A new heat propagation velocity prevails over Brownian particle velocities in determining the thermal conductivities of nanofluids
© Kihm et al; licensee Springer. 2011
Received: 4 December 2010
Accepted: 27 April 2011
Published: 27 April 2011
An alternative insight is presented concerning heat propagation velocity scales in predicting the effective thermal conductivities of nanofluids. The widely applied Brownian particle velocities in published literature are often found too slow to describe the relatively higher nanofluid conductivities. In contrast, the present model proposes a faster heat transfer velocity at the same order as the speed of sound, rooted in a modified kinetic principle. In addition, this model accounts for both nanoparticle heat dissipation as well as coagulation effects. This novel model of effective thermal conductivities of nanofluids agrees well with an extended range of experimental data.
Findings on nanofluid thermal conductivity
A nanofluid  is defined as a mixture of nanosized particles suspended in liquid as the base fluid. The nanofluid is perceived as an extended scope of earlier efforts to study the effective thermal conductivity of multiphase systems containing microscale particle-embedded solid materials [2–4] and a solid dispersion in liquid .
Since the first article on measurements of the enhanced thermal conductivity of nanofluids (suspension of Al2O3 and CuO nanoparticles in either water or ethylene glycol) using the transient hot-wire technique was published in 1999 , a number of successive measurement studies have supplemented the original findings and extended the parametric variations affecting the level of conductivity enhancement [7–23]. These experimental examinations have revealed the parametric importance of thermal conductivity enhancement, including the volume concentration of nanoparticles and their sizes, clustering or aggregation effect, pH effect, surfactant effect, and the base fluid temperature. As a systematic approach, Chon et al.  have constructed an experimentally extrapolated equation that predicts the nanofluid conductivity in terms of the related parameters.
A number of alternative models have been proposed with the use of the Brownian motion-induced micro-convection in a nanofluid. By adding the second term to the Maxwell model, Xuan et al.  proposed a model incorporating the Brownian motion of nanoparticles in 2003. A year later, Jang and Choi  introduced the Brownian-motion-driven convection model and attempted to describe the temperature-dependency of nanofluid thermal conductivity. They assumed the Nusselt number (Nu) to be the product of Reynolds number (Re) and Prandtl number (Pr), i.e., Nu = Re 2 Pr 2, based on the postulation of Reynolds number of the order of unity. However, this assumption is invalid because it is incorrect to neglect the first two terms, i.e., lower degree terms of Re·Pr, in the expression for the Nusselt number that Acrivos and Taylor  have derived for heat transfer from a spherical particle at low values of the Reynolds number.
Kumar et al.  also attempted to incorporate the nanoparticle thermal conductivity based on the Brownian velocity. However, their model failed as Keblinski et al.  asserted that "the Brownian motion mean free path of a nanoparticle in fluid (by Kumar et al.) is on the order of 1 cm, which is unphysical."
In 2005, Prasher et al.  developed a model combining the Maxwell-Garnett model  incorporating both the Kapitza resistance effect of particles with the surrounding medium and the effect of the Brownian motion-induced convection. Later, they expanded their theoretical prediction for nanofluid thermal conductivity by adding aggregation conductivity contributions for the convection enhancements . However, they assumed a less justifiable Brownian velocity of nanoparticles as based on the kinetic theory of gas, which is valid just for fine particles suspended in a dilute gas (Boltzmann constant k b = 1.3807 × 10-23J/K, the base fluid temperature T, the nanoparticle densify ρ p, and its diameter d p )--but not quite valid for nanoparticles suspended in liquid. Quite possibly because of this conflict, their model fits only to a subset of experimental data, e.g., agrees fairly well with Al2O3 nanofluid data, but fails to fit to CuO nanofluid data.
The effect of the Brownian motion-induced microconvection remains controversial among different research groups. Eapen et al.  strongly argued that microconvection around randomly moving nanoparticles does not influence the thermal conductivity of the nanofluid. In 2007, Das group proposed a nanofluid thermal conductivity model based on a cell model . Their cell model tried to explain the nonlinear dependence of thermal conductivity of nanofluids on particle volume fraction. However, their empirical constants were defined only to fit to their experimental data. In fact, their model constants did not show consistency for an identical Al2O3 nanofluid.
