d T i i γ + γ d K + K d + K d + a v h γ + γ d + K dd ] T i i { γ K dd ] + γ d K ] γ γ d v + v d d ] } T i i v d d = τ q = τ T = γ + γ d γ v + γ d v d d, γ γ d a v h γ + γ d, γ k dd + γ d k a v hk + k d + k d + k dd, k = k + k d + k d + k dd, α = k + k d + k d + k dd γ + γ d, 9

5 4 Advanes in Mehanial Engineering Fr, t Fr, t + τ q = { k d k d k k dd Δ 2 T i i a v h + γ udd T i i + γ d u T i i γ γ d v + v d d ] T i i { kdd Δ u + γ v ] + k Δ u dd + γ d v d d ] k d Δu d k d Δu d } T i i { u + γ v ] u dd udd + γ d v d d ] u u d + γ v ] ud + γ d v d d } ]} T i i. 0 This is a dual-phase-lagging heat-ondution equation with τ q and τ T as the phase lags of the heat flux and the temperature gradient, respetively 5, 2, 22]. Here, Fr, t is the volumetri heat soure. k, ρ, α, and v d d are the effetive thermal ondutivity, apaity, diffusivity, and veloity of nanofluids, respetively. Therefore, the presene of nanopartiles shifts the lassial energy equation for the heat transfer in the base fluid into the dual-phase-lagging energy equation in nanofluids at the marosale. This is signifiant beause all results regarding dual-phase-lagging heat transfer an thus be applied to study heat transfer in nanofluids 2, 22]. For the ase of heat ondution in nanofluids, the inlusion of the solid-fluid interfae heat transfer also leads to dual-phase-lagging heat-ondution equation, whih resolves the onflit between experimental data of nanofluid thermal ondutivity and lassial theories of effetive thermal ondutivity of suspensions 23]. It is interesting to note that there are nontraditional onvetive terms u T u d T d d and u d T u dd T d d in 2. In 8, however, suh terms disappear beause of the onstraint from the mass onservation. Therefore, the mirosale physis does not manifest itself as the marosale onvetion. The veloity-like terms appear only in the soure term in 8. The presene of nanopartiles gives rise to variations of thermal apaity, ondutivity, and diffusivity, whih are given by, in terms of ratios over those of the base fluid, ρ ρ = ε d + ε d d, ρ ρ k = k + k d + k d + k dd, k k α = k ρ α k ρ. Therefore, ρ/ρ depends only on the volume fration of nanopartiles and the nanopartile-fluid apaity ratio. However, both k/k and α/α are affeted by the geometry, property and dynami proess of nanopartile-fluid interfaes. Consider τ T = + γ2 k dd + γd 2k γ γ d k d + k d. 2 τ q γ γ d k + k d + k d + k dd It an be larger, equal, or smaller than depending on the sign of γ 2 k dd + γ 2 d k γ γ d k d + k d. Therefore, by the ondition for the existene of thermal waves that requires τ T /τ q < 22, 24], we may have thermal waves in nanofluid heat transfer when γ 2 k dd + γ 2 d k γ γ d k d + k d 2 = γ k dd γ d k + γ γ d 2 k k dd k d k d < 0. 3 A neessary but not suffiient ondition for 3 isk d + k d > 2 k k dd. Note also that for heat transfer in nanofluids the mirosale physis yields a time-dependent soure term Fr, t in the dual-phase-lagging energy equation 8 and 0. Therefore, the resonane an also our. These thermal waves and possibly resonane are believed to be the driving fore for the enhanement of heat transfer. When k d + k d = 0 so that τ T /τ q is always larger than, thermal waves and resonane would not appear. The sum k d + k d is thus responsible for thermal waves and resonane in nanofluid heat transfer. It is also interesting to note that although eah τ q and τ T is a v h-dependent, the ratio τ T /τ q is not. Therefore the evaluation of τ T /τ q will be muh simpler than τ q or τ T. Therefore, the moleular physis and the mirosale physis interations between nanopartiles and base fluids at the mirosale in partiular manifest themselves as heat diffusion and thermal waves/resonane at the marosale, respetively. Their overall marosopi manifestation shifts the lassial energy equation for the heat transfer in the base fluid into the dual-phase-lagging energy equation in nanofluids. When τ T /τ q <, thermal waves dominate and 8 is of a hyperboli type 22]. When τ T /τ q, however, heat diffusion dominates and 8 is paraboli 22]. Depending on fators like material properties of nanopartiles and base fluids, nanopartiles geometrial struture and their distribution in the base fluids, and interfaial properties and dynami proesses on partile-fluid interfaes, the heat diffusion, thermal waves/resonane, and onvetion may either enhane or ounterat eah other. Consequently, the heat transfer in nanofluids is endowed with muh riher features than that in the base fluid. 4. Conluding Remarks In an attempt to determine how the presene of nanopartiles affets the heat transfer at the marosale and isolates the mehanism responsible for the reported variation of thermal properties, a marosale energy equation is developed and

