# Thermal Conductivity of Nanofluids: A Review on Prediction Models, Controversies and Challenges

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## Abstract

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## 1. Introduction

## 2. Thermal Conductivity Measurements Overview

#### 2.1. Transient Hot Wire Method

#### 2.2. Transient Plane Source Theory

#### 2.3. Temperature Oscillation

#### 2.4. 3ω Method

#### 2.5. Laser Flash Method

#### 2.6. Parallel Plate

#### 2.7. Coaxial Cylinders

## 3. Parameters That Affect Thermal Conductivity

#### 3.1. Nanoparticles (NPs)

#### 3.1.1. Type

_{2}NPs than when using Al

_{2}O

_{3}NPs, despite the lower values of the thermal conductivity of the TiO

_{2}NPs.

#### 3.1.2. Size

_{2}O

_{3}NPs of different sizes and registered a reduction of the thermal conductivity with increasing sizes. Anoop et al. [31] also compared NFs with alumina NPs of 45 and 150 nm of dimensions. The higher enhancement was reported for the smaller particles. Although most of the results presented in the available literature agree with that proposition, a few others suggest opposite conclusions. Jang and Choi [32] registered an increase of the thermal conductivity with increasing particle size for different types of NPs. Timofeeva et al. [33] compared water-based NFs with four different sizes of SiC NPs and obtained higher values of thermal conductivity with the bigger particles. It should also be noted that for nearly micrometer-sized particles, the Brownian motion is not present, and the thermal conductivity remains unchanged. For a small range of particle size, some authors reported a linear increase of conductivity with increasing particle size due to the reduced interfacial thermal resistance of the larger particles [5,26].

#### 3.1.3. Shape

_{2}NFs when using rod-shape NPs than when using spherical NPs. Jeong et al. [35] obtained similar results when comparing spherical and nearly rectangular-shaped ZnO NPs. Timofeeva et al. [36] compared NFs with four different shapes of Al

_{2}O

_{3}NPs and concluded that the NF with higher thermal conductivity value was the one with cylindrical NPs, followed by the one with brick shape NPs. In addition, the NFs with blade shape and platelet shape particles presented similar values and were also the lower ones among the testing group. Glory et al. [37] investigated the increase of the thermal conductivity using Multi-Walled Carbon Nanotubes (MWCNT)of different lengths, obtaining greater values for the NFs with the longest nanotubes.

#### 3.2. Suspension

#### 3.2.1. Base Fluids

#### 3.2.2. Concentration

_{2}O

_{3}and CuO, measured experimentally the thermal conductivity of the NFs at low volume fraction range (lower than 0.05%) and found that the thermal conductivity ratios increase almost linearly with volume fraction, but with different rates of increase for each group of NPs tested. However, some reports noted a reduction of the thermal conductivity for high concentration values. The stability of the NF is likely to play a major role in the determination of the ideal concentration [5,26].

#### 3.2.3. Agglomeration

#### 3.3. Nanofluids

#### 3.3.1. Preparation Methods

#### 3.3.2. Temperature

_{2}NFs. The enhancement of the thermal conductivity with increasing temperature is thought to be caused by the improvement of the Brownian motion and the reduction of the surface energy of the particles. On the contrary, a reduction of the thermal conductivity with increasing temperature was reported for NFs with non-spherical particles, which might indicate that the aspect ratio of the particle has a relevant influence [5,26].

#### 3.3.3. pH

_{2}and TiO

_{2}NPs.

#### 3.3.4. Additives, Surfactants and Solvents

_{2}O

_{3}and CuO NPs dispersing in water, ethylene glycol, and water with SDBS as a surfactant.

#### 3.3.5. Sonication Time

#### 3.3.6. External Magnetic Field

_{2}O

_{3}or Ni and carbon Nanotubes [68,69]. Also, these studies have demonstrated that as the magnitude of the applied magnet field decreases, the time to reach the maximum peak value of the thermal conductivity will increase.

#### 3.3.7. Aggregation

## 4. Prediction Models for Thermal Conductivity and Other Properties of the NFs

#### 4.1. Thermal Conductivity Empirical Models

#### 4.2. Thermal Conductivity Machine Learning-Based Models

#### 4.3. Density

#### 4.4. Specific Heat Capacity

#### 4.5. Viscosity

## 5. Controversies in Thermal Conductivity Measurements

_{2}O

_{3}or TiO

_{2}prepared with ethylene glycol/water, studied the effect of adding a dispersant to evaluate the influence in the specific heat. They observed an increase of the specific heat but, it was gradually decreased with the increase of wt% of Al

_{2}O

_{3}and TiO

_{2}. Without addition of the dispersants, some researchers have shown that the specific heat of NFs decreases with the increase of volume fraction [101,102,103].