The kinetic principle well describes the thermal conductivity of gas, as the gas molecules are assumed to be freely moving due to their relatively lean distributions . For liquids, however, their stronger intermolecular forces, primarily because of the higher packing density, make it necessary to modify the kinetic theory. In addition, the molecular collision velocities of gases are too low to explain liquid thermal conductivities that are at least one order of magnitude higher than the gas conductivities. Hence, the thermal conductivities of denser liquids are conjectured to be more properly expressed by the faster sound propagation in the case of liquids, and by the phonon velocity in the case of solids.
In this article, a novel theoretical model describing the nanofluid thermal conductivities, considering all major effective parameters including the size, density and volume concentration of nanoparticles, the fluid temperature and viscosity, and relevant thermal parameters such as thermal conductivity of base fluid and heat capacity of nanoparticles, is proposed and examined for its validity against available experimental data.
Introduction of heat propagation velocity
Note that the combined term V ht·l ht relates to the increase of thermal diffusivity of nanofluid as .
where μ is the dynamic viscosity of the base fluid and c 1 is a dimensionless proportional constant. In the case of nanofluid, if l ht is assumed to have the same order of magnitude as the mean free path of water molecules, one can estimate l ht ~ 0.170 nm.
The heat propagation velocity can be estimated by examining the order-of-magnitudes of the involved parameters in Equation 2. For example, for 47-nm Al2O3 at 1 vol.% concentration (f·ρ p·c p ~ 3.2 × 104), the thermal conductivity enhancement Δk enh is found to range from 0.025 to 0.100 W/K m . Thus, the heat propagation velocity V ht is estimated to be on the order of 103 m/s. While a more rigorous analysis to determine the heat transfer velocity is yet to be discussed, this estimation is consistent with the conjectures of the characterisitc heat propagation velocity being on the scale of the sound propagation velocity of an order 103 m/s of both in a liquid medium  and in a colloidal medium [37, 38].
where k b denotes the Boltzmann constant, h and R are the Planck constant and the specific gas constant, respectively, and T is the fluid temperature. The free energy of activation, , is assumed to be constant for a specified fluid and also assumed to be directly related to the internal energy of vaporization at the normal boiling point .
The propagation length scale λ ht , is calculated based on the assumption that the base fluid moledules and nanoparticles are arranged in a cubic lattice, with a center-to-center spacing given by , where is molar mass of the base fluid.
New model for nanofluid thermal conductivity
Two additional modifications of Equation 6 are implemented. First, the volume fraction f is modified to a reduced volume fraction f a (a < 1) to account for the coagulation of nanoparticles that effectively reduce the original volume fraction . The coagulation becomes more severe to require a smaller exponent a with increasing particle concentration because of the decreased inter-particle distance. For example, The surface-to-surface distance of nanoparticles is twice the particle size at 1 vol.%; however, it can decrease to half the particle size at 5 vol.%. Secondly, the effective thermal conductivity of Equation 6 is modified by multiplying the heat capacity ratio of the base fluid to nanoparticles, . It is known that shorter heat dissipation time from nanoparticles into the base fluid enhances the effective thermal conductivity of nanofluid [41, 42]. The heat dissipation time decreases with increasing heat capacity of the base fluid and decreasing heat capacity of the nanoparticles. In other words, nanoparticles with a smaller heat capacity require shorter heat dissipation time to the base fluid, and this results in greater thermal diffusion and higher effective thermal conductivity. The effective conductivity increases in consistency with the heat capacity ratio .
where C is a modified constant and c BF is the base fluid specific heat. The heat transfer length scale λ ht is difficult to be calculated directly, but may be determined by order analysis and merged into the constant C. The exponents a and b are empirical constants that represent the effect of nanoparticle coagulation and of nanoparticle heat dissipation, respectively. A regression analysis of published experimental data by the authors  provides a = 0.70, b = 1.5, and C = 3.58 × 10-14 m for the case of Al2O3 nanoparticles of three different sizes (11 nm, 47 nm, 150 nm diameters) suspended in water under various experimental conditions of a volume concentration range of 1 to 4 vol.% and a tested temperature range of 21 to 71°C.
