Nanofluids, i.e., well-dispersed (metallic) nanoparticles at low- volume fractions in liquids, may enhance the mixture's thermal conductivity, k_{nf}, over the base-fluid values. Thus, they are potentially useful for advanced cooling of micro-systems. Focusing mainly on dilute suspensions of well-dispersed spherical nanoparticles in water or ethylene glycol, recent experimental observations, associated measurement techniques, and new theories as well as useful correlations have been reviewed.

It is evident that key questions still 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 on whether the anomalous enhancement is real or not endure, as well as discussions on what are repeatable correlations between k_{nf} and temperature, nanoparticle size/shape, and aggregation state. Clearly, benchmark experiments are needed, using the same nanofluids subject to different measurement methods. Such outcomes would validate new, minimally intrusive techniques and verify the reproducibility of experimental results. 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.

Introduction

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.

Experimental studies

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., Al_{2}O_{3}-water, SiO_{2}-water, and TiO_{2}-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, Al_{2}O_{3}-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 [4, 5], 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 [6–9], 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 [10, 11], 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.

Experimental measurement methods

The most common techniques for measuring the thermal conductivity of nanofluids are the transient hot-wire method [9, 12–15], temperature oscillation method [16, 17], and 3-ω method [18, 19]. As an example of a non-intrusive (optical) technique, forced Rayleigh scattering is discussed as well.

Transient hot-wire method

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 [20, 21]. 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:

(1)

with initial condition and boundary conditions

(2a)

and

(2b-c)

The analytic solution reads:

(3)

where γ = 0.5772 is Euler's constant. Hence, if the temperature of the hot wire at time t_{1} and t_{2} are T_{1} and T_{2}, then by neglecting higher-order terms the thermal conductivity can be approximated as:

(4)

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 [19, 23].

Other thermal measurement methods

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 [16, 17, 28], micro-hot-strip method [29], steady-state cut-bar method [30], 3-ω method [18, 31, 32], radial heat-flow method [33], photo-thermal radiometry method [34], and thermal comparator method [19, 35].

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% Al_{2}O_{3}-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 Al_{2}O_{3}-water nanofluid and 10% enhancement for MWCNT-silicone oil nanofluid. Such enhancements are very modest compared to the experimental data obtained by THW methods.

Table 1

Summary of experimental studies on convective heat transfer properties of nanofluids

Nu increases with increasing the Peclet number and φ, Brownian motion may play role in convective heat transfer enhancement

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.

Optical measurement methods

In recent years, optical measurement methods have been proposed as non-invasive techniques for thermal conductivity measurements to improve accuracy [6–9, 13, 11, 27, 37]. 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 [6, 38] 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 [8, 39]. Again, their results did not show any anomalous enhancement either for Au-or Al_{2}O_{3}-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 [7, 40]. 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 Al_{2}O_{3}-PAO and C_{10}H_{22}-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 [6, 7]. 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.

Experimental observations

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 [16–19], 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.

Thermal conductivity k_{nf}vs. volume fraction φ

Most experimental observations of nanofluids with just small nanoparticle volume fractions showed that k_{nf} will significantly increase when compared to the base fluid. For example, Lee and Choi [42] investigated CuO-water/ethylene glycol nanofluids with particle diameters 18.6 and 23.6 nm as well as Al_{2}O_{3}-water/ethylene glycol nanofluids with particle diameters 24.4 and 38.4 nm and discovered a 20% thermal conductivity increase at a volume fraction of 4%. Wang [43] measured a 12% increase in k_{nf} for 28-nm-diameter Al_{2}O_{3}-water and 23 nm CuO-water nanofluids with 3% volume fraction. Li and Peterson [44] provided thermal conductivity expressions in terms of temperature (T) and volume fraction (φ) by using curve fitting for CuO-water and Al_{2}O_{3}-water nanofluids. For non-metallic particles, i.e., SiC-water nanofluids, Xie [45] showed a k_{nf} enhancement effect. Recently, Mintsa [26] provided new thermal conductivity expressions for Al_{2}O_{3}-water and CuO-water nanofluids with particle sizes of 47, 36, and 29 nm by curve fitting their in-house experimental data obtained by the THW method. Murshed [46] measured a 27% increase in 4% TiO_{2}-water nanofluids with particle size 15 nm and 20% increase for Al_{2}O_{3}-water nanofluids. However, Duangthongsuk [14] reported a more moderate increase of about 14% for TiO_{2}-water nanofluids. Quite surprising, Moghadassi [47] observed a 50% increment of thermal conductivity for 5% CuO-monoethylene glycol (MEG) and CuO-paraffin nanofluids.

