Contribution of Brownian Motion in Thermal Conductivity of Nanofluids

Transcription

1 , July 6-8, 20, London, U.K. Contribution of Brownian Motion in Thermal Conductivity of Nanofluids S. M. Sohel Murshed and C. A. Nieto de Castro Abstract As a hot research toic nanofluids have attracted great interest from researchers worldwide. Although nanofluids are found to exhibit higher thermal conductivity comared to their base fluids, the underlying mechanisms for the enhancement are still debated and not fully understood. In addition, there has been little agreement among different studies and no widely acceted model is also available for the rediction of the effective thermal conductivity of nanofluids. In this study, an imroved Brownian motion-based combined thermal conductivity model is reorted and a renovated Brownian motion term is used in this model. Besides the Brownian motion of nanoarticles, this model also takes into account several other imortant factors such as article size interfacial nanolayer and fluid temerature that are believed to intensify the enhancement of the effective thermal conductivity of nanofluids. The resent model shows reasonably good agreement with the exerimental results of various aqueous nanofluids and gives better redictions comared to classical and recently develoed models used for nanofluids. Index Terms Brownian motion, interfacial nanolayer, nanofluids, nanoarticles, thermal conductivity, volume fraction O I. INTRODUCTION VER the last several decades, scientists and engineers have attemted to develo fluids, which offer better cooling or heating erformance. However, it is only in 995 that Steve Choi [] at Argonne National Laboratory of USA coined the novel concet of nanofluids to meet the cooling challenges facing many high-tech industries. Nanofluids are a new class of heat transfer fluids which are engineered by disersing nanometer-sized solid articles in conventional fluids articles, rods or tubes in conventional heat transfer fluids such as water and engine oil. This is a raidly emerging interdiscilinary field where nanoscience, nanotechnology, and thermal engineering meet. This research toic has attracted tremendous interest from researchers due to their exciting thermal roerties and otential alications in numerous imortant fields such as microelectronics, microfluidics, transortation, manufacturing, medical, and so on [2]-[4]. The imact of nanofluid technology is exected to be great considering Manuscrit received March 2, 20; revised Aril 5, 20. S. M. S. Murshed is with the Centre for Molecular Sciences and Materials, Faculty of Sciences of the University of Lisbon, Lisbon, Portugal (hone: ; fax: ; smmurshed@ fc.ul.t). C. A. Nieto de Castro is with the Centre for Molecular Sciences and Materials, Faculty of Sciences of the University of Lisbon, Lisbon, Portugal ( cacastror@fc.ul.t). that heat transfer erformance of heat exchangers or cooling devices is vital in numerous industries. When nanoarticles are roerly disersed, nanofluids can offer numerous benefits besides the anomalously high effective thermal conductivity. Some of these benefits are imroved heat transfer and stability, microchannel cooling without clogging, miniaturized systems, and reduction in uming ower. The better stability of nanofluids will revent raid settling and reduce clogging in the walls of heat transfer devices. The high thermal conductivity of nanofluids translates into higher energy efficiency, better erformance, and lower oerating costs. They can reduce energy consumtion for uming heat transfer fluids. Miniaturized systems require smaller inventories of fluids where nanofluids can be used. Thermal systems can be smaller and lighter. In vehicles, smaller comonents result in better gasoline mileage, fuel savings, lower emissions, and a cleaner environment. With these highly desired thermal roerties and otential benefits, nanofluids are thought to have a wide range of alications including transortation sector (because of the higher thermal conductivity nanofluids would allow for smaller, lighter engines, ums, radiators, and other comonents) and microelectromechanical systems [2], []. Although significant rogress has been made, variability and controversies in the reorted data and heat transort mechanisms still exist with nanofluids [2]. Furthermore, there has been little agreement among different studies and no widely acceted model is also available due to inconclusive heat transfer mechanisms of nanofluids. Nevertheless, fundamental understanding of the underlying mechanisms and develoment of a unanimous theoretical model are crucial for exloiting aforementioned otential benefits and alications of nanofluids. In this aer, a new and imroved Brownian motion (BM)-based model is introduced for the rediction of the enhanced thermal conductivity of nanofluids. In addition to the Brownian motion of nanoarticles, this model also takes into account several other key factors such as article size, fluid temerature and interfacial nanolayer that also contribute to the enhancement of the effective thermal conductivity of nanofluids. The conventional kinetic theory-based Brownian motion term has been renovated using effective diffusion coefficient concet. Besides roviding a brief review on theoretical studies and various heat transfer mechanisms of nanofluids, details of the resent model develoment and its validation with the exerimental results are also discussed in this aer.

