A nanofluid is a dilute suspension of nanometer-size particles and fibers dispersed in a liquid. As a result, when compared to the base fluid, changes in physical properties of such mixtures occur, e.g., viscosity, density, and thermal conductivity. Of all the physical properties of nanofluids, the thermal conductivity (k_nf) is the most complex and for many applications the most important one. Interestingly, experimental findings have been controversial and theories do not fully explain the mechanisms of elevated thermal conductivity. In this paper, experimental and theoretical studies are reviewed for nanofluid thermal conductivity and convection heat transfer enhancement. Specifically, comparisons between thermal measurement techniques (e.g., transient hot-wire (THW) method) and optical measurement techniques (e.g., forced Rayleigh scattering (FRS) method) are discussed. Recent theoretical models for nanofluid thermal conductivity are presented and compared, including the authors' model assuming well-dispersed spherical nanoparticles subject to micro-mixing effects due to Brownian motion. Concerning theories/correlations which try to explain thermal conductivity enhancement for all nanofluids, not a single model can predict a wide range of experimental data. However, many experimental data sets may fit between the lower and upper mean-field bounds originally proposed by Maxwell where the static nanoparticle configurations may range from a dispersed phase to a pseudo-continuous phase. Dynamic k_nf models, assuming non-interacting metallic nano-spheres, postulate an enhancement above the classical Maxwell theory and thereby provide potentially additional physical insight. Clearly, it will be necessary to consider not only one possible mechanism but combine several mechanisms and compare predictive results to new benchmark experimental data sets.

Nanofluids are a new class of heat transfer fluids by dispersing nanometer-size particles, e.g., metal-oxide spheres or carbon nanotubes, with typical diameter scales of 1 to 100 nm in traditional heat transfer fluids. Such colloidal dispersions may be uniform or somewhat aggregated. Earlier experimental studies reported greater enhancement of thermal conductivity, k_nf, than predicted by the classical model of Maxwell 1, known as the mean-field or effective medium theory. For example, Masuda 2 showed that different nanofluids (i.e., Al₂O₃-water, SiO₂-water, and TiO₂-water combinations) generated a k_nf increase of up to 30% at volume fractions of less than 4.3%. Such an enhancement phenomenon was also reported by Eastman and Choi 3 for CuO-water, Al₂O₃-water and Cu-Oil nanofluids, using the THW method. In the following decades, it was established that nanofluid thermal conductivity is a function of several parameters 45, i.e., nanoparticle material, volume fraction, spatial distribution, size, and shape, as well as base-fluid type, temperature, and pH value. In contrast, other experimentalists 6789, reported that no correlation was observed between k_nf and nanofluid temperature T. Furthermore, no k_nf enhancement above predictions based on Maxwell's effective medium theory for non-interacting spherical nanoparticles was obtained 5. Clearly, this poses the question if nanofluids can provide greater heat transfer performance, as it would be most desirable for cooling of microsystems. Some scientists argued that the anomalous k_nf enhancement data are caused by inaccuracies of thermal measurement methods, i.e., mainly intrusive vs. non-intrusive techniques. However, some researchers 1011, relying on both optical and thermal measurements, reported k_nf enhancements well above classical model predictions. When comparing different measurement methods, error sources may result from the preparation of nanofluids, heating process, measurement process, cleanliness of apparatus, and if the nanoparticles stay uniformly dispersed in the base fluid or aggregate 12. Thus, the controversy is still not over because of those uncertainties.

The most common techniques for measuring the thermal conductivity of nanofluids are the transient hot-wire method 912131415, temperature oscillation method 1617, and 3-ω method 1819. As an example of a non-intrusive (optical) technique, forced Rayleigh scattering is discussed as well.

THW method is the most widely used static, linear source experimental technique for measuring the thermal conductivity of fluids. A hot wire is placed in the fluid, which functions as both a heat source and a thermometer 2021. Based on Fourier's law, when heating the wire, a higher thermal conductivity of the fluid corresponds to a lower temperature rise. Das 22 claimed that during the short measurement interval of 2 to 8 s, natural convection will not influence the accuracy of the results.

The relationship between thermal conductivity k_nf and measured temperature T using the THW method is summarized as follows 20. Assuming a thin, infinitely long line source dissipating heat into a fluid reservoir, the energy equation in cylindrical coordinates can be written as:

with initial condition and boundary conditions

and

The analytic solution reads:

where γ = 0.5772 is Euler's constant. Hence, if the temperature of the hot wire at time t₁ and t₂ are T₁ and T₂, then by neglecting higher-order terms the thermal conductivity can be approximated as:

For the experimental procedure, the wire is heated via a constant electric power supply at step time t. A temperature increase of the wire is determined from its change in resistance which can be measured in time using a Wheatstone-bridge circuit. Then the thermal conductivity is determined from Eq. 4, knowing the heating power (or heat flux q) and the slope of the curve ln(t) versus T.

The advantages of THW method are low cost and easy implementation. However, the assumptions of an infinite wire-length and the ambient acting like a reservoir (see Eqs. 1 and 2c) may introduce errors. In addition, nanoparticle interactions, sedimentation and/or aggregation as well as natural convection during extended measurement times may also increase experimental uncertainties 1923.