Thermal conductivity k_{nf}vs. temperature T

Das [16] systematically discussed the relationship between thermal conductivity and temperature for nanofluids, noting significant increases of k_{nf} (T). More recently, Abareshi et al. [48] measured the thermal conductivity of Fe_{3}O_{4}-water with the THW method and asserted that k_{nf} increases with temperature T. Indeed, from a theoretical (i.e., kinetics) view-point, with the increment of the nanofluid's bulk temperature T, molecules and nanoparticles are more active and able to transfer more energy from one location to another per unit time.

In contrast, many scientists using optical measurement techniques found no anomalous effective thermal conductivity enhancement when increasing the mixture temperature [[6–9, 29, 30, 37, 49], etc.]. Additionally, Tavman et al. [32] measured SiO_{2}-water, TiO_{2}-water, and Al_{2}O_{3}-water by the 3-ω method and claimed, without showing actual data points, that there is no anomalous thermal conductivity enhancement with increment of both volume fraction and temperature. Whether anomalous enhancement relationship between k_{nf} and temperature T exist or not is still open for debate.

Dependence of k_{nf}on other parameters

Potentially influential parameters on thermal conductivity, other than volume fraction and temperature, include pH value, type of base fluid, nanoparticle shape, degree of nanoparticle dispersion/interaction, and various additives. For example, Zhu et al. [50] showed that the pH of a nanofluid strongly affects the thermal conductivity of suspensions. Indeed, pH value influence the stability of nanoparticle suspensions and the charges of the particle surface thereby affect the nanofluid thermal conductivity. For pH equal to 8.0-9.0, the thermal conductivity of nanofluid is higher than other situations [50] Of the most common base fluids, water exhibits a higher thermal conductivity when compared to ethylene glycol (EG) for the same nanoparticle volume fraction [43, 44, 51–53]. However, thermal conductivity enhancement of EG-based nanofluids is stronger than for water-based nanofluids [42, 43]. Different particle shapes may also influence the thermal conductivity of nanofluids. Nanoparticles with high aspect ratios seem to enhance the thermal conductivity further. For example, spherical particles show slightly less enhancement than those containing nanorods [54], while the thermal conductivity of CuO-water-based nanofluids containing shuttle-like-shaped CuO nanoparticles is larger than those for CuO nanofluids containing nearly spherical CuO nanoparticles [55]. Another parameter influencing nanofluid thermal conductivity is particle diameter. Das [16], Patel [56] and Chon [57] showed the inverse dependence of particle size on thermal conductivity enhancement, considering three sizes of alumina nanoparticles suspended in water. Beck et al. [58] and Moghadassi et al. [47] reported that the thermal conductivity will increase with the decrease of nanoparticle diameters. However, Timofeeva et al. [53] reported that k_{nf} increases with the increment of nanoparticle diameter for SiC-water nanofluids without publishing any data. Other factors which may influence the thermal conductivity of nanofluids are sonification time [32] and/or surfactant mass fraction [32] to obtain well-dispersed nanoparticles.

