INTRODUCTION

Heat exchange plays an important role in various industrial processes, in transport, in electronics, and in computers [1]. Enhancement of the power of internal combustion engines and high heat release in microelectronic devices in combination with their small size make it necessary to take off a large amount of heat from the heated units and parts. In the field of optical devices, an increase in the power in combination with a decrease in the size also requires innovative cooling technologies. The heat transfer is often performed using a liquid that flows in the laminar or turbulent mode. Liquid coolants used today such as water, synthetic oils, glycols, and perfluorinated hydrocarbons exhibit low thermal conductivity. Many solids surpass liquids in the thermal conductivity by orders of magnitude; therefore, addition of solid particles to coolants can improve the heat exchange [2].

The idea of filling a liquid with solid particles to enhance its thermal conductivity was put forward more than a century ago [3]. Numerous studies were performed on increasing the thermal conductivity of liquids by suspending in them fine particles with a size from several micrometers to hundreds of micrometers. An increase in the volume fraction of the filler increased the thermal conductivity of suspensions, but led to the particle agglomeration and sedimentation. Sedimentation of the particles led to plugging of flow-through channels and ultimately increased the power consumption for pumping the coolant. At fast circulation of the suspension, the particle sedimentation decreased, but a risk of damage of heat exchanger walls by the solid particles arose. Therefore, the approach based on suspending particles in a liquid was initially rejected in the practice of heat exchange [4].

The appearance of solid nanoparticles in the modern materials technology revived the idea of suspending. It was expected that the heat dissipation in a liquid containing such particles would occur faster owing to larger specific surface area of such particles and to their microconvection. In addition, it was anticipated that dispersions based on nanoparticles would be more stable because of small particle size, so that the probability of the channel plugging and heat exchanger erosion would decrease. Colloidal suspensions of nanoparticles (size 1–100 nm) were termed nanofluids [5, 6]. The first experiments have shown that nanoparticles can significantly increase the thermal conductivity of the base liquid [7–9] and that with the addition of a stabilizer they do not undergo sedimentation for several months [6, 8]. In addition, the thermal conductivity was improved at a low particle concentration with the preservation of the Newtonian behavior of the liquid and of the possibility of its flow through channels of very small diameter without erosion and plugging [10, 11]. Thus, the use of nanofluids allows improvement of the heat exchange; in addition, nanofluids can be used in miniature system.

The initial studies that demonstrated good prospects for using nanofluids for the heat transfer stimulated the start of extensive studies in this field. For example, the number of papers dealing with nanofluids was more than 100 in 2006 [12] and more than 2000 in 2019. Such growth of the publication activity is associated with the organization of research groups for studying nanofluids at leading institutions throughout the world; in addition, studies were performed at small enterprises and large companies in various branches of industry for solving specific problems. As a result, the main approaches to preparing nanofluids were developed, and numerous dispersions differing in the nanoparticle material, size, and shape, in the volume filling, and in the kind of the liquid were examined [13–21]. On the whole, filling of liquids with nanoparticles to a volume concentration of 0.5–5% increases the thermal conductivity of the liquid by 15–40% [12]. However, the extent to which the thermal conductivity of nanofluids was improved relative to the base liquid was different in different studies; both abnormally high growth of the thermal conductivity, inconsistent with the results of theoretical calculations, and normal or even unexpectedly low growth were noted. Therefore, different models for describing the thermal conductivity of nanofluids and different mechanisms accounting for their behavior were suggested. However, there is still no common opinion on the efficiency of using nanoparticles for enhancing the thermal conductivity of liquid media, and nanofluids still attract researchers’ interest as coolants.

This review is aimed at summarizing numerous theoretical and experimental studies of the thermal conductivity of disperse systems and at evaluating the potential of enhancing the thermal conductivity of liquids by filling them with solid particles.

THEORETICAL MODELS

The classical description of the thermal conductivity of suspensions is Maxwell’s theory predicting the effective thermal conductivity of a continuous medium with well dispersed solid particles on the basis of the electrical conductivity theory [3, 22]. According to Maxwell, the effective thermal conductivity of a suspension of resting spherical solid particles, keff, depends on their volume fraction φ and on the initial thermal conductivities of the particles, kp, and liquid, kf:

$$ {{k}_{\text{eff}}}={{k}_{\text{f}}}\frac{{{k}_{\text{p}}}+2{{k}_{\text{f}}}+2\text{ }\!\!\varphi\!\!\text{ }\left( {{k}_{\text{p}}}-{{k}_{\text{f}}} \right)}{{{k}_{\text{p}}}+2{{k}_{\text{f}}}-\text{ }\!\!\varphi\!\!\text{ }\left( {{k}_{\text{p}}}-{{k}_{\text{f}}} \right)}. $$
(1)

This equation is a first-order approximation applicable to mixtures with a low volume concentration of particles. Maxwell became a pioneer in the theory of the thermal conductivity of disperse systems, and his studies were followed by numerous theoretical and experimental studies involving calculation and measurement of the effective thermal conductivity of diverse systems.