2 , July 6-8, 20, London, U.K. II. MODELING FOR THE EFFECTIVE THERMAL CONDUCTIVITY OF NANOFLUIDS A. Existing Models Previous studies showed that the classical models such as those attributed to Maxwell [5] and Hamilton-Crosser [6] are unable to redict the anomalously high thermal conductivity of nanofluids. This is because these models were develoed for continuum medium of well-disersed mili- or micro-sized solid articles and they do not include the nanoscale effects of article size such as the interfacial nanolayer at the article/liquid interface, and motion of articles, which are considered as imortant factors for the enhancement of the thermal conductivity of nanofluids. For sherical articles, the Hamilton-Crosser model [6] is the same as the [5]. Therefore, the Maxwell model is used as the reresentative of classical models and it is given as ( k 2k f ) 2( k k f ) keff / k f ( k 2k f ) ( k k f ) where is the article volume fraction and k eff, k f and k are the effective thermal conductivity of susensions, thermal conductivity of base fluid and thermal conductivity of nanoarticle, resectively. Recently, many theoretical studies have been carried out to redict the anomalously increased thermal conductivity of nanofluids. Several models have also been roosed by considering various mechanisms. However, there has been little agreement among different studies and no widely acceted model is still available. A detailed review of numerous models roosed for the rediction of effective thermal conductivity of nanofluids is rovided in recent review article [2] and it will not be elaborated here. Very few theoretical efforts have been made on the combined static and dynamic effects of nanoarticles. In an attemt to develo such a combined model, Prasher et al. [7] considered dynamic contribution of nanoarticles in their thermal conductivity model for nanofluids. However, their model contains three unknown emirical or fitting arameters. In addition, dynamic contribution was couled with Maxwell s model without taking into account the effect of interfacial layer in their model. Besides article volume fraction, articles size, temerature, articles disersions and article movement should be taken into account in develoing model for the effective thermal conductivity of nanofluids. Prasher et al. s [7] model is exressed as ( 2) 2 ( ) m 0. eff / k f ( A Re Pr ) ( 2) ( ) k (2) () Thus, this study focuses on develoment of model by the combination of static and Brownian motion-based dynamic mechanisms of nanoarticles in base fluids. B. Present Modeling The effective thermal conductivity of nanofluids is considered to be from both the static and dynamic mechanisms. The effects of these mechanisms are treated to be additive. A static thermal conductivity model for the nanofluids (non-interacting nanoarticles) was reviously develoed by considering nanofluids as a mixture of three comonentsthe nanoarticle (radius r ), the interfacial layer between article/fluid medium (thickness h), and the fluid medium and it (k st ) is exressed as [8] k st k f k kf k kf k kf k kf ( )[2 ] ( 2 ) [ ( ) ] ( 2 ) ( ) [ ] () where rc/ r h/ r, r c is the radius of comlex nanoarticle (nanoarticle with interfacial layer), klr / kf (where ), and k lr is the thermal conductivity of interfacial layer. The detailed discussion and mathematical derivations for this static model containing sherical nanoarticles with interfacial nanolayer can be found elsewhere [8]. As each nanoarticle is considered to have a nanolayer at the article/fluid interface and its thickness, h remains in the model, the value of a nanolayer thickness is required to calculate the thermal conductivity of nanofluids. Hashimoto et al. [9] established a new definition of the interfacial layer thickness at the surface of sherical micro-domains, which is given as h 2 where σ is a arameter characterizing the diffuseness of the interfacial boundary and its tyical value falls in the range of 0. to 0.6 nm. For σ = 0.4 nm, h = nm. In fact, the exerimental results of Yu et al. [0] and the molecular dynamics simulations erformed by Xue et al. [] showed that the tyical interfacial layer thickness between the solid (nanoarticles) and liquid hases is of the order of a few atomic distances namely, nm. Hence, an interfacial layer thickness h of nm can reasonably be used to redict the thermal conductivity of nanofluids. Fig. resents concet of Brownian motion-based dynamic mechanisms of nanoarticles in base fluid. In order Comlex nanoarticle (c) Fluid molecule Identical comlex nanoarticles (c) where 2R bk f / d, d is the article diameter, R b is the interfacial resistance, and A and m are emirical arameters. Since nanoarticles in base fluids can easily exerience Brownian force, it is lausible that the observed thermal conductivity of nanofluids is from the combined static (thermal roerties and interfacial layer) and dynamic (e.g., Brownian motion) mechanisms of disersed nanoarticles. r F B d s h r r c Fig.. Concet of BM-based mechanism of nanoarticles in base fluids.