For other new experimental data, Wei X. et al. [59] reported nonlinear correlation between k_{nf} and synthesis parameters of nanoparticles as well as temperature T. Li and Peterson [60] showed natural convection deterioration with increase in nanoparticle volume fraction. This may be because the nanoparticle's Brownian motion smoothen the temperature gradient leading to the delay of the onset of natural convection. Also, higher viscosity of nanofluids can also induce such an effect. Wei et al. [61] claimed that the measured apparent thermal conductivity show time-dependent characteristics within 15 min when using the THW method. They suggested that measurements should be made after 15 min in order to obtain accurate data. Chiesa et al. [23] investigated the impact of the THW apparatus orientation on thermal conductivity measurements; however, that aspect was found not to be significant. Shalkevich et al. [62] reported no abnormal thermal conductivity enhancement for 0.11% and 0.00055% of gold nanoparticle suspensions, which are rather low volume fractions. Beck et al. [63] and Teng et al. [15] provided curve-fitted results based on their in-house experimental data, reflecting correlations between k_{nf} and several parameters, i.e., volume fraction, bulk temperature and particle size. Both models are easy to use for certain types of nanofluids. Ali et al. [27] proposed hot wire-laser probe beam method to measure nanofluid thermal conductivity and confirmed that particle clustering has a significant effect on thermal conductivity enhancement.

Theoretical studies

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.

Classical models

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.

Table 2

Classical models for effective thermal conductivity of mixtures

Models

Expressions

Remarks

Maxwell

Spherical particles

Hamilton-Crosser

n = 3 for spheres n = 6 for cylinders

Jeffrey

Spherical particles

Davis

High-order terms represent pair interaction of randomly dispersed sphere

Lu-Lin

Spherical and non-spherical particles

Dynamical models and comparisons with experimental data

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:

(5)

where C_{1} is an empirical constant and d_{bf} is the base fluid molecule diameter. Re_{dp} is the Reynolds number, defined as:

(6)

with

(7)

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:

(8)

Here, Re is defined by Eq. 7a, α = 2R_{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 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:

(9)

where k_{static} is the static thermal conductivity after Maxwell [1], i.e.,

(10)

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:

(11)

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 km^{2}/W the original k_{p} in Eq. 10 was replaced by a new k_{p,eff} in the form:

(12)

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

(13)

where g(T,φ,d_{p}) is:

(14)

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.

(15)

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:

(16)

The static part is given by Maxwell's model [1], while the micro-mixing part is given by:

(17)

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 [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.

Summary and future work

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.

Declarations

Authors’ Affiliations

(1)

Department of Mechanical and Aerospace Engineering, NC State University

References

Maxwell JC: A Treatise on Electricity and Magnetism. Oxford: Clarendon; 1891.

Masuda H, Ebata A, Teramea K, Hishinuma N: Alteration of thermal conductivity and viscosity of liquid by dispersing ultra-fine particles. Netsu Bussei 1993, 4: 227–233.View Article

Eastman JA, Choi US, Li S, Thompson LJ, Lee S: Enhanced thermal conductivity through the development of nanofluids. In Nanophase and Nanocomposite Materials II. Edited by: Komarneni S, Parker JC, Wollenberger HJ. Pittsburg: Materials Research Society; 1997:3–11.

Yu WH, France DM, Routbort JL, Choi SUS: Review and comparison of nanofluid thermal conductivity and heat transfer enhancements. Heat Transfer Engineering 2009, 29: 432–460. 10.1080/01457630701850851View Article

Eapen J, Rusconi R, Piazza R, Yip S: The classical nature of thermal conduction in nanofluids. Journal of Heat Transfer 2010., 132: 102402–1-102402–14. 102402-1-102402-14. 10.1115/1.4001304

Rusconi R, Rodari E, Piazza R: Optical measurements of the thermal properties of nanofluids. Applied Physics Letters 2006., 89: 261916–1-2619163. 261916-1-2619163. 10.1063/1.2425015

Putnam SA, Cahill DG, Braun PV: Thermal conductivity of nanoparticle suspensions. Journal of Applied Physics 2006., 99: 084308–1-084308–6. 084308-1-084308-6. 10.1063/1.2189933

Venerus DC, Kabadi MS, Lee S, Perez-Luna V: Study of thermal transport in nanoparticle suspensions using forced Rayleigh scattering. Journal of Applied Physics 2006., 100: 094310–1-094310–5. 094310-1-094310-5. 10.1063/1.2360378

Buongiorno J, Venerus DC, Prabhat N: A benchmark study on the thermal conductivity of nanofluids. Journal of Applied Physics 2009., 106: 094312–1-094312–14. 094312-1-094312-14. 10.1063/1.3245330

Shaikh S, Lafdi K: Thermal conductivity improvement in carbon nanoparticle doped PAO oil: An Experimental Study. Journal of Applied Physics 101: 064302–1-064302–7. 064302-1-064302-7.