Von Bruggeman suggested a model for calculating the thermal conductivity of concentrated suspensions of spherical particles [23],

$$ \varphi \left( \frac{{{k}_{\text{p}}}-{{k}_{\text{eff}}}}{{{k}_{\text{p}}}+2{{k}_{\text{eff}}}} \right)+\left( 1-\varphi \right)\left( \frac{{{k}_{\text{f}}}-{{k}_{\text{eff}}}}{{{k}_{\text{f}}}+2{{k}_{\text{eff}}}} \right)=0, $$
(2)

coinciding with Maxwell’s model at low volume concentration of particles. Later Hamilton and Crosser obtained an equation taking into account the influence of the particle shape through coefficient n (equal to 3/Ψ, where Ψ is the sphericity) [24]:

$$ {{k}_{\text{eff}}}={{k}_{\text{f}}}\frac{{{k}_{\text{p}}}+\left( n-1 \right){{k}_{\text{f}}}-\left( n-1 \right)\varphi \left( {{k}_{\text{f}}}-{{k}_{\text{p}}} \right)}{{{k}_{\text{p}}}+\left( n-1 \right){{k}_{\text{f}}}-\varphi \left( {{k}_{\text{f}}}-{{k}_{\text{p}}} \right)}. $$
(3)

For spherical particles, n  = 3, and the Hamilton–Crosser equation transforms into Maxwell’s relationship. Both models are the most widely used models for comparing the experimental data with theoretical expectations. In addition, other researchers augmented the models to take into account the particle size distribution [25], structure of their shell [26–28], contact resistance [29], etc. All the suggested equations adequately describe the behavior of common suspensions, but fail to predict the behavior of nanofluids. Therefore, various hypotheses on specific mechanisms of the heat transfer in nanofluids were put forward. Enhancement of the thermal conductivity by colloidal particles is often attributed to microconvection resulting from their Brownian motion [30–32] and to the complex effect of other forces: electrostatic (due to the presence of an electrical double layer on the particle surface), van der Waals, and hydrodynamic [33]. The particle collision and high drift velocity can also increase the thermal conductivity [34]. In addition, the thermal conductivity of nanofluids is also influenced by the formation of an ordered liquid layer around particles [35]. It is believed that this layer can both favor the thermal conductivity [36] and, on the contrary, increase the thermal resistance of the interface [37, 38]. As shown in numerous studies, one of the main causes of the anomalous thermal conductivity of nanofluids is structuring of the primary dispersed particles with the formation of fractal aggregates [39] and clusters of different density [40–43]. Nevertheless, there is still no common opinion on causes of the enhancement of the thermal conductivity by solid nanoparticles; various theories and mechanisms are described in [4, 12, 44].

When considering the thermal conductivity of nanofluids, attention is usually paid to dispersions of solid particles; however, numerous theoretical and experimental studies of the thermal conductivity of emulsions have also been performed. There are three main equations for predicting the effective thermal conductivity of two-phase mixtures of fluid media from the thermal conductivities of the constituent liquids [45]. One of them is the equation identical to Eq. (1), suggested by Tareev, who, like Maxwell, proceeded from the assumption that a thermal field in a two-phase system is similar to an electric field [46]. The other two equations for predicting the thermal conductivity of emulsions can be obtained only assuming an idealized state of the mixture, i.e., that the mixture consists of homogeneous oil and water layers parallel (Fig. 1a) or perpendicular (Fig. 1b) to the thermal flux direction. In such cases, the thermal conductivity of the system can be calculated from the volume fractions and specific thermal conductivities of the pure components. For the layers parallel to the heat transfer direction,

$$ {{k}_{\text{eff}}}={{k}_{1}}\left( \frac{{{A}_{1}}}{{{A}_{1}}+{{A}_{2}}} \right)+{{k}_{2}}\left( \frac{{{A}_{2}}}{{{A}_{1}}+{{A}_{2}}} \right)={{k}_{1}}{{\text{ }\!\!\varphi\!\!\text{ }}_{1}}+{{k}_{2}}{{\text{ }\!\!\varphi\!\!\text{ }}_{2}}, $$
(4)

whereas for the layers perpendicular to this direction,

$$ \frac{1}{{{k}_{\text{eff}}}}=\frac{1}{{{k}_{1}}}\left( \frac{{{L}_{1}}}{{{L}_{1}}+{{L}_{2}}} \right)+\frac{1}{{{k}_{2}}}\left( \frac{{{L}_{2}}}{{{L}_{1}}+{{L}_{2}}} \right)=\frac{{{\text{ }\!\!\varphi\!\!\text{ }}_{1}}}{{{k}_{1}}}+\frac{{{\text{ }\!\!\varphi\!\!\text{ }}_{2}}}{{{k}_{2}}}. $$
(5)

Subscripts 1 and 2 refer to the first and second phases; A and L are the layer sizes in the plane perpendicular to the heat transfer direction and coinciding with it, respectively (Fig. 1).