## 6. Backdraws and Future

- Complexity and high cost of NFs preparation;
- Long-term stable NFs are difficult to produce. The influence of the production should be further studied, namely the sonication time, volume fraction, and type of the NFS in order to avoid the sedimentation and agglomeration of the NPs in the BF and to achieve optimal performance;
- Lack of a common protocol for the manufacture and analysis of the thermal transport mechanisms in this type of fluids;
- Methods to scale-up production for commercialization are still in development;
- Some of the classical experimental fitting prediction models for thermal properties are not the most suitable for the estimation of the thermal conductivity, dynamic viscosity and density of the NFs;
- The classical modeling does not provide a fast prediction of the thermal properties, which may slow down the study and overall applicability of the NFs;
- Nowadays, it is clear the need to use the recent statistical data-driven machine learning models to obtain faster predictions of the thermal properties of the NFs. Those models are also more stable and sensitive than the empirical ones. However, for a in-depth knowledge on how the recent models work, the researchers must get proficient skills in machine learning methodology and in the most common data-driven models and algorithms;
- The proper machine learning model and algorithms should be chosen according to the dimension of the sample, the available computing resources, and the required prediction performance (modeling stability and sensibility).

- The stability and durability of the nanofluids should be improved by optimizing the concentration of NPs and base fluid characteristics (e.g., chemical, viscosity). The stability of nanofluids should also be predicted by further analysis of the surface tension of the nanofluid vs. time;
- The general properties of the NFs should be improved by optimizing the preparation procedures (e.g., sonication time);
- The influence of the solvents should be further studied: the use of high polar solvents like the DMF (Dimethylformamide) and THF (Tetrahydrofuran) and non-polar solvents as hexane and heptane could be the right way to fully understand how polarity influences the alignment and thermal conductivity of the NFs;
- The influence of the polarity in the alignment of the NPS on the base fluid can also be assessed by the use of different surfactants with negative charge, such as for instance CATB (Cetyltrimethylamonium bromide);
- The impact of an applied magnetic field should be further studied, namely the value or range of values that make possible to reach the thermal conductivity maximum peak in less time;
- Additives for decreasing the NFs viscosity while maintaining the same level of thermal conductivity are of paramount importance to achieve the best performance of the NFs;
- The systems using nanofluids as working fluid should become more cost-effective, without the need of extra pumping power and expensive maintenance. This should be accomplished with the optimization of the microchannels configuration (e.g. number of channels, inlet/outlet positioning);
- Machine learning prediction models should be increasingly used in the future, since they provide faster and less expensive modeling of thermal properties of the NFs. Those data-driven models achieve more accurate and stable estimations than the classical ones. The path created by machine learning is beginning to clear up several doubts and backdraws, and it is a secure one for future studies and developments on the nanomaterials field of research.

## 7. Conclusions

- the thermal conductivity of an NF is greater than the one of the respective base fluids;
- the methods used to measure the thermal properties of NFs are still in an embryonic stage of development;
- the heat transfer coefficient of an NF is higher than the one of the respective base fluid;
- viscosity of NFs increases with the concentration of the NPs leading to higher pumping power requirements;
- NF long-term stability is mandatory for miniaturized fluidic systems, mainly mini or microchannels devices;
- the production costs for systems using nanofluids as working fluid are still high.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

AC-ANN | Ant Colony Artificial Neural Network |

ANN | Artificial Neural Network |

CART | Category and Regression Tree |

CS-GMDH | Group Method of Data Handling |

CFNN | Correlation Filter Neural Network |

GA-RBF | Genetic Algorithm Radial Basis Function |

GMDH | Group Method of Data Handling |

GRNN | General Regression Neural Network |

GS-GMDH | Generalized Structure Group Method of Data Handling |

LS-SVM | Least Square Support Vector Machine |

NSGA II | Non-dominated Sorting Genetic Algorithm |

PSO-ANN | Particle Sworn Optimization Artificial Neural Network |

RBF | Radial Basis Function |

RF | Random Forest |

RWLS-SVM | Recursive Weighted Least Squares Support Vector Machine |

SA-ANN | Simulate Anneal Artificial Neural Network |

SVM | Support Vector Machine |

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**Figure 1.**Number of scientific journal articles presented in the Science Direct database published between January of2000 and September of 2020.(Search date: 10.09.2020) (Please read the data availability statement).