3 , July 6-8, 20, London, U.K. to determine the Brownian motion of nanoarticles in susensions containing a secific volumetric loading of nanoarticles, it is imortant to emloy the Brownian motion term which is also a function of the article volume fraction than the conventional kinetic theory-based formulation for Brownian motion as used by various researchers [7], [2]. Thus, a modified Brownian motion term (U MBM ) which was reviously deduced by incororating nanoarticle volume fraction has the form [] U MBM 2 KBT(.5 ) (4) m where m is the mass of the article, T is the fluid temerature, and K B is the Boltzmann s constant. It can be seen from (4) that the higher the article volume fraction, the smaller the diffusion coefficient and thus the weaker the Brownian motion. For uniform comlex nanoarticles (c) in a base fluid as shown in Fig., the net axial heat flux due to the movement of nanoarticles (U MBM ) resulting from the Brownian force (F B ) imacting on them can be written as [], [4] qdy nmcccdsumbm T (5) 2 c where n is the number density, m c, c and d s are the mass, secific heat, and average searation distance of comlex nanoarticles, resectively. Using nmc c in (5), the dynamic art of thermal conductivity of susensions due to Brownian motion of nanoarticles (k dy ) can be exressed as kdy cccdsumbm (6) 2 Alying U MBM for comlex nanoarticles from (4) into (6) and making use of c, the following final form of k dy TiO 2 (5 nm)/diw Al 2 O (80nm)/DIW Fig. 2. Dynamic thermal conductivity of nanofluids with nanoarticle volume fraction. Brownian motion-contributed thermal conductivity is obtained as KBT(.5 ) kdy cccds[ 2 2 where ( c c r / / s 0.89 c c 0.89 (7) d r r [5] and density ) and secific heat ( c c ) of comlex nanoarticles can be obtained from the formulations given elsewhere []. Fig. 2 deicts the effect of nanoarticle volume fraction on the dynamic thermal conductivity comonent given by (7) for two deionized water (DIW)-based nanofluids. As exected, Fig. 2 shows that the smaller the article the larger the dynamic (Brownian motion-based) thermal conductivity contribution (k dy ). Mainly for smaller size and low volume fraction of nanoarticles, the dynamic contribution of thermal conductivity is significant. The reason is that the smaller the article size the greater the movement of articles in the fluid. It is also noted that this k dy increases nonlinearly with decreasing article volume fraction from due to large interarticle searation distance (d s ) at such small article volume fraction. For 0002., the interarticle searation distance (d s ) is too large to cause any interaction through Brownian force of articles. Thus, the dynamic contribution of the thermal conductivity (k dy ) is not alicable for The effective thermal conductivity of nanofluids is considered to be from both the static and dynamic mechanisms. As mentioned before, the effects of these mechanisms are treated to be additive and the final model for the effective thermal conductivity of nanofluids ( ) by combining the static art given by () and dynamic art given by (7) has the form k TABLE I CONTRIBUTIONS OF STATIC AND DYNAMIC MECHANISMS TO THE EFFECTIVE THERMAL CONDUCTIVITY OF TIO 2 /DIW-NANOFLUIDS PREDICTED BY (8) Nanoarticle vol.% (W/m-K) k st (%) k dy (%) ( k kf )[2 ] ( k 2 kf ) [ ( ) ] k ( k 2 kf ) ( k kf ) [ ] KBT(.5 ) cccds[ 2 2c r eff nf f (8) The imortant features of this model are summarized as follows: i. The model takes into account the effect of nanolayer together with the static and dynamic mechanisms of nanoarticles in the base fluid. The article size effect is also included in the model. ii. The second term on the right hand side of (8) is the dynamic contribution of thermal conductivity (k dy ), which