Fig. 1.
figure 1

Idealized structure of two-phase systems: homogeneous layers of oil and water, arranged (a) parallel or (b) perpendicular to the thermal flux direction [45]Footnote

Publication permission of October 2, 2020, © 1958 American Chemical Society.

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These equations are also applicable to calculating the thermal conductivity of suspensions. The variant of the parallel arrangement of the layers of different phases relative to the heat transfer axis (Fig. 1a) is identical to the variant of the parallel arrangement of threads of the same phases. To a first approximation, the set of threads can be considered as a percolation structure of filler particles that arises under definite conditions owing to particle sticking to each other [47, 48], fully pierces the continuous phase matrix, and imparts to it the gel properties [49]. It can be expected that the formation of the percolation network from filler particles will ensure the maximal thermal conductivity of the mixture. The perpendicular arrangement of filler layers in a liquid matrix (Fig. 1b) is difficult to implement; it can be reached only at interfacial activity of the particles and only in a polymer matrix tending to microphase segregation with the formation of lamellar morphology, e.g., in styrene–isoprene–styrene triblock copolymer [50]; in this case, the modified material will be characterized by the anisotropic thermal conductivity, and its increment will be maximal in two directions of the three-dimensional space [according to Eq. (4)] and, on the contrary, minimal in the third direction [according to Eq. (5)].

Equations (4) and (5) are nothing else than the equations for calculating the weighted mean quantities: arithmetic mean and harmonic mean, respectively; they can be presented by a common equation [51, 52]

$$ {{k}_{\text{eff}}}={{({{\text{ }\!\!\varphi\!\!\text{ }}_{1}}k_{1}^{n}+{{\text{ }\!\!\varphi\!\!\text{ }}_{2}}k_{2}^{n})}^{{}^{1}/{}_{n}}}, $$
(6)

in which the exponent n is equal to 1 and –1 in calculation of the arithmetic mean and harmonic mean, respectively. In addition, by taking n → 0, we can find the weighted geometric mean; it is convenient to present the equation for its calculation in the logarithmic form (as a rule of the logarithmic additivity of the component thermal conductivities):

$$ \text{log}~{{k}_{\text{eff}}}~=\text{ }{{\varphi }_{1}}\text{log }{{k}_{1}}+\text{ }{{\varphi }_{2}}\text{log }{{k}_{2}}. $$
(7)

Numerically, Maxwell’s equation (1) corresponds to the case when the exponent n in Eq. (6) depends on the dispersed phase concentration (as the weighted harmonic mean) and is equal to −0.34 at φ → 0 (i.e., without filler) and to −0.76 at φ → 1.

Let us compare the thermal conductivities calculated as the arithmetic, geometric, and harmonic mean values and by Maxwell’s equation for a model suspension in which the specific thermal conductivities of the particles and continuous medium strongly differ: 1000 and 1 W m–1 K–1, respectively (Fig. 2). In the semilog coordinates, the concentration dependence of the geometric mean is a straight line connecting the points that correspond to the specific conductivities of the pure components. Such dependence of the thermal conductivity can be expected for homogeneous mixtures, whereas in the case of heterogeneous mixtures the thermal conductivity will show positive or negative deviations from this dependence. The negative deviation is characteristic of random distribution of particles in the continuous phase, whereas the positive deviation is possible only in the case of particle structuring. In the first case, significant increase in the thermal conductivity cannot be expected even at high degrees of filling, whereas in the second case such increase is possible.

Fig. 2.
figure 2

Theoretical calculations of the thermal conductivity of a model suspension in which the thermal conductivity of the particles is 1000 times higher than that of the continuous medium. The thermal conductivity is calculated as weighted (1) arithmetic, (2) geometric, and (4) harmonic mean and (3) using Maxwell’s equation.

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As a rule, the researchers’ goal is to radically (by orders of magnitude) increase the thermal conductivity of a liquid medium by small additions of heat-conducting particles (to a total concentration within 1–5 vol %). When the thermal conductivities of the medium and particles differ by a factor of 1000 (this is close to the maximal practically attainable difference), simple dispersion of the particles will not allow the thermal conductivity to be increased by more than 16% even at their 5% content (Table 1). Significant increase in the thermal conductivity at low volume fraction of particles can be reached only under the conditions of arithmetic averaging. This can be hoped for only when the particles form a three-dimensional percolation structure. In this case, the contacts between particles in the system should be phase contacts rather than coagulation contacts (through a thin interlayer of the continuous medium) [53].