A number of improved hot-wire methods and experimental designs have been proposed. For example, Zhang 24 used a short-hot-wire method (see also Woodfield 25) which can take into account boundary effects. Mintsa 26 inserted a mixer into his THW experimental devices in order to avoid nanoparticle aggregation/deposition in the suspensions. Ali et al. 27 combined a laser beam displacement method with the THW method to separate the detector and heater to avoid interference.

Alternative static experimental methods include the temperature oscillation method 161728, micro-hot-strip method 29, steady-state cut-bar method 30, 3-ω method 183132, radial heat-flow method 33, photo-thermal radiometry method 34, and thermal comparator method 1935.

It is worth mentioning that most of the thermal measurement techniques are static or so called "bulk" methods (see Eq. 4). However, nanofluids could be used as coolants in forced convection, requiring convective measurement methods to obtain thermal conductivity data. Some experimental results of convective nanofluid heat transfer characteristics are listed in Table 1. For example, Lee 36 fabricated a microchannel, D_h = 200 μm, to measure the nanofluid thermal conductivity with a modest enhancement when compared to the result obtained by the THW method. Also, Kolade et al. 37 considered 2% Al₂O₃-water and 0.2% multi-wall carbon nano-tube (MWCNT)-silicone oil nanofluids. By measuring the thermal conductivities of nanofluids in a convective environment, Kolade et al. 37 obtained 6% enhancement for Al₂O₃-water nanofluid and 10% enhancement for MWCNT-silicone oil nanofluid. Such enhancements are very modest compared to the experimental data obtained by THW methods.

Actually, "convective" k_nf values are not directly measured. Instead, wall temperature T_w and bulk temperature T_b are obtained and the heat transfer coefficient is then calculated as h = q_w/(T_w - T_b). From the definition of the Nusselt number, k_nf = hD/Nu where generally D is the hydraulic diameter. With h being basically measured and D known, either an analytic solution or an iterative numerical evaluation of Nu is required to calculate k_nf. Clearly, the accuracy of the "convective measurement method" largely depends on the degree of uncertainties related to the measured wall and bulk temperatures as well as the computed Nusselt number.

In recent years, optical measurement methods have been proposed as non-invasive techniques for thermal conductivity measurements to improve accuracy 678913112737. Indeed, because the "hot wire" is a combination of heater and thermometer, interference is unavoidable. However, in optical techniques, detector and heater are always separated from each other, providing potentially more accurate data. Additionally, measurements are completed within several microseconds, i.e., much shorter than reported THW-measurement times of 2 to 8 s, so that natural convection effects are avoided.

For example, Rusconi 638 proposed a thermal-lensing (TL) measurement method to obtain k_nf data. The nanofluid sample was heated by a laser-diode module and the temperature difference was measured by photodiode as optical signals. After post-processing, the thermal conductivity values were generated, which did not exceed mean-field theory results. Similar to the TL method, FRS have been used to investigate the thermal conductivity of well-dispersed nanofluids 839. Again, their results did not show any anomalous enhancement either for Au-or Al₂O₃-nanofluids. Also, based on their data, no enhancement of thermal conductivity with temperature was observed. In contrast, Buongiorno et al. 9 presented data agreement when using both the THW method and FRS method. Another optical technique for thermal conductivity measurements of nanofluids is optical beam deflection 740. The nanofluid is heated by two parallel lines using a square current. The temperature change of nanofluids can be transformed to light signals captured by dual photodiodes. For Au-nanofluids, Putnam 7 reported significantly lower k_nf enhancement than the data collected with the THW method.

However, other papers based on optical measurement techniques showed similar enhancement trends for nanofluid thermal conductivities as obtained with the thermal measurement methods. For example, Shaikh et al. 10 used the modern light flash technique (LFA 447) and measured the thermal conductivity of three types of nanofluids. They reported a maximum enhancement of 161% for the thermal conductivity of carbon nanotube (CNT)-polyalphaolefin (PAO) suspensions. Such an enhancement is well above the prediction of the classical model by Hamilton and Crosser 41. Also, Schmidt et al. 13 compared experimental data for Al₂O₃-PAO and C₁₀H₂₂-PAO nanofluids obtained via the Transient Optical Grating method and THW method. In both cases, the thermal conductivities were greater than expected from classical models. Additionally, Bazan 11 executed measurements by three different methods, i.e., laser flash (LF), transient plane source, and THW for PAO-based nanofluids. They concluded that the THW method is the most accurate one while the LF method lacks precision when measuring nanofluids with low thermal conductivities. Also, no correlation between thermal conductivity and temperature was observed. Clearly, materials and experimental methods employed differ from study to study, where some of the new measurement methods were not verified repeatedly 67. Thus, it will be necessary for scientists to use different experimental techniques for the same nanofluids in order to achieve high comparable accuracy and prove reproducibility of the experimental results.