4 , July 6-8, 20, London, U.K. takes into account the effect of article Brownian motion as well as temerature. This term (k dy ) is only alicable for article loading of In this art (k dy ), the conventional kinetic theory-based Brownian motion term has been renovated using effective diffusion coefficient concet to incororate the effect of volume fraction on Brownian motion of nanoarticles. iii. The model can also redict the temeraturedeendent thermal conductivity of nanofluids. iv. If there is no interfacial layer, the static art of the model reduces to the [5]. C. Determination of contribution to Table demonstrates the contributions of static and dynamic mechanisms to the effective thermal conductivity of TiO 2 (5 nm)/diw-based nanofluids redicted by the resent model i.e., (8). As can be seen from Table, while the major contributions arise from static mechanisms such as volume fraction, article size and interfacial nanolayer, Brownian motion-based dynamic mechanism can also lay significant role in enhancing the thermal conductivity of nanofluids articularly at low volume fraction of nanoarticles. This dynamic mechanisms can also be significant for smaller-sized nanoarticles, tyically < 20nm (Fig. 2). III. RESULTS AND DISCUSSION The resent model is validated by comaring its results with exerimental data. The thermal conductivity of nanofluids was obtained from using transient hot-wire technique which has been described elsewhere [6]. The redictions by the resent model are also comared with the results from i.e., () as well as results obtained from a reresentative recent model develoed for nanofluids i.e., (2). Figs. to 5 demonstrate that the resent model shows fairly good agreement with the exerimental results and gives far better redictions comared to Maxwell s model as well as model develoed by Prasher et al. [7]. As shown in Fig., the redictions by the resent model for TiO 2 (5 nm)/diw-based nanofluids are in good agreement with the exerimental results which are comletely under-redicted / k f Present model [ =.] = 40000, m = 2.5 and R b = K m 2 /W] Exeriments [Al 2 O (80 nm)/diw] Fig. 4. Comarison of resent model s redictions with exerimental results and results from other models for Al 2 O /DIW-based nanofluids. by Maxwell s [5] and Prasher et al. s [7] models. The resent model shows better agreement with the exerimental results for Al 2 O /water-based nanofluids as shown in Fig. 4. It can also be seen that the thermal conductivity of this nanofluids redicted by Prasher et al. s [7] model are much lower than the exerimental data. Fig. 5 demonstrates that for CuO/water-based nanofluids, the resent model fits very well with the results obtained from Eastman et al. [7] and gives better redictions comared to other models. Like other tyes of nanofluids, Prasher et al. s model [7] also under-redicts the thermal conductivity of this nanofluid. Although their model considers the effect of article size and microconvection, it could not redict the anomalously high thermal conductivity of nanofluids. This may be because their model includes effect of interfacial resistance and does not consider the effect of interfacial layer, article movement and surface chemistry, which lay significant roles in susensions containing nanoarticles. It is also noted that suggested values of three unknown arameters i.e. R b, A c, and m were needed in the calculation of thermal conductivity when using Prasher et al. s [7] model. Currently, these emirical arameters cannot be obtained by exerimental or theoretical mean..4. = 40000, m = 2.6 and R b = K m 2 /W] Present model [ = 2] Exeriments [TiO 2 (5 nm)/diw] = 40000, m = 2.5 and R b = K m 2 /W] CuO (6 nm)/diw [7] Present model [ = 2.5] /k f.2 / k f Fig.. Comarison of resent model s redictions with exerimental results and redictions of other models for TiO 2 /DIW-based nanofluids Fig. 5. Comarison of resent model s redictions with exerimental data [7] and other models redictions for CuO/DIW-based nanofluids.