Table 1. Theoretical thermal conductivity of a suspension with low thermal conductivity of the continuous medium (k =1 W m–1 K–1) and high thermal conductivity of the dispersed phase (k =1000 W m–1 K–1)
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The majority of researchers do not consider the possibility of structuring of filler particles in the matrix, and this is not their aim. Furthermore, as far as circulating coolants are concerned, it is impossible to enhance their thermal conductivity by structuring, because the percolation structure of the dispersed phase is broken in a flow, irrespective of the phase state of the particles [54]. Under such conditions, there are no grounds to hope that the thermal conductivity of the mixture will exceed the value predicted by Maxwell’s equation. The possibility of increasing the thermal conductivity of the medium by using particles of materials with the highest thermal conductivity seems to remain in this case. However, even when particles differ in the thermal conductivity by one or two orders of magnitude, their theoretical effect on the thermal conductivity of a liquid is still similar (Table 2). In other words, we cannot expect a significant increase in the thermal conductivity of a system due solely to the difference between the thermal conductivities of the components: The particle concentration is the major factor determining the thermal conductivity of the suspension.

Table 2. Thermal conductivity of suspensions of different particles in water (k = 0.607 W m–1 K–1), according to Maxwell equation
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Furthermore, when considering properties of solid particles, we should take into account possible anisotropy of their properties. Graphene and graphene oxide attract attention of researchers dealing with enhancement of the thermal conductivity of media owing to their extremely high specific thermal conductivity, which is estimated for both compounds at 2000 to 4000 W m–1 K–1. However, this value characterizes the thermal conductivity in two directions in the graphene particle plane, whereas in the third, perpendicular direction the thermal conductivity is relatively low, 6 m–1 K–1 [56]. The particle dispersion leads to their random arrangement in the medium; therefore, the averaged specific thermal conductivity of graphene should be used for calculating the thermal conductivity of the mixture. This quantity can be calculated using Eq. (6), taking into account that the thermal conductivity of graphene in two directions (φ1 = 0.667, k1 = 3000 W m–1 K–1) differs from that in the third direction (φ2 = 0.333, k2 = 6 W m–1 K–1). The use of arithmetic, geometric, and harmonic mean values allows the mean thermal conductivity of graphene to be estimated at 2012, 390, and 18.1 W m–1 K–1, respectively. We have measured the thermal conductivity of a pressed graphene pellet by the steady-state method and obtained the value of 21.3 W m–1 K–1. Thus, when properties of particles are anisotropic, their effective thermal conductivity is the weighted harmonic mean of the components of the thermal conductivity tensor, which allows estimation of the thermal conductivity of other anisotropic materials, e.g., carbon nanotubes (the value of 8.9 W m–1 K–1 is obtained). In addition, this fact allows a conclusion that wide use of allotropic modifications of carbon for enhancing the thermal conductivity of liquid media is not justified, because they are inferior in the effective thermal conductivity to pure metals and to many binary compounds of metals with nonmetals (oxides, nitrides, etc.).

Thus, from the viewpoint of theory, a manifold increase in the thermal conductivity can be reached only for a resting liquid when a specific structure of the filler is formed in it; otherwise, the gain cannot exceed several tens of percents, and even this level can be reached only at high (>10 vol %) content of particles with high thermal conductivity. Let us consider available data of different research teams on the main methods and results of studying the thermal conductivity of dispersions.

EXPERIMENTAL

Methods of Experimental Studies

There are numerous methods for measuring the thermal conductivity. Each of them is suitable for a definite temperature interval. On the whole, they can be subdivided into steady-state and non-steady-state methods differing in that the temperature of the material either does not vary or varies in the course of the measurement [57].

Steady-state methods for measuring the thermal conductivity are based on the Fourier law; these methods are characterized by direct contact of the sample with the heating element. The use of steady-state methods for determining the thermal conductivity is restricted by the following factors: First, the method is mainly used for friable materials; hence, large contact area is required, which makes the method sensitive to heat loss and decreases the measurement accuracy; second, long time (up to several hours) is required to attain the steady state [58]. Absolute and relative steady-state methods are distinguished [59]. Absolute methods are mainly applied to specimens of rectangular or cylindrical shape, and the thermal conductivity is calculated in accordance with the Fourier law. The main problem of the absolute method for measuring the thermal conductivity consists in accurate measurement of both the thermal flux through the sample (taking into account the heat loss) and the arising temperature differential. The heat loss is caused by convection, heat exchange with the environment, and heating of thermocouple wires [60, 61].

