Matter of Cooling

What really matters in cooling applications


Beyond the Maxwell-Garnett limit for the spherical nanoparticles



MAXWELL-GARNETT MODEL OVERTAKEN

What really matter in cooling applications using nanofluid is not

the thermal conductivity per se,

but rather their heat transfer properties in both laminar and turbulent flow regimes





After over 100 years theMaxwell-Garnet Model has been overtaken


Lord Maxwell [Maxwell, 1878] provided a simple explanation of the enhancement in thermal conductivity of fluids-solid particle dispersions, assuming that the dispersion could indeed be

treated as a continuum

In channels, such as those to cool semiconductor devices, the effects are complicated by issues such as flow regime, boundary layer effects (gradients, transport anisotropy) and possibly micro-bubble formation due to locally high heat fluxes (but also to other unexplained effects) at the interface between nanofluid and channel wall. Experiments show that the relationship between conductivity and percentage of nanoparticle is far from linear, e.g., see  [Murshed et al, 2005; Chopkar et al, 2006]. Similarly nonlinear is the dependence on temperature [Das et al, 2003]. The HT capabilities of nanofluids tend to increase with the Re of  laminar flows, with a factor of order several units [Faulkner et al, 2004]. However, using disc-shaped nanoparticles, other investigators [Yang et al, 2005] found this factor to be much less, pointing to the strong effect of particle shape. Lower enhancement of HT was observed in the turbulent regime [Xuan and Li, 2003]. When tested in natural convection flows, nanofluids showed lesser HT than pure fluids, a puzzling fact to date. Equally interesting is the increase (by a factor of order 3 to 4) in critical heat flux in boiling fluids when using certain nanofluids [You et al, 2003; Vassallo et al, 2004].


This synthetic list of recent results indicates that the simple explanation by Maxwell is definitely incomplete, even with the modifications and updating by later researchers, e.g., see [Hashin and Shtrikman, 1962; Jackson, 1975; Davis, 1986; Lu and Lin, 1996]. As in the case of Maxwell,  theories proposed by these researchers assume the nanoparticles + fluid combination to behave as a single continuum, that accordingly can modelled by a simple convection-diffusion transport equation. These models predict reasonably well conductivity in fluids with large particles (that is, > 1 micrometer), but not that when nanosize copper or nanotube particles are used: in this case the thermal conductivity predicted is in fact about an order of magnitude less than observed experimentally.


Thus particle scale is critical, suggesting mechanisms other than those devised by Maxwell are at work.

In essence, quoting S.U.S. Choi, the investigator who coined the term “nanofluid”[Das et al, 2008, p. 20], “…nanoscale particle motion is more complex than that of conventional solid-liquid suspensions, and cannot be explained by the diffusive heat transport mechanism alone.”

Bewildering array of mechanisms investigated, all hard to separate from others and even harder to measure, may be divided in two classes. Some of the proposed mechanisms assume that particles and liquid form structures,  others that it is the motion of the particles that shapes the conductivity behaviour. Examples of the first class of explanations is in [Yu and Choi, 2004; Xue, 2003; Xie et al, 2002; Nan et al, 2003; Ju and Li, 2006; Xue, 2006]. Dynamic models, that assume motion is paramount in explaining the nanofluid effects, are by [Wang et al, 1999; Keblinski et al, 2002; Bhattacharya et al, 2004; Jang and Choi, 2004; Prasher et al, 2005; Koo and Kleinstreuer, 2005]. In these works the role of Brownian motion has been alternatively rejected and revaluated, and no clear general conclusion has yet  been drawn. An extensive list of all the effects that have been invoked so far to explain the property of nanofluids is in Chapter 1 of [Das et al, 2008].


No single effect or approach can satisfactory simulate the thermal conductivity behaviour  of nanofluids as a function of nature and geometry of particles, temperature and concentration. On the other hand, empirical attempts at modifying nanofluids to achieve the desired performance, although sometimes successful, are generally too costly and time consuming, since the matrix of parameters to investigate is impractically large.

Theories proposed by researchers assume that the nanoparticles and carrier fluid combinations behave as a single continuum that can be modeled by a simple convection-diffusion transport equation. These models predict reasonably good thermal conductivity in fluids with large particles (that is, > 1 micrometer), but not when nano-size particles or nanotubes are used: in this case the thermal conductivity predicted is in fact much lower than that observed experimentally. Therefore, particle size is critical, suggesting mechanisms other than those devised by Maxwell are at work.

In essence, to quote S.U.S. Choi [Das et al, 2008, p. 20], “…nanoscale particle motion is more complex than that of conventional solid-liquid suspensions, and cannot be explained by the diffusive heat transport mechanism alone.”


Array of mechanisms may be divided in two classes. 1) Some of the proposed mechanisms assume that particles and liquid form structures, 2) others that it is the motion of the particles that shapes the conductivity behavior. Examples of the first class of explanations have been discussed by Yu and Choi, 2004; Xue, 2003; Xie et al, 2002; Nan et al, 2003; Ju and Li, 2006; Xue, 2006. Dynamic models, that assume motion are paramount in explaining the thermal effects of nanofluid and have been discussed by a number of researchers including Wang et al, 1999; Keblinski et al, 2002; Bhattacharya et al, 2004; Jang and Choi, 2004; Prasher et al, 2005; Koo and Kleinstreuer, 2005. In their work the role of Brownian motion has been rejected and revaluated. Recently, near field radiative heat transfer has been assumed to play an important role when the mean distance between nanoparticles starts to be of the same order as the particle size. This also does not appear to explain the full behavior of nanofluid coolants.


It should be also noted that what really matters in cooling applications using nanofluid is not the thermal conductivity per se, but rather their heat transfer properties in both laminar and turbulent flow regimes


After over 100 years the

Maxwell-Garnet Model has been overtaken


The new proposed model is strongly supported by extensive numerical simulations, demonstrating a potential increase of the heat flux far beyond the  Maxwell-Garnett limit for the spherical nanoparticles.


(Acknowledgement-Financial support from the EU FP7 project “Enhanced nano-fluid heat exchange”, Contract No. 228882).