Differently defined Brownian velocities and heat propagation velocities, and their magnitudes calculated for the range from 20 to 71°C
Calculated velocity (m/s)
Brownian velocity of nanoparticles 
Brownian velocity of nanoparticles 
Brownian velocity of nanoparticles 
Brownian velocity of water molecules 
Sound propagation velocity in water 
Heat propagation velocity [Present model, Equation 5]
Nevertheless, we do not mean that the Brownian motion is not related to the thermal conductivity enhancement. Nor do we mean that Brownian convection is not significant. What we imply is that the assumption in , i.e., the Nusselt number can be expressed as Nu = Re 2 Pr 2, is invalid because it is incorrect to neglect the first two terms, i.e., lower degree terms of Re·Pr, in the expression for the Nusselt number that Acrivos and Taylor  have derived.
In addition, in order to have significant convection effect by wavelength mode of long molecular motion, the bulk fluid needs externally imposed gradients such as pressure, gravity or temperature. However, a nanofluid has quiescent condition, which cannot support any convection [34, 46]. The Brownian velocity, as shown in Figure 1, is several orders of magnitude lower than the required velocity scale of 103 in modeling nanofluid conductivity enhancement.
Prasher et al. [28, 29] show fairly good agreement with the experiments for the Al2O3 nanofluid, as shown in Figure 2a, c. However, for the CuO nanofluid (Figure 2b), their model underestimates the corresponding experimental data . When completely different model parameters were imposed for CuO from that of Al2O3, the model agrees well with the data; however, the model then lacks comprehensiveness because different model parameters need to be determined for different types of nanofluids. Finally, Patel et al.  agrees fairly well with the experimental data at higher concentrations (Figure 2c) but overestimate the thermal conductivities for low volume concentrations (Figure 2a, b).
can more accurately and comprehensively describe the effective thermal conductivities of nanofluids with different types (Al2O3 and CuO nanofluids) and sizes of nanoparticles (ranging from 10 to 150 nm), for a relatively wider range of temperatures in comparison with the most popular range of up to 50°C of published studies.
As similar conceptual studies, the recent thermal-wave  and the dual-phase lagging heat conductions  are attracted by researchers because both models can explain the high-rate heat flux in microscale and also can be applied to the thermal conductivity of nanofluid. Thermal-wave and dual-phase lagging heat conduction are developed analytically, however the new model is approached by physical manner and it considers more practical factors such as particle coagulation effect and heat dissipation effect. Therefore our new model will be bridging the practical thermal conductivity enhancement of nanofluid and theoretical concept of the high-rate heat flux of nanofluid such as thermal-wave dual-phase lagging heat conduction of nanofluid.
This work was supported by the U.S. Department of Energy, Office of Basic Energy Science under Contract No. DE-FG02-05ER46182 to the University of Tennessee (KK, CC, SC) as well as by the WCU (World Class University) Program at Seoul National University through the Korea Research Foundation funded by the Ministry of Education, Science and Technology under Contract No. R31-2008-000-10083-0 (KK, JL).
- Choi US: Enhancing thermal conductivity of fluids with nanoparticles. In Developments and Applications of Non-Newtonian Flows. Volume 231. New York: ASME; 1995:99–105.
- Maxwell JC: A Treatise on Electricity and Magnetism. Volume 1. 3rd edition. New York: Dover; 1954.
- Hamilton RL, Crosser OK: Thermal conductivity of heterogeneous two-components systems. Ind Eng Chem Fundam 1962, 1: 187–191. 10.1021/i160003a005View Article
- Bionnecaze RR, Brady JF: The effective conductivity of random suspensions of spherical particles. Proc R Soc Lond A 1991, 432: 445–465. 10.1098/rspa.1991.0025View Article
- Adler PM: Porous Media: Geometry and Transports. Boston: Butterworth-Heinemann; 1992:434.