Bazan JAN: Thermal conductivity of poly-aelpha-olefin (PAO)-based nanofluids. Ph.D. Thesis, University of Dayton, Dayton, OH, USA. 2010.

Yoo DH, Hong KS, Yang HS: Study of thermal conductivity of nanofluids for the application of heat transfer fluids. Thermochim Acta 2007, 455.

Schmidt AJ, Chiesa M, Torchinsky DH, Johnson JA, Nelson KA, Chen G: Thermal conductivity of nanoparticle suspension in insulating media measured with a transient optical grating and a hotwire. Journal of Applied Physics 2008., 103: 083529–1-083529–5. 083529-1-083529-5.

Duangthongsuk W, Wongwises S: Measurement of temperature-dependent thermal conductivity and viscosity of TiO_{2}-water nanofluids.Experimental Thermal and Fluid Science 2009, 33: 706–714. 10.1016/j.expthermflusci.2009.01.005View Article

Teng TP, Hung YH, Teng TC, Mo HE, Hsu HG: The effect of alumina/water nanofluid particle size on thermal conductivity. Applied Thermal Engineering 2010, 30: 2213–2218. 10.1016/j.applthermaleng.2010.05.036View Article

Das SK, Putra N, Theisen P, Roetzel W: Temperature dependence of thermal conductivity enhancement for nanofluids. Journal of Heat Transfer 2003, 125: 567–574. 10.1115/1.1571080View Article

Czarnetzki W, Roetzel W: Temperature oscillation techniques for simultaneous measurement of thermal diffusivitiy and conductivity. International Journal of Thermophysics 1995, 16: 413–422. 10.1007/BF01441907View Article

Choi TY, Maneshian MH: Measurement of the thermal conductivity of a water-based single-wall carbon nanotube colloidal suspension with a modified 3-w method. Nanotechnology 2009., 20: 315706–1-315706–6. 315706-1-315706-6.

Paul G, Chopkar M, Manna I, Das PK: Techniques for measuring the thermal conductivity of nanofluids: a review. Renewable and Sustainable Energy Reviews 2010, 14: 1913–1924. 10.1016/j.rser.2010.03.017View Article

Feng Y: A new thermal conductivity model for nanofluids with convection heat transfer application. MS thesis, North Carolina State University, Raleigh, NC, USA. 2010.

Vadasz P: Rendering the transient hot wire experimental method for thermal conductivity estimation to two-phase systems-theoretical leading order results. Journal of Heat Transfer 2010., 132: 081601–1-081601–7. 081601-1-081601-7.

Das SK, Choi SUS, Yu W, Pradeep T: Nanofluids: Science and Technology. New Jersey: Wiley; 2008.

Chiesa M, Simonsen AJ: The importance of suspension stability for hot-wire measurements of thermal conductivity of colloidal suspensions. 16th Australasian Fluid Mechanics Conference, Gold Coast, Australia 2010.

Zhang X, Gu H, Fujii M: Experimental study on the effective thermal conductivity and thermal diffusivity of nanofluid. AIAA Journal 2006, 41: 831–840. 10.2514/2.2044View Article

Woodfield PL: A two-dimensional analytical solution for the transient short-hot-wire method. International Journal of Thermophysics 2008, 29: 1278–1298. 10.1007/s10765-008-0469-yView Article

Mintsa HA, Roy G, Nguyen CT, Doucet D: New temperature dependent thermal conductivity data for water-based nanofluids. International Journal of Thermal Sciences 2009, 48: 363–371. 10.1016/j.ijthermalsci.2008.03.009View Article

Ali FM, Yunus WMM, Moksin MM, Talib ZA: The effect of volume fraction concentration on the thermal conductivity and thermal diffusivity of nanofluids: numerical and experimental. Review of Scientific Instruments 2010., 81: 074901–1-074901–9. 074901-1-074901-9. 10.1063/1.3458011

Patel HE, Sundararajan T, Das SK: An experimental investigation into the thermal conductivity enhancement in oxide and metallic nanofluids. Journal of Nanoparticle Research 2010, 12: 1015–1031. 10.1007/s11051-009-9658-2View Article

Ju YS, Kim J, Hung MT: Experimental study of heat conduction in aqueous suspensions of aluminum oxide nanoparticles. Journal of Heat Transfer 2008., 130: 092403–1-092403–6. 092403-1-092403-6.