In contrast to the absolute method, the relative method does not require accurate evaluation of the thermal flux and is based on the use of a reference sample with the known thermal conductivity [62–64]. This method allows reaching higher accuracy when the thermal conductivity of the sample is comparable to that of the reference. The relative methods include, in particular, the method of radial thermal flux [65], allowing the heat loss in high-temperature tests to be decreased, and the method of parallel measurements, intended for samples of small size [66].

Non-steady-state methods for measuring the thermal conductivity are free of such drawback typical of steady-state methods as high role of the heat loss [67]. In these methods, the heat is fed either by a pulsed external action or by a periodic thermal flux. One of the most demanded non-steady-state methods for measuring the thermal conductivity of solids is the hot-wire method [68] based on recording a short increase in the temperature of a thin vertical metal wire of infinite length upon stepwise application of electric voltage [69–71]. The wire is immersed in a liquid and acts simultaneously as a heating element and as a resistivity thermometer. The theory of the non-steady-state hot-wire method has been developed comprehensively, and the thermal conductivity is measured with high accuracy [72]. There are no convection heat exchange and no need for calibration; the experiment time is as short as 1 s, and the method itself is suitable for measuring the properties of materials with low thermal conductivity.

One more widely used non-steady-state method is the planar source method, in which a metal plate or disk is heated instead of a wire [73]. This method allows evaluation of the thermal conductivity of materials in the range from 0.05 to 500 W m–1 K–1 with the upper temperature limit of 500 K [59]. Non-steady-state methods for measuring the thermal conductivity also include the method of temperature oscillations [74, 75] and the three-frequency (3ω) method [76–78]. The method of temperature oscillations is based on measuring the temperature conductivity of a liquid placed between two heaters, followed by calculating the thermal conductivity taking into account the volume heat capacity of the sample. In the three-frequency method, alternating current of frequency ω is passed through a metal wire suspended in a liquid. This current generates periodic heating of the medium and wire with frequency 2ω, which, in turn, causes oscillation of the voltage on the wire with frequency 3ω; its amplitude is measured and used for the subsequent calculation of the thermal conductivity [79]. This method is well suited for measuring the temperature dependence of the thermal conductivity of nanofluids.

The thermal conductivity of various media can also be measured using some other methods, which have found limited use: linear source [80, 81], optical [82], and diffusion [83] methods. Thus, the methods used are diverse and are based on different physical principles of studying properties. This is rather a drawback; it can lead to different results of measuring the thermal conductivity of the same samples by different research groups. Furthermore, many methods are based on using reference samples whose thermal conductivity can differ by orders of magnitude from that of the media studied, which can lead to high measurement errors.

Preparation of Nanofluids

Preparation of stable nanofluids with good dispersion of nanoparticles is of decisive importance for their use. There are two main methods for preparing nanofluid: one-step and two-step [44]. In the two-step process, nanoparticles are first synthesized by the vapor condensation in an inert medium [84], after which they are dispersed in a liquid. In this case, the nanoparticle agglomeration is possible, and the preparation of stable nanofluids may require effective mixing and low nanoparticle concentration. The advantage of the two-step process is that the gas-phase procedure allows preparation of nanoparticles in large volumes, which reduces the production cost of the final nanofluid.

In the direct evaporation method, nanoparticles are synthesized and dispersed in a liquid in one step. As in the case of the gas-phase synthesis, this technology involves evaporation of the starting material of future particles, which is performed in a vacuum. This is followed by condensation of the vapor directly in the medium of a flowing liquid. The agglomeration of the nanoparticles formed is thus reduced to a minimum [85]. The drawbacks of the method are that the vapor pressure of the liquid should be very low and that this procedure is suitable for preparing only a limited amount of nanofluids. Therefore, the majority of researchers use the two-step process. As a result, research teams operating with the same starting materials can obtain dispersions with different properties due to different mixing procedures leading to different structural organization of nanoparticles in the continuous phase.

Effect of Various Factors on the Thermal Conductivity of Nanofluids

The results of numerous studies allow revealing several factors that influence the gain in the thermal conductivity of nanofluids. These include the volume concentration of particles, their material, size, and shape, the kind of the base liquid, the temperature, and the presence of additives [12].

Particle material. Suspensions of nanoparticles of metal oxides Al2O3 and CuO [6, 33] were the first nanofluids for which a slight but abnormal increase in the thermal conductivity was revealed (2–18% higher than the values predicted by Maxwell’s model). Preparation of nanofluids based on metal particles, e.g., of copper particles in transformer oil, became a large step forward from the viewpoint of the thermal conductivity enhancement [8]. However, the best results were obtained when using carbon nanotubes. The most pronounced (160%) gain in the thermal conductivity was observed when filling motor oil with multiwall carbon nanotubes to a concentration of 1 vol % [9]; similar results were also noted in other studies [86–88]. In the general case, when using particles with low thermal conductivity (e.g., oxides), their material does not noticeably influence the gain in the thermal conductivity of the medium, whereas the use of particles with higher thermal conductivity (metal or carbon-containing particles) leads to a similar gain in the thermal conductivity at considerably lower degree of filling. Nevertheless, nanoparticles with high conductivity are not always effective in enhancement of the thermal conductivity of nanofluids [89, 90].

