When using the classical models, it is implied that the nanoparticles are stationary to the base fluid. In contrast, dynamic models are taking the effect of the nanoparticles' random motion into account, leading to a "micro-mixing" effect [

70]. In general, anomalous thermal conductivity enhancement of nanofluids may be due to:

• Brownian-motion-induced micro-mixing;

• heat-resistance lowering liquid-molecule layering at the particle surface;

• higher heat conduction in metallic nanoparticles;

• preferred conduction pathway as a function of nanoparticle shape, *e.g.*, for carbon nanotubes;

• augmented conduction due to nanoparticle clustering.

Up front, while the impact of micro-scale mixing due to Brownian motion is still being debated, the effects of nanoparticle clustering and preferred conduction pathways also require further studies.

Oezerinc et al. [

71] systematically reviewed existing heat transfer mechanisms which can be categorized into conduction, nano-scale convection and/or near-field radiation [

22], thermal waves propagation [

67,

72], quantum mechanics [

73], and local thermal non-equilibrium [

74].

For a better understanding of the micro-mixing effect due to Brownian motion, the works by Leal [

75] and Gupte [

76] are of interest. Starting with the paper by Koo and Kleinstreuer [

70], several models stressing the Brownian motion effect have been published [

22]. Nevertheless, that effect leading to micro-mixing was dismissed by several authors. For example, Wang [

43] compared Brownian particle diffusion time scale and heat transfer time scale and declared that the effective thermal conductivity enhancement due to Brownian motion (including particle rotation) is unimportant. Keblinski [

77] concluded that the heat transferred by nanoparticle diffusion contributes little to thermal conductivity enhancement. However, Wang [

43] and Keblinski [

77] failed to consider the surrounding fluid motion induced by the Brownian particles.

Incorporating indirectly the Brownian-motion effect, Jang and Choi [

78] proposed four modes of energy transport where random nanoparticle motion produces a convection-like effect at the nano-scale. Their effective thermal conductivity is written as:

where

*C*_{1 }is an empirical constant and

*d*_{bf }is the base fluid molecule diameter. Re

_{dp }is the Reynolds number, defined as:

where

*D *is the nanoparticle diffusion coefficient,

*κ*_{Boltzmann }= 1.3807e-23 J/K is the Boltzmann constant,

is the root mean square velocity of particles and

*λ*_{bf }is the base fluid molecular mean free path. The definition of

(see Eq. 7b) is different from Jang and Choi's 2006 model [

79]. The arbitrary definitions of the coefficient "random motion velocity" brought questions about the model's generality [

78]. Considering the model by Jang and Choi [

78], Kleinstreuer and Li [

80] examined thermal conductivities of nanofluids subject to different definitions of "random motion velocity". The results heavily deviated from benchmark experimental data (see Figure ), because there is no accepted way for calculating the random motion velocity. Clearly, such a rather arbitrary parameter is not physically sound, leading to questions about the model's generality [

80].

Prasher [

81] incorporated semi-empirically the random particle motion effect in a multi-sphere Brownian (MSB) model which reads:

Here, Re is defined by Eq. 7a,

*α *= 2

*R*_{b}*k*_{m}/

*d*_{p }is the nanoparticle Biot number, and

*R*_{b }= 0.77 × 10

^{-8 }Km

^{2}/W for water-based nanofluids which is the so-called thermal interface resistance, while

*A *and

*m *are empirical constants. As mentioned by Li [

82] and Kleinstreuer and Li [

80], the MSB model fails to predict the thermal conductivity enhancement trend when the particle are too small or too large. Also, because of the need for curve-fitting parameters

*A *and

*m*, Prasher's model lacks generality (Figure ).

Kumar [

83] proposed a "moving nanoparticle" model, where the effective thermal conductivity relates to the average particle velocity which is determined by the mixture temperature. However, the solid-fluid interaction effect was not taken into account.

Koo and Kleinstreuer [

70] considered the effective thermal conductivity to be composed of two parts:

where

*k*_{static }is the static thermal conductivity after Maxwell [

1],

*i.e.*,

Now,

*k*_{Brownian }is the enhanced thermal conductivity part generated by midro-scale convective heat transfer of a particle's Brownian motion and affected ambient fluid motion, obtained as Stokes flow around a sphere. By introducing two empirical functions

*β *and

*f*, Koo [

84] combined the interaction between nanoparticles as well as temperature effect into the model and produced:

Li [

82] revisited the model of Koo and Kleinstreuer (2004) and replaced the functions

*β *and

*f*(

*T*,

) with a new g-function which captures the influences of particle diameter, temperature and volume fraction. The empirical g-function depends on the type of nanofluid [

82]. Also, by introducing a thermal interfacial resistance

*R*_{f }= 4e - 8 km

^{2}/W the original

*k*_{p }in Eq. 10 was replaced by a new

*k*_{p,eff }in the form:

Finally, the KKL (Koo-Kleinstreuer-Li) correlation is written as:

where

*g*(

*T*,

,

*d*_{p}) is:

The coefficients

*a*-

*k *are based on the type of particle-liquid pairing [

82]. The comparison between KKL model and benchmark experimental data are shown in Figure .

In a more recent paper dealing with the Brownian motion effect, Bao [

85] also considered the effective thermal conductivity to consist of a static part and a Brownian motion part. In a deviation from the KKL model, he assumed the velocity of the nanoparticles to be constant, and hence treated the ambient fluid around nanoparticle as steady flow. Considering convective heat transfer through the boundary of the ambient fluid, which follows the same concept as in the KKL model, Bao [

85] provided an expression for Brownian motion thermal conductivity as a function of volume fraction

, particle Brownian motion velocity

*v*_{p }and Brownian motion time interval

*τ*. Bao asserted that the fluctuating particle velocity

*v*_{p }can be measured and

*τ *can be expressed via a velocity correlation function based on the stochastic process describing Brownian motion. Unfortunately, he did not consider nanoparticle interaction, and the physical interpretation of

*R*(

*t*) is not clear. The comparisons between Bao's model and experimental data are shown in Figure . For certain sets of experimental data, Bao's model shows good agreement; however, it is necessary to select a proper value of a matching constant

*M *which is not discussed in Bao [

85].

Feng and Kleinstreuer [

86] proposed a new thermal conductivity model (labeled the F-K model for convenience). Enlightened by the turbulence concept,

*i.e.*, just random quantity fluctuations which can cause additional fluid mixing and not turbulence structures such as diverse eddies, an analogy was made between random Brownian-motion-generated fluid-cell fluctuations and turbulence. The extended Langevin equation was employed to take into account the inter-particle potentials, Stokes force, and random force.

Combining the continuity equation, momentum equations and energy equation with Reynolds decompositions of parameters,

*i.e.*, velocity and temperature, the F-K model can be expressed as:

The static part is given by Maxwell's model [

1], while the micro-mixing part is given by:

The comparisons between the F-K model and benchmark experimental data are shown in Figures , , . Figure also provides comparisons between F-K model predictions and two sets of newer experimental data [

26,

32]. The F-K model indicates higher

*k*_{nf }trends when compared to data by Tavman and Turgut [

32], but it shows a good agreement with measurements by Mintsa et al. [

26]. The reason may be that the volume fraction of the nanofluid used by Tavman and Turgut [

32] was too small,

*i.e.*, less than 1.5%. Overall, the F-K model is suitable for several types of metal-oxide nanoparticles (20 <

*d*_{p }< 50 nm) in water with volume fractions up to 5%, and mixture temperatures below 350 K.