**Figure 2.**Schematic diagram of steady and transient state methods to measure the thermal conductivity of nanofluids (NFs) (Adapted from [5]).

**Figure 3.**Different properties that affect the thermal conductivity of NFs (Adapted from [5]).

**Figure 4.**Determination of the function β derived from experimental data, adapted from [76].

**Figure 5.**Behavior of the different models for thermal conductivity prediction of an NF in function of the nanoparticle(NP) fraction [84].

**Figure 6.**Categories of prediction and regression machine learning algorithms (Adapted and abridged from [92].

**Figure 7.**Schematic structure of the Artificial Neural Network (ANN) machine learning model topology (Adapted and abridged from [92]).

Researchers | Mathematical Expressions | Included Parameters/Observations |
---|---|---|

Maxwell [73] | ${k}_{nf}={K}_{f}\left(\frac{{K}_{np}+2{K}_{bf}+2\phi \left({K}_{np}-{K}_{bf}\right)}{{K}_{np}+2{K}_{bf}-2\phi \left({K}_{np}-{K}_{bf}\right)}\right)$ | Includes the thermal conductivities of BF and NPs |

Hamilton and Crosser [72] | ${k}_{nf}={K}_{bf}\left(\frac{{K}_{np}+\left(n-1\right){K}_{bf}-\phi \left(n-1\right)\left({K}_{bf}-{K}_{np}\right)}{{K}_{np}+\left(n-1\right){K}_{bf}+\phi \left({K}_{bf}-{K}_{np}\right)}\right)$ | Includes the particle shape and composition and the thermal conductivities of BF and NPs |

Wasp [74] | ${k}_{nf}={K}_{bf}{\left(\frac{{K}_{np}+\left(n-1\right){K}_{bf}-\phi \left(n-1\right)\left({K}_{bf}-{K}_{np}\right)}{{K}_{np}+\left(n-1\right){K}_{bf}+\phi \left({K}_{bf}-{K}_{np}\right)}\right)}_{}$ | Includes the thermal conductivities of BF and NPs and the sphericity of the particles assumption |

Xue [71] | ${k}_{nf}={K}_{bf}{\left(\frac{1-\phi +2\phi \frac{{K}_{np}}{{K}_{np}-{K}_{bf}}ln\left(\frac{{K}_{np}+{K}_{bf}}{2{K}_{f}}\right)}{1-\phi +2\phi \frac{{K}_{f}}{{K}_{np}-{K}_{bf}}ln\left(\frac{{K}_{np}+{K}_{bf}}{2{K}_{bf}}\right)}\right)}_{}$ | Includes the particle shape and composition and the logarithmic progressions of the thermal conductivities of BF and NPs |

Xuan [75] | ${k}_{nf}={K}_{nf}{}_{Maxwell}+\frac{1}{2}{\rho}_{np}{c}_{np}{}_{p}\phi {\sqrt{2{D}_{B}}}_{}$ | Includes the temperature and viscosity of the BF, the average radius and viscosity of the clusters, and the Brownian motion |

Kleinstreuer [76] | ${k}_{nf}={K}_{nf}{}_{Maxwell}+\frac{5\times {10}^{4}}{{k}_{bf}}\beta \left(\phi \right){\rho}_{bf}{c}_{np}{}_{bf}f\left(T,\phi \right){\sqrt{\frac{{k}_{B}T}{{\rho}_{np}{d}_{np}}}}_{}$ | Includes the temperature and viscosity of the BF, the volume fraction and type of NPs, and the Brownian diffusion |

Xie [77] | ${k}_{nf}=1+3{\Theta}_{\phi T}+{\frac{3{\Theta}_{{\phi}^{2}T}^{2}}{1-{\Theta}_{\phi T}}}_{}$ | Includes the nanolayer thickness, the NPs volume fraction and radius and the thermal conductivity ratio between the NPs and the BF |

