- About this Journal ·
- Abstracting and Indexing ·
- Aims and Scope ·
- Annual Issues ·
- Article Processing Charges ·
- Articles in Press ·
- Author Guidelines ·
- Bibliographic Information ·
- Citations to this Journal ·
- Contact Information ·
- Editorial Board ·
- Editorial Workflow ·
- Free eTOC Alerts ·
- Publication Ethics ·
- Reviewers Acknowledgment ·
- Submit a Manuscript ·
- Subscription Information ·
- Table of Contents

Advances in Mechanical Engineering

Volume 2010 (2010), Article ID 719406, 5 pages

http://dx.doi.org/10.1155/2010/719406

## Is Classical Energy Equation Adequate for Convective Heat Transfer in Nanofluids?

Department of Mechanical Engineering, The University of Hong Kong, Pokfulam Road, Hong Kong

Received 20 August 2009; Accepted 17 September 2009

Academic Editor: Oronzio Manca

Copyright © 2010 Jing Fan and Liqiu Wang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

To address whether the heat transfer in nanofluids still satisfies the classical energy equation, we theoretically examine the macroscale manifestation of the microscale physics in nanofluids. The microscale interaction between nanoparticles and base fluids manifests itself as thermal waves/resonance at the macroscale. The energy equation that governs the heat transfer in nanofluids is of a dual-phase-lagging type instead of the *postulated* and *commonly-used* classical energy equation. The interplays among diffusion, convection, and thermal waves/resonance enrich the heat transfer in nanofluids considerably.

#### 1. Introduction

Choi coined the term “nanofluids” for the fluids with nanoelements (nanoparticles, nanotubes, or nanofibers) suspended in them [1]. Recent experiments on nanofluids have shown, for example, twofold increases in thermal conductivity [2], strong temperature dependence of thermal conductivity [3], substantial increases in convective heat transfer coefficient [4, 5], and threefold increases in critical heat flux (CHF) in boiling heat transfer [2, 3, 6]. State-of-the-art expositions of major advances on the synthesis, characterization, and application of nanofluids are available, for example, in [2, 3, 6–12]. These characteristics make them very attractive for a large number of industries such as transportation, electronics, defense, space, nuclear systems cooling, and biomedicine.

The study of nanofluids is still in its infancy [2, 3, 6–12]. The precise nature and mechanism of the significant improvement of thermal performance are still not known. There is also a lack of agreement between experimental results and between theoretical models. The fact that the enhancement in thermal properties comes from the presence of nanoparticles has directed research efforts nearly exclusively towards thermal transport at nanoscale. The classical conservation equations including the energy equation have been postulated as the macroscale model of nanofluid convective heat transfer but without adequate justification. Thermal conductivity and convective heat transfer coefficient are a macroscale phenomenological characterization of heat transfer and their measurements are not performed at the nanoscale, but rather at the macroscale. Therefore, interest should focus not only on what happens at the nanoscale but also on how the presence of nanoelements affects the heat transport at macroscale.

We attempt to examine whether the classical energy equation is adequate for describing convective heat transfer in nanofluids at macroscale based on a macroscale heat transfer model in nanofluids, which is rigorously developed by scaling-up the microscale model for the heat transfer in the nanoparticles and in the base fluids. The approach for scaling-up is the volume averaging [13–15] with help of multiscale theorems [15].

#### 2. Macroscale Energy Equation

The microscale model for heat transfer in nanofluids is well known. It consists of the field equation and the constitutive equation. The field equation comes from the conservation laws of mass, momentum, and energy. The commonly-used constitutive equation includes the Newton law of viscosity and the Fourier law of heat conduction [16].

For transport in nanofluids, the macroscale is a phenomenological scale that is much larger than the microscale and much smaller than the system length scale. Interest in the macroscale rather than the microscale comes from the fact that a prediction at the microscale is complicated due to the complex microscale structure of nanofluids, and also because we are usually more interested in large scales of transport for practical applications. Existence of such a macroscale description equivalent to the microscale behavior requires a good separation of length scales and has been well discussed in [15, 17].