Volume concentration of particles. With an increase in the volume concentration of particles, the thermal conductivity of the modified medium increases (as can be expected in accordance with Maxwell’s equation). However, experiments demonstrate an abnormal increase in the thermal conductivity of nanofluids at a very low volume fraction of nanoparticles [9, 91]. In particular, nonlinear dependence of the gain in the thermal conductivity on the nanoparticle concentration was noted [92]. In addition, negative deviation of the thermal conductivity from the values predicted by Maxwell’s theory was noted in some papers [93, 94]; this was attributed to aggregative instability of the systems [94].

Particle size. On the one hand, the particle size does not influence the thermal conductivity of the mixture in accordance with Maxwell’s model. On the other hand, a decrease in the effective diameter of particles not only leads to an increase in the specific area of contact between the particles and liquid medium, but also can give rise to specific mechanisms of the enhancement of the heat transfer [36, 95, 96]. For example, an abnormal (3.2–7%) increase in the thermal conductivity of toluene on introducing gold nanoparticles (size ~10–20 nm, concentration 0.0013–0.011 vol %) has been reported [97]. Similar result was obtained for copper nanoparticles in ethylene glycol: 40% increase in the thermal conductivity at small particle size (10 nm) and low particle concentration (0.3%) [7].

An increase in the thermal conductivity of a medium with a decrease in the particle size is reported most frequently, e.g., an 11–24% increase in the thermal conductivity of a 0.1% aqueous dispersion of copper nanoparticles with a decrease in their diameter from 300 to 50 nm [98] or a 9–15% increase in the thermal conductivity of a 1% aqueous dispersion of aluminum oxide nanoparticles with a decrease in their diameter from 150 to 11 nm [99, 100]. On the other hand, the gain in the thermal conductivity of ethylene glycol containing 1.8% Al2O3 decreases from 10 to 6% with a decrease in the particle size from 60 to 15 nm [101]. Presumably, the properties of nanofluids are influenced by nanoparticle agglomeration, which is enhanced with a decrease in the particle size and it not taken into account by many authors who believe that the particle size and shape in the dispersion coincide with producer’s data.

Particle shape. According to Hamilton–Crosser equation (3), the thermal conductivity of dispersion should increase with a decrease in the particle sphericity. Indeed, experiments show that elongated particles enhance the thermal conductivity more efficiently than spherical particles do. For example, in going from spherical (15 nm) to rodlike (10 × 40 nm) titanium dioxide nanoparticles, the thermal conductivity gain for their 5% aqueous dispersions increases from 30 to 33% [10]. Nevertheless, the available experimental data on the enhancement of the thermal conductivity of media by anisometric nanoparticles are scarce. On the other hand, the latter nanoparticles are able to form percolation solid-like structures at low concentrations [102–104], which makes them promising for preparing stationary media with high thermal conductivity or media whose thermal conductivity can be varied stepwise by a factor of several times by applying an external mechanical field.

Base liquid. According to Maxwell’s equation (1), the thermal conductivity of dispersion is primarily determined by that of the base liquid. In addition, the affinity of the liquid for the particle surface influences the agglomeration and structuring of the particles: Nanoparticles of the same material can both undergo deagglomeration to primary particles and form a solid-like structure, depending on the medium [105]. Experimental studies show that a larger gain in the thermal conductivity with the same nanoparticles is observed when using them for the modification of a medium with a lower thermal conductivity. For example, introduction of 1% multiwall carbon nanotubes (15 nm in diameter, 30 μm long) increases the thermal conductivity of decene (k = 0.131 W m–1 K–1), ethylene glycol (0.256 W m–1 K–1), and water (0.607 W m–1 K–1) by 19.6, 12.7, and 7%, respectively [106]. Similar result was obtained for 5% dispersions of spherical particles of aluminum oxide (60.4 nm), increasing the thermal conductivity of water, ethylene glycol, and mineral oil by 21, 30%, and 39%, respectively [107, 108]. On the other hand, upon introduction of the same amount of Al2O3 particles but of smaller diameter (28 nm), this trend is broken: The thermal conductivity of water, ethylene glycol, and mineral oil increases by 15, 25, and 13%, respectively [33]. Apparently, the latter result is due to agglomeration of nanoparticles in water and mineral oil and, on the contrary, to their efficient dispersion in ethylene glycol, as demonstrated previously for silicon dioxide particles exhibiting similar properties [47, 105].