Avsec et al. [78] | ${k}_{nf}={k}_{bf}\left[\frac{{k}_{np}+\left(n-1\right){k}_{bf}-\left(n-1\right){\phi}_{nf}\left({k}_{np}-{k}_{bf}\right)}{{k}_{np}+\left(n-1\right){k}_{bf}+{\phi}_{nf}\left({k}_{np}-{k}_{bf}\right)}\right]$ ${\phi}_{nf}=\phi {\left(1+\frac{h}{r}\right)}^{3}$ | Includes the liquid layer thickness, thermal conductivities of BF and NPs, but not the particle size and the interface between the particles |

Jang and Choi [32] | ${k}_{nf}={k}_{bf}\left(1-\phi \right)+0.01{k}_{nano}\phi +18\times {10}^{6}\frac{{d}_{bf}}{{d}_{np}}\xb7{k}_{bf}R{e}_{d}^{2}P{r}_{bf}\phi $ | Includes the thermal condutivity and the diameter of the molecules of the BF, the particle fraction and diameter, the thermal conductivity of the NPs involving the Kapitza resistance (surface resistance),and the numbers of Reynolds and Prandtl |

Pak and Cho [79] | ${k}_{nf}=1+7.47\phi $ | Includes the geometry, diameter, and the surface resistance of the NPs |

Timofeeva et al. [33] | ${k}_{nf}={k}_{f}\left(1+3\phi \right)$ | Includes the geometry, agglomeration state, and the surface resistance of the NPs |

Yu and Choi et al. [80] | ${k}_{nf}={k}_{bf}\left[\frac{{k}_{pe}+2{k}_{bf}+2\left({k}_{pe}-{k}_{bf}\right){\left(1+\beta \right)}^{3}\phi}{{k}_{pe}+2{k}_{bf}-\left({k}_{pe}-{k}_{bf}\right){\left(1+\beta \right)}^{3}\phi}\right]$ ${k}_{pe}={k}_{np}\left[\frac{\left[2\left(1-\gamma \right)+{\left(1+\beta \right)}^{3}\left(1+2\gamma \right)\right]\gamma}{-\left(1-\gamma \right)+{\left(1+\beta \right)}^{3}\left(1+2\gamma \right)}\right]$ | Modified Maxwell model.Includes the nanolayer thickness |

Wang et al. [81] | ${k}_{nf}={k}_{bf}\left[1+\frac{\frac{3{f}_{q}\left(p\right)}{{p}_{0}}}{1-\frac{{f}_{q\left(p\right)}}{{p}_{0}}}\right]$ | Includes nanolayer thickness, particle size, temperature, volume fraction, and interaction between adjacent particles |

Chandrasekar et al. [82] | $MODELI:{k}_{nf}={k}_{bf}\left[\left(\frac{{c}_{np,nf}}{{c}_{p}}\right){\left(\frac{{\rho}_{nf}}{\rho}\right)}^{1.33}{\left(\frac{M}{{M}_{nf}}\right)}^{0.33}\right]MODELII:{k}_{nf}\phantom{\rule{0ex}{0ex}}={k}_{bf}\left[\frac{{k}_{np}+\left(n-1\right){k}_{bf}+\left(n-1\right)\left({k}_{np}-{k}_{bf}\right){\left(1+\beta \right)}^{3}\phi}{{k}_{np}+\left(n-1\right){k}_{bf}-\left({k}_{np}-{k}_{bf}\right){\left(1+\beta \right)}^{3}\phi}\right]$ | Model I is applicable over a wide range of particlenature, fraction and size, and different base fluids. Model II help to determine the contribution layer thickness, particle shape, and Brownian motion |

Corcione [83] | ${k}_{nf}=1+4.4R{e}^{0.4}P{r}^{0.66}{\left(\frac{T}{{T}_{fre}}\right)}^{10}\xb7{\left(\frac{{k}_{np}}{{k}_{bf}}\right)}^{0.03}\xb7{\phi}^{0.66}$ | Applicable for the temperature range of 294–324 K, nanoparticle diameter of 10–150 nm and volume fraction of 0.002–0.9 |

**Table 2.**Representative review papers on nanofluid heat transfer research using machine learning prediction models.

Reference | Work Focus | Machine Learning Involviment |
---|---|---|

Zhao et al., 2016 [93] | Prediction of thermal conductivity and viscosity based on ANN and applications in automotive radiators | ANN data-driven modeling |