Li CH, Williams W: Transient and steady-state experimental comparison study of effective thermal conductivity of Al_{2}O_{3}-water nanofluids.Journal of Heat Transfer 2008., 130: 042407–1-042407–7. 042407-1-042407-7.

Turgut A, Tavman I, Chirtoc M, Schuchmann HP, Sauter C, Tavman S: Thermal conductivity and viscosity measurements of water-based TiO2 nanofluids. International Journal of Thermophysics 2009, 30: 1213–1226. 10.1007/s10765-009-0594-2View Article

Tavman I, Turgut A: An investigation on thermal conductivity and viscosity of water based nanofluids. Microfluidics Based Microsystems 2010, 0: 139–162. full_textView Article

Iygengar AS, Abramson AR: Comparative radial heat flow method for thermal conductivity measurement of liquids. Journal of Heat Transfer 2009., 131: 064502–1-064502–3. 064502-1-064502-3.

Kusiak A, Pradere C, Battaglia JL: Measuring the thermal conductivity of liquids using photo-thermal radiometry. Measurement Science and Technology 2010., 21: 015403–1-015403–6. 015403-1-015403-6. 10.1088/0957-0233/21/1/015403

Rousan AA, Roy DM: A thermal comparator method for measuring thermal conductivity of cementitious materials. Industrial and Engineering Chemistry Product Research and Development 1983, 22: 349–351. 10.1021/i300010a035View Article

Lee JH: Convection Performance of Nanofluids for Electronics Cooling, Ph. D. Dissertation. Stanford University, CA, USA; 2009.

Kolade B, Goodson KE, Eaton JK: Convective performance of nanofluids in a laminar thermally developing tube flow. Journal of Heat Transfer 2009., 131: 052402–1-052402–8. 052402-1-052402-8. 10.1115/1.3013831

Rusconi R, Isa L, Piazza R: Thermal-lensing measurement of particle thermophoresis in aqueous dispersions. Journal of Optical Society of America 2004, 21(3):605–616. 10.1364/JOSAB.21.000605View Article

Venerus DC, Schieber JD, Iddir H, Guzman JD, Broerman AW: Measurement of thermal diffusivity in polymer melts using forced rayleigh light scattering. Journal of Polymer Science: Part B: Polymer Physics 1999, 37: 1069–1078. 10.1002/(SICI)1099-0488(19990601)37:11<1069::AID-POLB3>3.0.CO;2-UView Article

Putnam SA, Cahill DG: Micron-scale apparatus for measurements of thermodiffusion in liquids. Review of Scientific Instruments 2004, 75(7):2368–2372. 10.1063/1.1765761View Article

Hamilton RL, Crosser OK: Thermal conductivity of heterogeneous two-component systems. Industrial Engineering and Chemistry Fundamentals 1962, 1(3):187–191. 10.1021/i160003a005View Article

Lee S, Choi SUS: Measuring thermal conductivity of fluids containing oxide nanoparticles. Journal of Heat Transfer 1999, 121: 280–289. 10.1115/1.2825978View Article

Wang XW, Xu XF, Choi SUS: Thermal conductivity of nanoparticle-fluid mixture. Journal of Thermalphysics and Heat Transfer 1999, 13: 474–480. 10.2514/2.6486View Article

Li CH, Peterson GP: Experimental investigation of temperature and volume fraction variations on the effective thermal conductivity of nanoparticle suspensions (nanofluids). Journal of Applied Physics 2006, 99(8):084314. 10.1063/1.2191571View Article