Temperature. With increasing temperature, the thermal conductivity of polar liquids increases and that of nonpolar liquids decreases [109]. The experimental results are, on the whole, consistent with this trend; however, more pronounced increase in the thermal conductivity of nanofluids with temperature was noted in some studies; this effect was attributed to the acceleration of the Brownian motion of particles [6, 75, 97].

Effect of additives. To prevent agglomeration of nanoparticles and stabilize dispersions, surfactants are added such as thioglycolic acid, sodium dodecyl sulfonate, hexadecyltrimethylammonium bromide, etc. Their addition leads to a certain increase in the thermal conductivity of the dispersions, which may be due to a decrease in the particle size [7, 86, 110].

Behavior of nanofluids in a flow. As noted above, the thermal conductivity of a liquid at rest or in a flow can differ significantly if the nanoparticles are capable of structuring to form a percolation network leading to viscoplastic behavior [111]. The viscoplasticity means solid-like behavior of a system at low stresses and its fluidity at high stresses that exceed the yield stress and break the three-dimensional structure formed by the particles [112]. However, the flow of a nanofluid can be both laminar and turbulent; in addition, its use under the conditions of boiling is possible [113, 114]. Therefore, it is important to measure the heat-transfer characteristics of nanofluids directly under the conditions of their use [115], although experimental studies in this field are scarce. Despite the fact that the enhancement of the thermal conductivity of a liquid is an important parameter of the improvement of the heat transfer, in practice different coolants are compared using the heat transfer coefficient. Its increase favors a decrease in the size of thermal power systems and improvement of the efficiency of the power and fuel utilization [12].

When using aqueous dispersions of aluminum oxide in the laminar flow mode, both the heat transfer and the thermal conductivity increase with an increase in the volume concentration of particles. However, the gain in the heat transfer is more pronounced and reaches 40% (at 1.6% particle content), whereas the gain in the thermal conductivity in all the cases does not exceed 15% [116]. Different influence on the thermal characteristics is attributed to microconvection of nanoparticles and to their interaction with the liquid. Similar experiments with carbon nanotubes in water also demonstrate enhancement of the heat transfer [117].

Under the conditions of turbulent flow of water-based nanofluids containing Al2O3, TiO2, or Cu nanoparticles, the heat transfer is not appreciably enhanced [118, 119]. At equal volume concentrations (e.g., 1 vol %), the gain in the heat transfer is the highest for Cu particles (15%), lower for Al2O3 particles (10%), and the lowest for TiO2 particles (3%), which agrees with a decrease in the thermal conductivity in this series (400, 40, and 8.6 W m–1 K–1, respectively [55]). In this case, the thermal conductivity gain is comparable to the heat transfer gain.

When using aqueous dispersions of Al2O3 under the conditions of boiling, the presence of particles does not lead to an increase in the heat transfer coefficient [120]. Furthermore, the addition of particles decreases the heat transfer irrespective of the particle size, although it slightly enhances the thermal conductivity. Poor characteristics of the nanofluid are attributed to the particle deposition on heated surfaces. However, a decrease in the particle concentration (to 0.31 vol % and less) and their electrostatic stabilization (by decreasing pH to 7.0) change the pattern and allow the heat exchange under the conditions of boiling to be enhanced by 20–40% [121].

Generalized Experimental Data

The experimental data from numerous studies on the thermal conductivity of dispersions of various nanoparticles in water, ethylene glycol, mineral oil, and other media are summarized in Table 3. Silicon compounds (SiO2, SiC), metal oxides (TiO2, Fe3O4, Al2O3, CuO), metals (Fe, Al, Au, Cu, Ag), and allotropic modifications of carbon (single- and multiwall carbon nanotubes, diamond, graphene) were used as nanoparticle materials. Stabilizers such as thioglycolic acid, sodium dodecyl sulfonate and hexadecyltrimethylammonium bromide were added to some dispersions. Data for Table 3 were taken from different sources and are not complete for the whole period of studying heat-conducting nanofluids, but they representatively reflect the main trends of research in this field. Nanoparticles are ranked in the order of increasing the thermal conductivity of the material. All the measurements were made at room temperature unless otherwise indicated.

Table 3. Experimental data on thermal conductivity of dispersions of various nanoparticles
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SWCNT, single-wall, and MWCNT, multiwall carbon nanotubes; EG, ethylene glycol; TA, thioglycolic acid; SDS, sodium dodecyl sulfonate; HTAB, hexadecyltrimethylammonium bromide; I, 3ω method; II, hot-wire method; III, steady-state method; IV, diffusion method; V, method of temperature oscillations; VI, otical method; VII, linear source method; the temperature is indicated if it is other than room temperature.