- Lee S, Choi SUS, Li S, Eastman JA: Measuring thermal conductivity of fluids containing oxide nanoparticles. J Heat Transfer 1999, 121: 280–289. 10.1115/1.2825978View Article
- Eastman JA, Choi SUS, Li S, Yu W, Thompson LJ: Anomalously increased effective thermal conductivities of ethylene glycol-based nanofluids containing copper nanoparticles. Appl Phys Lett 2001, 78: 718–720. 10.1063/1.1341218View Article
- Choi SUS, Zhang ZG, Yu W, Lockwood FE, Grulke EA: Anomalous thermal conducitivity enhancement in nanotube suspensions. Appl Phys Lett 2001, 79: 2252–2254. 10.1063/1.1408272View Article
- Xie H, Lee H, Youn W, Choi M: Thermal conductivity enhancement of suspensions containing nanosized alumina particles. J Appl Phys 2002, 91: 4568–4572. 10.1063/1.1454184View Article
- Das SK, Putra N, Thiesen P, Roetzel W: Temperature dependence of thermal conductivity enhancement for nanofluids. J Heat Transfer 2003, 125: 567–574. 10.1115/1.1571080View Article
- Patel HE, Das SK, Sundararajan T, Nair AS, George B, Pradeep T: Thermal conductivities of naked and monolayer protected metal nanoparticle based nanofluids: Manifestation of anomalous enhancement and chemical effects. Appl Phys Lett 2003, 83: 2931–2933. 10.1063/1.1602578View Article
- Wang X, Xu X, Choi SUS: Thermal conductivity of nanoparticle-fluid mixture. J Thermophys Heat Transfer 1999, 13: 474–480. 10.2514/2.6486View Article
- Kim SH, Choi SR, Kim D: Thermal conductivity of metal-oxide nanofluids: particle size dependence and effect of laser irradiation. J Heat Transfer 2007, 129: 298–307. 10.1115/1.2427071View Article
- Jang SP, Choi SUS: Effects of various parameters on nanofluid thermal conductivity. J Heat Transfer 2007, 129: 617–623. 10.1115/1.2712475View Article
- Mintsa HA, Roy G, Nguyen CT, Doucet D: New temperature dependent thermal conductivity data for water-based nanofluids. Int J Therm Sci 2009, 48: 363–371. 10.1016/j.ijthermalsci.2008.03.009View Article
- Hwang Y, Lee JK, Lee CH, Jung YM, Cheong SI, Lee CG, Ku BC, Jang SP: Stability and thermal conductivity characteristics of nanofluids. Thermochim Acta 2007, 455: 70–74. 10.1016/j.tca.2006.11.036View Article
- Hong KS, Hong T-K, Yang H-S: Thermal conductivity of Fe nanofluids depending on the cluster size of nanoparticles. Appl Phys Lett 2006, 88: 031901. 10.1063/1.2166199View Article
- Prasher R, Evans W, Meakin P, Fish J, Phelan P, Keblinski P: Effect of aggregation on thermal conduction in colloidal nanofluids. Appl Phys Lett 2006, 89: 143119. 10.1063/1.2360229View Article
- Karthikeyan NR, Philip J, Ray B: Effect of clustering on the thermal conductivity of nanofluids. Mater Chem Phys 2008, 109: 50–55. 10.1016/j.matchemphys.2007.10.029View Article
- Li XF, Zhu DS, Wang XJ, Wang N, Gao JW, Li H: Thermal conductivity enhancement dependent pH and chemical surfactant for Cu-H 2 O nanofluids. Thermochim Acta 2008, 469: 98–103. 10.1016/j.tca.2008.01.008View Article
- Evans W, Prasher R, Fish J, Meakin P, Phelan P, Keblinski P: Effect of aggregation and interfacial thermal resistance on thermal conductivity of nanocomposites and colloidal nanofluids. Int J Heat Mass Transfer 2008, 51: 1431–1438. 10.1016/j.ijheatmasstransfer.2007.10.017View Article
- Li CH, Peterson GP: The effect of particle size on the effective thermal conductivity of Al 2 O 3 -water nanofluids. J Appl Phys 2007, 101: 044312. 10.1063/1.2436472View Article
- Timofeeva EV, Gavrilov AN, McCloseky JM, Tolmachev YV: Thermal conductivity and particle agglomeration in alumina nanofluids: experiment and theory. Phys Rev E 2007, 76: 061203.View Article
- Chon CH, Kihm KD, Lee SP, Choi SUS: Empirical correlation finding the role of temperature and particle size for nanofluid (Al 2 O 3 ) thermal conductivity enhancement. Appl Phys Lett 2005, 87: 153107. 10.1063/1.2093936View Article