To develop a macroscale model of heat transfer in nanofluids, the method of volume averaging starts with a microscale description. Both conservation and constitutive equations are introduced at the microscale. The resulting microscale field equations are then averaged over a representative elementary volume (REV), the smallest differential volume resulting in statistically meaningful local averaging properties, to obtain the macroscale field equations. In the process of averaging, the *multiscale theorems* [15] are used to convert integrals of gradient, divergence, curl, and partial time derivatives of a function into some combination of gradient, divergence, curl, and partial time derivatives of integrals of the function and integrals over the boundary of the REV.

Consider heat transfer in nanofluids with the continuous base fluid and the dispersed nanoparticle denoted by - and -phases, respectively. Note that the dispersed nanoparticles can be liquid droplets for general nanofluids [10, 12]. As a two-component mixture (base fluid nanoparticles), its microscale model can be written as [18]

Here and are the velocity and the temperature, respectively. *ρ*, , , ,, and are the density, specific heat, thermal conductivity, pressure, viscosity, and gravitational acceleration, respectively. Subscripts *c* and *d* refer to the *c*- and *d*-phases, respectively. *A _{cd}* represents the area of the

*c-d*interface contained in the REV, and is the outward-directed surface normal from the -phase toward the -phase, (Figure 1). To be thorough, we must also specify the initial conditions and the boundary conditions at the entrances and exits of the REV; however, we need not do so for our discussion.

Applying the volume averaging and multiscale theorems to scale-up the microscale model yields a macroscale model [12, 19], where the energy equation reads Here is the volume fraction of the -phase (the index can take or ), with and as the volume of the -phase in REV, and the volume of the REV respectively:

The *intrinsic average* is defined by

,,,,,,,, and represent the effects of microscale physics on the macroscale heat transfer. The readers are referred to [12, 19, 20] for their governing equations and numerical computations.

#### 3. Results and Discussion

Rewrite (2) in their operator form:

where , , , and . We then obtain an uncoupled form by evaluating the operator determinant such that

where the index *i *can take* c *or *d*. and are the *c*-phase and *d*-phase effective thermal capacities, respectively. Its explicit form reads, after dividing by ,

When the system is isotropic and the physical properties of the two phases are constant, it reduces to

where

This is a dual-phase-lagging heat-conduction equation with and as the phase lags of the heat flux and the temperature gradient, respectively [15, 21, 22]. Here, is the volumetric heat source. *k*, , , and are the effective thermal conductivity, capacity, diffusivity, and velocity of nanofluids, respectively. Therefore, the presence of nanoparticles shifts the classical energy equation for the heat transfer in the base fluid into the dual-phase-lagging energy equation in nanofluids at the macroscale. This is significant because all results regarding dual-phase-lagging heat transfer can thus be applied to study heat transfer in nanofluids [21, 22]. For the case of heat conduction in nanofluids, the inclusion of the solid-fluid interface heat transfer also leads to dual-phase-lagging heat-conduction equation, which resolves the conflict between experimental data of nanofluid thermal conductivity and classical theories of effective thermal conductivity of suspensions [23].

It is interesting to note that there are nontraditional convective terms and in (2). In (8), however, such terms disappear because of the constraint from the mass conservation. Therefore, the microscale physics does not manifest itself as the macroscale convection. The velocity-like terms appear only in the source term in (8).

The presence of nanoparticles gives rise to variations of thermal capacity, conductivity, and diffusivity, which are given by, in terms of ratios over those of the base fluid,

Therefore, depends *only* on the volume fraction of nanoparticles and the nanoparticle-fluid capacity ratio. However, both and are affected by the geometry, property and dynamic process of nanoparticle-fluid interfaces.