6 Advanes in Mehanial Engineering 5 examined analytially for nanofluid heat transfer. The model is obtained by saling-up the mirosale model for the heat transfer in the nanopartiles and in the base fluids. The approah for saling-up is the volume averaging with help of multisale theorems. The result shows that the presene of nanopartiles leads to a dual-phase-lagging energy equation in nanofluids at the marosale. Therefore, the moleular physis and the mirosale physis manifest themselves as heat diffusion and thermal waves at the marosale, respetively. Depending on fators like material properties of nanopartiles and base fluids, nanopartiles geometrial struture and their distribution in the base fluids, and interfaial properties and dynami proesses on partilefluid interfaes, the heat diffusion, onvetion, and thermal waves may either enhane or ounterat eah other, whih will enrih heat-transfer performane signifiantly. Aknowledgment The finanial support from the Researh Grants Counil of Hong Kong GRF78009 and GRF77508 is gratefully aknowledged. Referenes ] S. U. S. Choi, Enhaning thermal ondutivity of fluids with nanopartiles, in Developments and Appliations of Non- Newtonian Flows, D.A.SingerandH.P.Wang,Eds.,vol.23 of FED, pp , ASME, New York, NY, USA, ] G. P. Peterson and C. H. Li, Heat and mass transfer in fluids with nanopartile suspensions, Advanes in Heat Transfer, vol. 39, pp , ] S.K.Das,S.U.S.Choi,W.H.Yu,andT.Pradeep,Nanofluids: Siene and Tehnology, John Wiley & Sons, Hoboken, NJ, USA, ] D. Wen, Y. Ding, and R. Williams, Nanofluids turn up the heat, The Chemial Engineer, no. 77, pp , ] D. Y. Tzou, Thermal instability of nanofluids in natural onvetion, International Heat and Mass Transfer, vol. 5, no. -2, pp , ] S. U. S. Choi, Z. G. Zhang, and P. Keblinski, Nanofluids, in Enylopedia of Nanosiene and Nanotehnology, H.S.Nalwa, Ed., vol. 6, pp , Amerian Sientifi Publishers, New York, NY, USA, ] J.A.Eastman,S.R.Phillpot,S.U.S.Choi,andP.Keblinski, Thermal transport in nanofluids, Annual Review of Materials Researh, vol. 34, pp , ] P. E. Phelan, P. Bhattaharya, and R. S. Prasher, Nanofluids for heat transfer appliations, Annual Reviews of Heat Transfer, vol. 4, pp , ] C. B. Sobhan and G. P. Peterson, Mirosale and Nanosale Heat Transfer: Fundamentals and Engineering Appliations, CRC Press, Boa Raton, Fla, USA, ] L. Q. Wang and X. H. Wei, Nanofluids: synthesis, heat ondution, and extension, Heat Transfer, vol. 3, no. 3, Artile ID 03302, 7 pages, ] S. U. S. Choi, Nanofluids: from vision to reality through researh, Heat Transfer, vol. 3, Artile ID 03306, 9 pages, ] L. Q. Wang and M. Quintard, Nanofluids of the future, in Advanes in Transport Phenomena, pp , Springer, Heidelberg, Germany, ] S. Whitaker, The Method of Volume Averaging, Kluwer Aademi Publishers, Dordreht, The Netherlands, ] L. Q. Wang, Flows through porous media: a theoretial development at marosale, Transport in Porous Media, vol. 39, no., pp. 24, ] L. Q. Wang, M. T. Xu, and X. H. Wei, Multisale theorems, Advanes in Chemial Engineering, vol. 34, pp , ] L. Q. Wang, A general theory of diffusion, Progress of Theoretial Physis, vol. 0, no. 3, pp , ] J. L. Auriault, Heterogeneous medium: is an equivalent marosopi desription possible? International Engineering Siene, vol. 29, no. 7, pp , 99. 8] R. B. Bird, W. E. Stewart, and E. N. Lightfoot, Transport Phenomena, John Wiley & Sons, New York, NY, USA, 3rd edition, ] M. Quintard and S. Whitaker, Theoretial analysis of transport in porous media, in Handbook of Heat Transfer in Porous Media, K. Vafai, Ed., pp. 52, Marel Dekker, New York, NY, USA, ] J. Fan and L. Q. Wang, Mirostrutural effets on marosale thermal properties in nanofluids, NANO, in press. 2] D. Y. Tzou, Maro-to Mirosale Heat Transfer: The Lagging Behavior, Taylor & Franis, London, UK, ] L. Q. Wang, X. S. Zhou, and X. H. Wei, Heat Condution: Mathematial Models and Analytial Solutions, Springer, Berlin, Germany, ] P. Vadasz, Heat ondution in nanofluid suspensions, Journal of Heat Transfer, vol. 28, no. 5, pp , ] M. T. Xu and L. Q. Wang, Thermal osillation and resonane in dual-phase-lagging heat ondution, International Journal of Heat and Mass Transfer, vol. 45, no. 5, pp , 2002.

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