First and foremost, data in Table 3 demonstrated very large scatter in the thermal conductivity values measured for dispersions of the same composition by different research groups. This illustrates the dependence of the thermal conductivity of nanofluids on the preparation procedure, initial nanoparticle size, addition of stabilizers, conditions and methods of tests, etc. Nevertheless, certain conclusions can be made.

The largest gain in the thermal conductivity (160%) was reached when modifying the oil with 1 vol % carbon nanotubes [9]. Fairly good results (34–125% increase in k) were also reached when using other allotropic modifications of carbon (diamond and graphene, 0.6–5 vol %). The gain in the thermal conductivity when using graphene decreases with a decrease in the number of nonexfoliated graphene layers in the nanoparticle, which may be due to the break of thin graphene plates under the conditions of mixing: The gain in the thermal conductivity of paraffin upon its filling with 5 vol % graphene particles with a thickness of 1–5, 6–8, and 11–15 nm is 6.3, 107.5, and 113.7%, respectively [141]. Abnormally high gain in the thermal conductivity with very small additions (0.0001–0.05 vol %) of copper and silver nanoparticles can also be noted. In all these cases of significant enhancement of the thermal conductivity, the gain is close to that expected in the case of arithmetic averaging of the thermal conductivities of the components [Eq. (4)], i.e., in the case of formation of a percolation structure from heat-conducting nanoparticles in the volume of a resting liquid. However, there are no experimental evidences of the formation of such structures in the samples because the researchers did not consider this possibility. On the other hand, the three-dimensional structuring could be readily detected by the method of rotation rheometry by the solid-like behavior of dispersions and/or presence of the yield stress, including samples with low (0.001–0.01%) concentrations of metal nanoparticles [140, 143].

Analysis of the experimental data on the influence of the nanoparticle concentration on the relative thermal conductivity of dispersions [144] in connection with the theoretical predictions by Eqs. (1), (4), (5), and (7) (Maxwell’s relationship and various alternatives of calculating the mean values) reveals no common universal dependence (Fig. 3). The majority of points (~60%) lie in the field bounded by two straight lines corresponding to the geometric mean values and Maxwell’s relationship (the minor fraction of points are slightly above or below these two straight lines, which may be due to the measurement error). Numerous hypotheses based on small size of nanoparticles (see Introduction) were put forward to account for positive deviations of the thermal conductivity values from Maxwell’s theory. Apparently, the gain in the thermal conductivity due to specific effect of nanoparticles cannot exceed the gain expected when nanoparticles and the medium form a homogeneous mixture (i.e., in the case of logarithmic additivity of the component thermal conductivities). On the other hand, approximately 10% of points are close to the straight line corresponding to the arithmetic mean. These points correspond to dispersions of metal (copper, gold, iron) nanoparticles whose structure was not studied in detail, which does allow unambiguous conclusions on the causes and mechanism of the abnormal increase in the thermal conductivity of these systems.

Fig. 3.
figure 3

Relative gain in the thermal conductivity of liquids containing different kinds of nanoparticles in different concentrations [144]Footnote

Publication permission of October 2, 2020, © 2007 Elsevier.

. The straight lines show the results of calculating the thermal conductivity of the dispersions as weighted (1) arithmetic, (2) geometric, and (4) harmonic mean and (3) using Maxwell’s equation.

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CONCLUSION

Liquid dispersions of nanoparticles (nanofluids) continue to attract researchers’ interest owing to high demand for them and diversity of potential application fields. However, their thermal characteristics leave much to be desired: In most cases, the relative gain in the thermal conductivity on adding nanoparticles does not exceed 10%, the 15–40% gain is a good result, and more significant gain (which, however, does not exceed 110–160%) was reported only in a few papers. In other words, introduction of nanoparticles allows only limited but not manifold increase in the thermal conductivity of liquids. Furthermore, the available extensive experimental data are contradictory, and the suggested mechanisms of the enhancement of the thermal conductivity of media by nanoparticles are diverse. Today there is no commonly accepted mechanism or theory for predicting the gain in the thermal conductivity of nanofluids. None of the models predicts their thermal conductivity accurately. This is cased by diversity of the measurement methods used (with unclear errors) and by the fact that the majority of researchers do not pay due attention to determination of the real size of nanoparticles in dispersions, to possible anisotropy of their properties, and to their three-dimensional structuring, which are influenced both by dispersion-stabilizing additives and by the mixing procedure. Without standardization of procedures for measuring the thermal conductivity of liquids and without compulsory characterization of their structure, this research field will remain on the low, mostly speculative level. Furthermore, it should be understood that the thermal conductivities of media that are resting, boiling, or flowing in different modes can differ essentially. Therefore, it is necessary to expand the experimental work toward studies of nanofluids under the conditions of their probable use.