Ramezanizadeh et al., 2019 [94] | Characteristics of different machine learning methods including MLP-ANN, GMDH, ANFIS, RBF, and LS-SVM combined with GA, PSO, and ICA. Applications of machine learning methods to dynamic viscosity modeling of nanofluids | Machine learning for viscosity prediction |

Bahiraei et al., 2019 [95] | AI algorithms inluding ANNs, fuzzy logic optimization methods and hybrid AI algorithms used for prediction and optimization of thermal properties of nanofluids | Machine learning algorithms for prediction and optimization |

Guo 2020 [96] | Overview on measured thermal properties, enhancement mechanisms, models for properties and heat transfer characteristics and applications of nanofluids to cooling, renewable energy, and energy and building technologies | ANN model for thermal conductivity prediction |

Sahaluddin et al., 2020 [86] | Development of a machine learning model for density prediction of nitrides in ethylene glycol. The developed is much more accurate than the Pak and Cho empirical model | SVM model for density prediction |

Zhang and Xu 2020 [89] | Machine learning glass transition temperature of polymers prediction using a GPR (Gauss Process Regression) data-driven model | GPR model for glass transition temperature prediction |

Zhang and Xu 2020 [91] | Machine learning decomposition onset temperature of lubricant additives prediction using a GPR data-driven model | GPR model for temperature prediction |

Shateri et al., 2020 [88] | CMIS (Comittee Machine Intelligent System) machine learning model for nanofluid viscosity estimation | CMIS model for viscosity prediction |

Alade et al., 2020 [87] | BSVR (Bayesian Support Vector Regression) and ANN machine learning models for nanofluid viscosity prediction | BSVR and ANN models for viscosity prediction |

Ma et al., 2021 [92] | Nanofluid heat transfer machine learning research applied to renewable energy | Machine learning description and applications |

NPs | BF | Size [nm] | Concentration | K_{eff} Increase [%] | Method | Ref. |
---|---|---|---|---|---|---|

$A{l}_{2}{O}_{3}$ | Water | 33 | 1 and 2 vol.% | 5.4 | Theory | [104] |

30–60 | 0.5, 1.0, 2, 3 and 4 vol.% | 1.96 | THW | [105] | ||

10, 20–30 and 150 | Up to 1.5 vol.% | 23 | Theory | [106] | ||

13 | 0.1, 0.15, 0.20 and 0.25 vol.% | 6.40 for 0.25 vol.% NF | THW | [107] | ||

41 | 18 vol.% | 31 | THW | [54] | ||

36 | 10 vol.% | 30 | SSCB | [108] | ||

36 | 6 vol.% | 28 | SSCB | [109] | ||

20 | 14.6 vol.% | 22 | TSHW | [110] | ||

282 | 4 vol.% | 17.7 | THW | [111] | ||

40 | 4 vol.% | 14.4 | Flash | [112] | ||

15–50 | 4 vol.% | 10.1 | TPS | [61] | ||

43 | 3 vol.% | 9.7 | THW | [113] | ||

11 | 1 vol.% | 9 | THW | [53] | ||

38 | 3 vol.% | 8 | THW | [114] | ||

43 | 2 vol.% | 7.52 | THW | [113] | ||

10 | 0.08 vol.% | 7.1 | THW | [115] | ||

12 | 4 vol.% | 5.4 | THW | [116] | ||

10 | 0.05 vol.% | 4.7 | THW | [115] | ||

43 | 0.75 vol.% | 3.28 | THW | [113] | ||

10 | 0.04 vol.% | 3.1 | THW | [115] | ||

43 | 0.33 vol.% | 1.64 | THW | [113] | ||

EG | 10 | 0.08 vol.% | 22 | THW | [115] | |

10 | 0.06 vol.% | 17.3 | THW | [115] | ||

282 | 3 vol.% | 16.3 | THW | [111] | ||

12 | 4 vol.% | 14.3 | THW | [116] | ||

38 | 3 vol.% | 10.6 | THW | [114] | ||

45 | 4 vol.% | 9.7 | 3ω | [117] | ||

EG/Water | 13 | 2 vol.% | 12.6 | THW | [118] | |

10 | 3 vol.% | 11.3 | THW | [116] | ||

50 | 3 vol.% | 10.4 | THW | [116] | ||

13 | 2 vol.% | 8.4 | THW | [118] | ||

13 | 2 vol.% | 16.2 | THW | [118] | ||

DI | 45 | 4 vol.% | 13.3 | 3ω | [117] | |

48 | 1 vol.% | 4 | THW | [28] | ||

$Ti{O}_{2}$ | Water | 21 | 0.1 vol% | About 11.1 | THW | [119] |

100 | 0.10, 0.15, 0.21 and 0.25 vol.% | 16.7 for TiO_{2}-0.25% | THW | [120] | ||