Xie H, Wang J, Xi T, Liu Y: Thermal conductivity of suspensions containing nanosized SiC particles. International Journal of Thermophysics 2002, 23(2):571–580. 10.1023/A:1015121805842View Article

Murshed SMS, Leong KC, Yang C: A combined model for the effective thermal conductivity of nanofluids. Applied Thermal Engineering 2009, 29: 2477–2483. 10.1016/j.applthermaleng.2008.12.018View Article

Moghadassi AR, Hosseini SM, Henneke DE: Effect of CuO nanoparticles in enhancing the thermal conductivities of monoethylene glycol and paraffin fluids. Industrial Engineering and Chemistry Research 2010, 49: 1900–1904. 10.1021/ie901060eView Article

Abareshi M, Goharshiadi EK, Zebarjad SM, Fadafan HK, Youssefi A: Fabrication, characterization and measurement of thermal conductivity of Fe_{3}O_{4}nanofluids. Journal of Magnetism and Magnetic Materials 2010, 322(24):3895–3901. 10.1016/j.jmmm.2010.08.016View Article

Williams W, Buongiorno J, Hu LW: Experimental investigation of turbulent convective heat transfer and pressure loss of alumina/water and zirconia/water nanoparticle colloids (nanofluids) in horizontal tubes. Journal of Heat Transfer 2008., 130: 042412–1-042412–7. 042412-1-042412-7. 10.1115/1.2818775

Zhu D, Li X, Wang N, Wang X, Gao J, Li H: Dispersion behavior and thermal conductivity characteristics of Al_{2}O_{3}-H_{2}O nanofluids. Current Applied Physics 2009, 9: 131–139. 10.1016/j.cap.2007.12.008View Article

Jang SP, Choi SUS: Role of Brownian motion in the enhanced thermal conductivity of nanofluids. Applied Physics Letters 2004, 84: 4316–4318. 10.1063/1.1756684View Article

Timofeeva EV, Gavrilov AN, McCloskey JM, Tolmachev YV: Thermal conductivity and particle agglomeration in alumina nanofluids: experiment and theory. Physical Review E 2007., 76: 061203–1-061203–16. 061203-1-061203-16. 10.1103/PhysRevE.76.061203

Timofeeva EV, Smith DS, Yu W, France DM, Singh D, Routbort JL: Particle size and interfacial effects on thermo-physical and heat transfer characteristics of water-based alpha-SiC nanofluids. Nanotechnology 2010., 21: 215703–1-215703–10. 215703-1-215703-10. 10.1088/0957-4484/21/21/215703

Murshed SMS, Leong KC, Yang C: Enhanced thermal conductivity of TiO2-water based nanofluids. International Journal of Thermal Sciences 2005, 44: 367. 10.1016/j.ijthermalsci.2004.12.005View Article

Zhu HT, Zhang CY, Tang YM, Wang JX: Novel synthesis and thermal conductivity of CuO nanofluid. Journal of Physical Chemistry C 2007, 111: 1646. 10.1021/jp065926tView Article

Patel HE, Das SK, Sundararajan T, Sreekumanran NA, George B, Pradeep T: Thermal conductivities of naked and monolayer protected metal nanoparticle based nanofluids: manifestation of anomalous enhancement and chemical effects. Applied Physics Letters 2003, 83: 2931–2933. 10.1063/1.1602578View 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.Applied Physics Letters 2005., 87: 153107–1-153107–3. 153107-1-153107-3. 10.1063/1.2093936

Beck MP, Yuan Y, Warrier P, Teja AS: The effect of particle size on the thermal conductivity of alumina nanofluids. Journal of Nanoparticle Research 2009, 11: 1129–1136. 10.1007/s11051-008-9500-2View Article

Wei X, Kong T, Zhu H, Wang L: CuS/Cu_{2}S nanofluids: synthesis and thermal conductivity. International of Heat and Mass Transfer 2010, 53: 1841–1843. 10.1016/j.ijheatmasstransfer.2010.01.006View Article

Li CH, Peterson GP: Experimental studies of natural convection heat transfer of Al_{2}O_{3}/DI water nanoparticle suspensions (nanofluids).Advances in Mechanical Engineering 2010., 2010: 742739–1-742739–10. 742739-1-742739-10.