- Xuan Y, Li Q, Hu W: Aggregation struture and thermal conductivity of nanofluids. AIChE J 2003, 49: 1038–1043. 10.1002/aic.690490420View Article
- Jang SP, Choi SUS: Role of Brownian motion in enhanced thermal conductivity of nanofluids. Appl Phys Lett 2004, 84: 4316–4318. 10.1063/1.1756684View Article
- Kumar DH, Patel HE, Kumar VRR, Sundararajan T, Das SK: Model for heat conduction in nanofluids. Phys Rev Lett 2004, 93: 144301.View Article
- Prasher R, Bhattacharya P, Phelan PE: Thermal conductivity of nanoscale colloidal solutions (nanofluids). Phys Rev Lett 2005, 94: 025901.View Article
- Prasher R, Phelan PE, Bhattacharya P: Effect of aggregation kinetics on the thermal conductivity of nanoscale colloidal solutions (nanofluids). Nano Lett 2006, 6: 1529–1534. 10.1021/nl060992sView Article
- Patel HE, Sundararajan T, Das SK: A cell model approach for thermal conductivity of nanofluids. J Nanopart Res 2008, 10: 87–97. 10.1007/s11051-007-9236-4View Article
- Acrivos A, Taylor TD: Heat and mass transfer from single spheres in Stokes flows. Phys Fluids 1962, 5: 387–394. 10.1063/1.1706630View Article
- Keblinski P, Eastman JA, Cahill DG: Nanofluids for thermal transport. Mater Today 2005, 8: 36–44.View Article
- Nan CW, Birringer R, Clarke DR, Gleiter H: Effective thermal conductivity of particulate composites with interfacial thermal resistance. J Appl Phys 1997, 81: 6692–6699. 10.1063/1.365209View Article
- Eapen J, Williams WC, Buongiorno J, Hu L, Yip S: Mean-field versus microconvection effects in nanofluid thermal conduction. Phys Rev Lett 2007, 99: 095901.View Article
- Carey VP: Statistical Thermodynamics and Microscale Thermophysics. New York: Cambridge; 1999:145.
- Glasstone S, Laidler KJ, Eyrinig H: Theory of Rate Processes. Volume Chapter 9. New York: McGraw-Hill; 1941.
- Bird RB, Stewart WE, Lightfoot EN: Transport Phenomena. 2nd edition. New York: Wiley & Sons; 2002:29–31.
- Keblinski P, Phillpot SR, Choi SUS, Eastman JA: Mechanisms of heat flow in suspensions of nano-sized particles (nanofluids). Int J Heat Mass Transfer 2002, 45: 855–863. 10.1016/S0017-9310(01)00175-2View Article
- Kincaid JF, Eyring H, Stearn AE: The theory of absolute reaction rates and its application to viscosity and diffusion in the liquid state. Chem Rev 1941, 28:;301–365. 10.1021/cr60090a005View Article
- Digilov RM, Reiner M: Trouton's rule for the law of corresponding states. Eur J Phys 2004, 25: 15–22. 10.1088/0143-0807/25/1/003View Article
- Wilson OM, Hu X, Cahill DG, Braun PV: Colloidal metal particles as probes of nanoscale thermal transport in fluids. Phys Rev B 2002, 66: 224301.View Article
- Ge Z, Cahill DG, Braun PV: AuPd metal nanoparticles as probes of nanoscale thermal transport in aqueous solution. J Phys Chem B 2004, 108: 18870–18875. 10.1021/jp048375kView Article
- Young HD, Freedman RA: University Physics. Volume Chapter 19. 9th edition. Reading, MA: Addison-Wesley; 1996.
- Balandin AA: Nanoscale thermal management. IEEE Potentials 2002, Feb/Mar: 11–15.View Article
- Pasquini L, Barla A, Chumakov AI, Leupold O, Rüffer R, Deriu A, Bonetti E: Size and oxidation effects on the vibrational properties of nanoscrystalline α-Fe. Phys Rev B 2002, 66: 073410.View Article
- Boon JP, Yip S: Molecular Hydrodynamics. New York: Dover; 1991.
- Vadasz JJ, Govender S: Thermal wave effects on heat transfer enhancement in nanofluids suspensions. Int J Therm Sci 2010, 49: 235–242. 10.1016/j.ijthermalsci.2009.06.002View Article
- Wang LQ, Wei X: Nanofluids: synthesis, heat conduction, and extension. ASME J Heat Trans 2009, 131: 033102. 10.1115/1.3056597View Article
This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.