Consider

It can be larger, equal, or smaller than 1 depending on the sign of . Therefore, by the condition for the existence of thermal waves that requires [22, 24], we may have thermal waves in nanofluid heat transfer when

A necessary (but not sufficient) condition for (13) is . Note also that for heat transfer in nanofluids the microscale physics yields a time-dependent source term in the dual-phase-lagging energy equation ((8) and (10)). Therefore, the resonance can also occur. These thermal waves and possibly resonance are believed to be the driving force for the enhancement of heat transfer. When so that is always larger than 1, thermal waves and resonance would not appear. The sum is thus responsible for thermal waves and resonance in nanofluid heat transfer. It is also interesting to note that although each *τ*_{q} and *τ** _{T}* is

*a*-dependent, the ratio

_{v}h*τ*

*/*

_{T}*τ*

_{q }is not. Therefore the evaluation of

*τ*

*/*

_{T}*τ*

_{q}will be much simpler than

*τ*

_{q}or

*τ*

*.*

_{T}Therefore, the molecular physics and the microscale physics (interactions between nanoparticles and base fluids at the microscale in particular) manifest themselves as heat diffusion and thermal waves/resonance at the macroscale, respectively. Their overall macroscopic manifestation shifts the classical energy equation for the heat transfer in the base fluid into the dual-phase-lagging energy equation in nanofluids. When, thermal waves dominate and (8) is of a hyperbolic type [22]. When, however, heat diffusion dominates and (8) is parabolic [22]. Depending on factors like material properties of nanoparticles and base fluids, nanoparticles’ geometrical structure and their distribution in the base fluids, and interfacial properties and dynamic processes on particle-fluid interfaces, the heat diffusion, thermal waves/resonance, and convection may either enhance or counteract each other. Consequently, the heat transfer in nanofluids is endowed with much richer features than that in the base fluid.

#### 4. Concluding Remarks

In an attempt to determine how the presence of nanoparticles affects the heat transfer at the macroscale and isolates the mechanism responsible for the reported variation of thermal properties, a macroscale energy equation is developed and examined analytically for nanofluid heat transfer. The model is obtained by scaling-up the microscale model for the heat transfer in the nanoparticles and in the base fluids. The approach for scaling-up is the volume averaging with help of multiscale theorems. The result shows that the presence of nanoparticles leads to a dual-phase-lagging energy equation in nanofluids at the macroscale. Therefore, the molecular physics and the microscale physics manifest themselves as heat diffusion and thermal waves at the macroscale, respectively. Depending on factors like material properties of nanoparticles and base fluids, nanoparticles’ geometrical structure and their distribution in the base fluids, and interfacial properties and dynamic processes on particle-fluid interfaces, the heat diffusion, convection, and thermal waves may either enhance or counteract each other, which will enrich heat-transfer performance significantly.

#### Acknowledgment

The financial support from the Research Grants Council of Hong Kong (GRF718009 and GRF717508) is gratefully acknowledged.