Wei Y, Xie H, Chen L, Li Y: Investigation on the thermal transport properties of ethylene glycol-based nanofluids containing copper nanoparticles. Powder Technology 2010, 197: 218–221. 10.1016/j.powtec.2009.09.016View Article

Shalkevich N, Escher W, Buergi T, Michel B, Ahmed L, Poulikakos D: On the thermal conductivity of gold nanoparticle colloid. Langmuir 2010, 26(2):663–670. 10.1021/la9022757View Article

Beck MP, Yuan Y, Warrier P, Teja AS: The thermal conductivity of alumina nanofluids in water, ethylene glycol, and ethylene glycol + water mixture. Journal of Nanoparticles Research 2010, 12: 1469–1477. 10.1007/s11051-009-9716-9View Article

Jeffrey DJ: Conduction through a random suspension of spheres. Proceedings of Royal Society, A 1973, 335: 355–367. 10.1098/rspa.1973.0130View Article

Davis RH: The effective thermal conductivity of a composite material with spherical inclusions. International Journal of Thermophysics 1986, 7: 609–620. 10.1007/BF00502394View Article

Bruggeman DAG: Berechnung verschiedener physikalischer konstanten von heterogenen substanzen, I-Dielektrizitatskonstanten und leitfahigkeiten der mischkorper aus isotropen substanzen. Annalen der Physik, Leipzig 1935, 24: 636–679. 10.1002/andp.19354160705View Article

Wang LQ, Zhou XS, Wei XH: Heat conduction mathematical models and analytical solutions. Berlin: Springer-Verlag; 2008.

Koo J, Kang Y, Kleinstreuer C: A nonlinear effective thermal conductivity model for carbon nanotube and nanofiber suspensions. Nanotechnology 2008., 19: 375705–1-375705–7. 375705-1-375705-7.

Koo J, Kleinstreuer C: A new thermal conductivity model for nanofluids. Journal of Nanoparticle Research 2004, 6: 577–588. 10.1007/s11051-004-3170-5View Article

Wang LQ, Fan J: Nanofluids research: key issues. Nanoscale Research Letter 2010, 5: 1241–1252. 10.1007/s11671-010-9638-6View Article

Prevenslik T: Nanoscale Heat Transfer by Quantum Mechanics. Fifth International Conference on Thermal Engineering: Theory and Applications, Marrakesh, Morocco 2010.

Nield DA, Kuznetsov AV: The effect of local thermal nonequilibrium on the onset of convection in a nanofluid. Journal of Heat Transfer 2010., 132: 052405–1-052405–7 052405-1-052405-7

Leal LG: On the effective conductivity of a dilute suspension of spherical drops in the limit of low particle Peclet number. Chem Eng Commun 1973, 1: 21–31. 10.1080/00986447308960412View Article

Gupte SK, Advani SG: Role of micro-convection due to non-affine motion of particles in a mono-disperse suspension. Int J Heat Mass Transfer 1995, 38(16):2945–2958. 10.1016/0017-9310(95)00060-MView Article

Keblinski P, Phillpot SR, Choi SUS, Eastman JA: Mechanisms of heat flow in suspensions of nanos-sezed particles (nanofluids). International Journal of Heat and Mass Transfer 2002, 45: 855–863. 10.1016/S0017-9310(01)00175-2View Article

Jang SP, Choi SUS: Effects of various parameters on nanofluid thermal conductivity. ASME J Heat Transfer 2007, 129: 617–623. 10.1115/1.2712475View Article

Jang SP, Choi SUS: Cooling performance of a microchannel heat sink with nanofluids. Appl Therm Eng 2006, 26: 2457–2463. 10.1016/j.applthermaleng.2006.02.036View Article

Kleinstreuer C, Li J: Discussion: effects of various parameters on nanofluid thermal conductivity. ASME Journal of Heat Transfer 2008., 130: 025501–1-025501–3. 025501-1-025501-3. 10.1115/1.2812307

Prasher R: Brownian-motion-based convective-conductive model for the effective thermal conductivity of nanofluids. Journal of Heat Transfer 2006, 128: 588–595. 10.1115/1.2188509View Article

Li J: Computational analysis of nanofluid flow in microchannels with applications to micro-heat sinks and bio-MEMS, PhD Thesis. NC State University, Raleigh, NC, the United States; 2008.