Nearly all experimental results before 2005 indicate an anomalous enhancement of nanofluid thermal conductivity, assuming well-dispersed nanoparticles. However, more recent efforts with refined transient hot-wire and optical methods spawned a controversy on whether the anomalous enhancement beyond the mean-field theory is real or not. Eapen et al. 5 suggested a solution, arguing that even for dilute nanoparticle suspensions k_nf enhancement is a function of the aggregation state and hence connectivity of the particles; specifically, almost all experimental k_nf data published fall between lower and upper bounds predicted by classical theories.

In order to provide some physical insight, benchmark experimental data sets obtained in 2010 as well as before 2010 are displayed in Figures 1 and 2. Specifically, Figure 1a,b demonstrate that k_nf increases with nanoparticle volume fraction. This is because of a number of interactive mechanisms, where Brownian-motion-induced micro-mixing is arguably the most important one when uniformly distributed nanoparticles can be assumed. Figure 2a,b indicate that k_nf also increases with nanofluid bulk temperature. Such a relationship can be derived based on kinetics theory as outlined in Theoretical studies section. The impact of nanoparticle diameter on k_nf is given in Figures 1 and 2 as well. Compared to older benchmark data sets 16171819, new experimental results shown in Figures 1 and 2 indicate a smaller enhancement of nanofluid thermal conductivity, perhaps because of lower experimental uncertainties. Nevertheless, discrepancies between the data sets provided by different research groups remain.

In summary, k_nf is likely to improve with nanoparticle volume fraction and temperature as well as particle diameter, conductivity, and degree of aggregation, as further demonstrated in subsequent sections.

Significant differences among published experimental data sets clearly indicate that some findings were inaccurate. Theoretical analyses, mathematical models, and associated computer simulations may provide additional physical insight which helps to explain possibly anomalous enhancement of the thermal conductivity of nanofluids.

The static model of Maxwell 1 has been used to determine the effective electrical or thermal conductivity of liquid-solid suspensions of monodisperse, low-volume-fraction mixtures of spherical particles. Hamilton and Crosser 41 extended Maxwell's theory to non-spherical particles. For other classical models, please refer to Jeffery 64, Davis 65 and Bruggeman 66 as summarized in Table 2. The classical models originated from continuum formulations which typically involve only the particle size/shape and volume fraction and assume diffusive heat transfer in both fluid and solid phases 67. Although they can give good predictions for micrometer or larger-size multiphase systems, the classical models usually underestimate the enhancement of thermal conductivity increase of nanofluids as a function of volume fraction. Nevertheless, stressing that nanoparticle aggregation is the major cause of k_nf enhancement, Eapen et al. 5 revived Maxwell's lower and upper bounds for the thermal conductivities of dilute suspensions (see also the derivation by Hashin and Shtrikman 68). While for the lower bound, it is assumed that heat conducts through the mixture path where the nanoparticles are well dispersed, the upper bound is valid when connected/interacting nanoparticles are the dominant heat conduction pathway. The effect of particle contact in liquids was analyzed by Koo et al. 69, i.e., actually for CNTs, and successfully compared to various experimental data sets. Their stochastic model considered the CNT-length as well as the number of contacts per CNT to explain the nonlinear behavior of k_nf with volume fraction.

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 6772, 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 C₁ is an empirical constant and d_bf is the base fluid molecule diameter. Re_dp is the Reynolds number, defined as:

with

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 3a,b), 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 = 2R_bk_m/d_p is the nanoparticle Biot number, and R_b = 0.77 × 10^-8 Km²/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 4).

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,φ) 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 km²/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,φ,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 5.

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 φ, 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 6. 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 4, 6, 7a,b. Figure 7a also provides comparisons between F-K model predictions and two sets of newer experimental data 2632. 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.

Nanofluids, i.e., well-dispersed metallic nanoparticles at low volume fractions in liquids, enhance the mixture's thermal conductivity over the base-fluid values. Thus, they are potentially useful for advanced cooling of micro-systems. Still, key questions linger concerning the best nanoparticle-and-liquid pairing and conditioning, reliable measurements of achievable k_nf values, and easy-to-use, physically sound computer models which fully describe the particle dynamics and heat transfer of nanofluids. At present, experimental data and measurement methods are lacking consistency. In fact, debates are still going on whether the anomalous enhancement is real or not, and what are repeatable correlations between k_nf and temperature, nanoparticle size/shape, and aggregation state. Clearly, additional benchmark experiments are needed, using the same nanofluids subject to different measurement methods as well as variations in nanofluid characteristics. This would validate new, minimally intrusive techniques and verify the reproducibility of experimental results.

Concerning theories/correlations which try to explain thermal conductivity enhancement for all nanofluids, not a single model can predict a wide range of experimental observations. However, many experimental data sets may fit between the lower and upper mean-field bounds originally proposed by Maxwell 1, where the static nanoparticle configurations may range between the two extremes of a dispersed phase to a continuous phase. Dynamic k_nf models postulate an enhancement above the classic Maxwell theory and thereby provide additional physical insight. Clearly, it will be necessary to consider not only one possible mechanism but combine several mechanisms and compare predictive results to new benchmark experimental data sets.

The authors declare that they have no competing interests.

YF conducted the extensive literature review and CK wrote the article. Both authors read and approved the final manuscript.