#### References

- S. U. S. Choi, “Enhancing thermal conductivity of fluids with nanoparticles,” in
*Developments and Applications of Non-Newtonian Flows*, D. A. Singer and H. P. Wang, Eds., vol. 231 of*FED*, pp. 99–105, ASME, New York, NY, USA, 1995. - G. P. Peterson and C. H. Li, “Heat and mass transfer in fluids with nanoparticle suspensions,”
*Advances in Heat Transfer*, vol. 39, pp. 257–376, 2006. - S. K. Das, S. U. S. Choi, W. H. Yu, and T. Pradeep,
*Nanofluids: Science and Technology*, John Wiley & Sons, Hoboken, NJ, USA, 2008. - D. Wen, Y. Ding, and R. Williams, “Nanofluids turn up the heat,”
*The Chemical Engineer*, no. 771, pp. 32–34, 2005. View at Scopus - D. Y. Tzou, “Thermal instability of nanofluids in natural convection,”
*International Journal of Heat and Mass Transfer*, vol. 51, no. 11-12, pp. 2967–2979, 2008. View at Publisher · View at Google Scholar · View at Scopus - S. U. S. Choi, Z. G. Zhang, and P. Keblinski, “Nanofluids,” in
*Encyclopedia of Nanoscience and Nanotechnology*, H. S. Nalwa, Ed., vol. 6, pp. 757–773, American Scientific Publishers, New York, NY, USA, 2004. - J. A. Eastman, S. R. Phillpot, S. U. S. Choi, and P. Keblinski, “Thermal transport in nanofluids,”
*Annual Review of Materials Research*, vol. 34, pp. 219–246, 2004. View at Publisher · View at Google Scholar · View at Scopus - P. E. Phelan, P. Bhattacharya, and R. S. Prasher, “Nanofluids for heat transfer applications,”
*Annual Reviews of Heat Transfer*, vol. 14, pp. 255–275, 2005. - C. B. Sobhan and G. P. Peterson,
*Microscale and Nanoscale Heat Transfer: Fundamentals and Engineering Applications*, CRC Press, Boca Raton, Fla, USA, 2008. - L. Q. Wang and X. H. Wei, “Nanofluids: synthesis, heat conduction, and extension,”
*Journal of Heat Transfer*, vol. 131, no. 3, Article ID 033102, 7 pages, 2009. - S. U. S. Choi, “Nanofluids: from vision to reality through research,”
*Journal of Heat Transfer*, vol. 131, Article ID 033106, 9 pages, 2009. - L. Q. Wang and M. Quintard, “Nanofluids of the future,” in
*Advances in Transport Phenomena*, pp. 179–243, Springer, Heidelberg, Germany, 2009. - S. Whitaker,
*The Method of Volume Averaging*, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1999. - L. Q. Wang, “Flows through porous media: a theoretical development at macroscale,”
*Transport in Porous Media*, vol. 39, no. 1, pp. 1–24, 2000. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - L. Q. Wang, M. T. Xu, and X. H. Wei, “Multiscale theorems,”
*Advances in Chemical Engineering*, vol. 34, pp. 175–468, 2008. View at Publisher · View at Google Scholar · View at Scopus - L. Q. Wang, “A general theory of diffusion,”
*Progress of Theoretical Physics*, vol. 101, no. 3, pp. 541–557, 1999. View at Scopus - J. L. Auriault, “Heterogeneous medium: is an equivalent macroscopic description possible?”
*International Journal of Engineering Science*, vol. 29, no. 7, pp. 785–795, 1991. View at Scopus - R. B. Bird, W. E. Stewart, and E. N. Lightfoot,
*Transport Phenomena*, John Wiley & Sons, New York, NY, USA, 3rd edition, 2007. - M. Quintard and S. Whitaker, “Theoretical analysis of transport in porous media,” in
*Handbook of Heat Transfer in Porous Media*, K. Vafai, Ed., pp. 1–52, Marcel Dekker, New York, NY, USA, 2000. - J. Fan and L. Q. Wang, “Microstructural effects on macroscale thermal properties in nanofluids,”
*NANO*, (in press). - D. Y. Tzou,
*Macro-to Microscale Heat Transfer: The Lagging Behavior*, Taylor & Francis, London, UK, 1997. - L. Q. Wang, X. S. Zhou, and X. H. Wei,
*Heat Conduction: Mathematical Models and Analytical Solutions*, Springer, Berlin, Germany, 2008. - P. Vadasz, “Heat conduction in nanofluid suspensions,”
*Journal of Heat Transfer*, vol. 128, no. 5, pp. 465–477, 2006. View at Publisher · View at Google Scholar · View at Scopus - M. T. Xu and L. Q. Wang, “Thermal oscillation and resonance in dual-phase-lagging heat conduction,”
*International Journal of Heat and Mass Transfer*, vol. 45, no. 5, pp. 1055–1061, 2002. View at Publisher · View at Google Scholar · View at Scopus