Kumar DH, Patel HE, Kumar VRR, Sundararajan T, Pradeep T, Das SK: Model for heat conduction in nanofluids. Physical Review Letters 2004., 93: 144301–1-144301–4. 144301-1-144301-4. 10.1103/PhysRevLett.93.144301

Koo JM: Computational nanofluid flow and heat transfer analyses applied to micro-systems, Ph.D Thesis. NC State University, Raleigh, NC, USA; 2005.

Bao Y: Thermal conductivity equations based on Brownian motion in suspensions of nanoparticles (nanofluids). Journal of Heat Transfer 130: 042408–1-042408–5. 042408-1-042408-5.

Feng Y, Kleinstreuer C: Nanofluid convective heat transfer in a parallel-disk system. International Journal of Heat and Mass Transfer 2010, 53: 4619–4628. 10.1016/j.ijheatmasstransfer.2010.06.031View Article

Chopkar M, Sudarshan S, Das PK, Manna I: Effect of particle size on thermal conductivity of nanofluid. Metallurgical and Materials Transactions A 2008, 39A: 1535–1542. 10.1007/s11661-007-9444-7View Article

Wu D, Zhu H, Wang L, Liu L: Critical issues in nanofluids preparation, characterization and thermal conductivity. Current Nanoscience 2009, 5: 103–112. 10.2174/157341309787314548View Article

Singh D, Timofeeva E, Yu W, Routbort J, France D, Smith D, Lopez-Cepero JM: An investigation of silicon carbide-water nanofluid for heat transfer applications. Journal of Applied Physics 2009., 15: 064306–1-064306–6. 064306-1-064306-6.

Jung YJ, Jung YY: Thermal conductivity enhancement of nanofluids in conjunction with electrical double layer (EDL). International Journal of Heat and Mass Transfer 2009, 52: 525–528. 10.1016/j.ijheatmasstransfer.2008.07.016View Article

Pak BC, Cho YI: Hydrodynamic and heat transfer study of dispersed fluids with submicron metallic oxide particles. Exp Heat Transfer 1998, 11: 151–170. 10.1080/08916159808946559View Article

Li Q, Xuan Y: Convective heat transfer and flow characteristics of Cu-water nanofluid. Science in China (Series E) 2002, 45: 408–416.

Wen D, Ding Y: Experimental investigation into convective heat transfer of nanofluids at the entrance region under laminar flow conditions. International Journal of Heat and Mass Transfer 2004, 47: 5181–5188. 10.1016/j.ijheatmasstransfer.2004.07.012View Article

Ding Y, Alias H, Wen D, Williams RA: Heat transfer of aqueous suspensions of carbon nanotubes (CNT nanofluids). International Journal of Heat and Mass Transfer 2006, 49: 240–250. 10.1016/j.ijheatmasstransfer.2005.07.009View Article

Heris SZ, Esfahany MN, Etemad SG: Experimental investigation of convective heat transfer of Al_{2}O_{3}/water nanofluid in circular tube.International Journal of Heat and Fluid Flow 2007, 28: 203–210. 10.1016/j.ijheatfluidflow.2006.05.001View Article

Rea U, McKrell T, Hu L, Buongiorno J: Laminar convective heat transfer and viscous pressure loss of alumina-water and zirconia-water nanofluids. International Journal of Heat and Mass Transfer 2009, 52: 2042–2048. 10.1016/j.ijheatmasstransfer.2008.10.025View Article

Heris SZ, Etemad SGh, Esfahany MN: Convective heat transfer of a Cu/water nanofluid flowing through a circular tube. Experimental Heat Transfer 2009, 22: 217–227. 10.1080/08916150902950145View 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.