5 , July 6-8, 20, London, U.K. IV. CONCLUDING REMARKS An imroved model for the rediction of the observed thermal conductivity of nanofluids is reorted in this aer. The model is develoed by incororating both static and dynamic mechanisms that are behind the anomalous thermal conductivity of nanofluids. In addition to the effect of Brownian motion of nanoarticles, the model also takes into account other imortant factors such as article size, fluid temerature, and interfacial nanolayer. Comared to existing classical and other recently develoed models, resent model shows much better agreement with exerimental results. This is attributed to the incororation of those key static and dynamic mechanisms in the resent model. In the dynamic art, the conventional kinetic theorybased Brownian motion term has been renovated using effective diffusion coefficient concet in order to formulate the Brownian motion of a susension containing a certain volumetric loading of nanoarticles. It is found that the major contribution to the enhanced thermal conductivity of nanofluids arise from the static mechanisms. However, the Brownian motion-based dynamic mechanism is also significant for nanofluids with smaller-size and low concentration of nanoarticles. It can be inferred that the thermal conductivity of nanofluids is due to both the static and dynamic mechanisms. [5] S. Chandrasekhar, Stochastic roblems in hysics and astronomy, Rev. Mod. Phys., vol.5,.-89, 94. [6] S. M. S. Murshed, K. C. Leong and C. Yang, Enhanced thermal conductivity of TiO 2 -water based nanofluids, Int. J. Therm. Sci., vol. 44,. 67-7, [7] J. A. Eastman, S. U. S. Choi, S. Li and L. J. Thomson, Enhanced thermal conductivity through the develoment of nanofluids, In Proc. Sym. Nanohase Nanocomosite Materials II, Boston, USA, 997, -. REFERENCES [] S. U. S. Choi, Enhancing thermal conductivity of fluids with nanoarticles, ASME FED, vol. 2, , 995. [2] S. M. S. Murshed, K. C. Leong and C. Yang, Thermohysical and electrokinetic roerties of nanofluids- A critical review, Al. Therm. Eng., vol. 28, , [] S. M. S. Murshed, K. C. Leong and C. Yang, Heat transfer of nanoarticle susensions (nanofluids), in Nanoarticles: New Research, S. L. Lombardi, Ed., New York: Nova Science Publishers Inc, 2008, [4] S. M. S. Murshed, C. A. Nieto de Castro, M. J. V. Lourenço, M. L. M. Loes and F. J. V. Santos, A review of boiling and convective heat transfer with nanofluids, Renew Sust. Energ. Rev., vol. 5, , 20. [5] J. C. Maxwell, A Treatise on Electricity and Magnetism. Oxford: Clarendon Press, 89. [6] R. L. Hamilton and O. K., Crosser, Thermal conductivity of heterogeneous two comonent systems, I&EC Fundamentals, vol.,. 87-9, 962. [7] R. Prasher, P. Bhattacharya and P. E. Phelan, Thermal conductivity of nanoscale colloidal solutions (nanofluids), Phys. Rev. Lett., vol. 94, , [8] K. C. Leong, C. Yang and S. M. S. Murshed, A model for the thermal conductivity of nanofluids the effect of interfacial layer, J. Nanoarticle Res., vol. 8, , [9] T. Hashimoto, M. Fujimura and H. Kawai, Domain-boundary structure of styrene-isorene block coolymer films cast from solutions. 5. Molecular-weight deendence of sherical microdomains, Macromolecules, vol., , 980. [0] C.-J. Yu, A. G. Richter, A. Datta, M. K. Durbin and P. Dutta, Molecular layering in a liquid on a solid substrate: an X-ray reflectivity study, Physica B, vol. 28,. 27-, [] L. Xue, P. Keblinski, S. R. Phillot, S. U. S. Choi and J. A. Eastman, Effect of liquid layering at the liquid-solid interface on thermal transort, Int. J. Heat Mass Transfer, vol. 47, , [2] D. H. Kumar, H. E. Patel, V. R. R Kumar, T. Sundararajan, T. Pradee and S. K. Das, Model for heat conduction in nanofluids, Phys. Rev. Lett., vol. 9, , [] S. M. S. Murshed, K. C. Leong and C. Yang, A combined model for the effective thermal conductivity of nanofluids, Al. Therm. Eng., vol. 29, , [4] R. B. Bird, W. E. Stewart and E. N. Lightfoot, Transort Phenomena. New York: John Wiley & Sons, Inc., 2002.