10, 30, and 50 | 0.005, 0.01, 0.1, 0.5, and 1 vol.% | 0.4 | Theory | [121] | ||

5 | 0, 0.1, 0.5 and 1.0 vol.% | 6.55 | TPS | [122] | ||

21 | 0.1–0.5 vol.% | 7.28 | [123] | |||

10 | 3 vol.% | 11.4 | THW | [114] | ||

34 | 3 vol.% | 8.7 | THW | [114] | ||

21 | 2 vol.% | 7 | THW | [56] | ||

40 | 2.6 vol.% | 6.5 | THW | [110] | ||

70 | 3 vol.% | 6.4 | THW | [114] | ||

20 | 2 vol.% | 4.2 | THW | [124] | ||

Water:EG | 40 | 0.2 to 0.8 vol.% | 24 at VF 0.8% and temperature 50 °C | THW | [125] | |

EG | 5 | 7 vol.% | 19.52 | THW | [126] | |

15 | 5 vol.% | 18 | THW | [127] | ||

10 | 3 vol.% | 14.4 | THW | [114] | ||

34 | 3 vol.% | 12.3 | THW | [114] | ||

70 | 3 vol.% | 7.5 | THW | [114] | ||

DI | Ø10×40 | 5 vol.% | 33 | THW | [34] | |

15 | 5 vol.% | 30 | THW | [34] | ||

20.5 | 1 vol.% | 14.4 | THW | [28] | ||

21 | 3 vol.% | 7.2 | 3ω | [128] | ||

$SiC$ | Water | 45–65 | 1, 1.5, 2, 3, and 4 wt% | 8.2 | THW | [129] |

45–65 | 0.5, 1.0, 2, 3 and 4 vol.% | 4.8 | THW | [105] | ||

CuO | Water | 35–45 | 0.5, 1.0, 2, 3 and 4 vol.% | 3.42 | THW | [105] |

29 | 6 vol.% | 52 | SSCB | [108] | ||

25 | 7.5 vol.% | 32.3 | THW | [130] | ||

55–66 | 2 vol.% | 24 | THW | [129] | ||

33 | 4.68 vol.% | 16.5 | TSHW | [131] | ||

33 | 1 vol.% | 5 | THW | [131] | ||

Water + 0.5 wt% CMC | 40 | 0.2–1.0 wt% | 29% | THW | [132] | |

EG | 55–66 | 2 vol.% | 21 | THW | [130] | |

33 | 1 vol.% | 9 | THW | [40] | ||

MEG | 25 | 7.5 vol.% | 21.3 | THW | [129] | |

EO | 55–66 | 2 vol.% | 14 | THW | [130] | |

CeO_{2} | EG | 10–30 | 2.5 vol.% | 22 | THW | [132] |

ZnO | Water | 10 | 3 vol.% | 14.2 | THW | [133] |

60 | 3 vol.% | 7.3 | THW | [134] | ||

EG | 30 | 3 vol.% | 21 | THW | [113] | |

50 | 2.4 vol.% | 13 | THW | [135] | ||

WO_{3} | EG | 38 | 0.3 vol.% | 13.8 | THW | [28] |

Fe_{3}O_{4} | Kerosene | 15 | 1 vol.% | 34.6 | THW | [136] |

Water | 15–23 | 3 vol.% | 11.5 | THW | [137] | |

15–20 | 4.8 vol.% | 2.9 | 3ω | [138] | ||

15–20 | 1 vol.% | 1.1 | 3ω | [138] | ||

NiFe_{2}O_{4} | DI | 8 | 2 vol.% | 17.2 | THW | [139] |

MgO | EG/Water | 40 | 3 vol.% | 34.43 | THW | [140] |

Cu | Water | 75–100 | 0.1 vol.% | 23.8 | THW | [141] |

Au | Toluene | 1.65 | 0.003 vol.% | 8 | TSHW | [110] |

2 | 0.024 vol.% | 1.4 | MSBD | [142] | ||

Ethanol | 4 | 0.018 vol.% | 1.3 | MSBD | [142] | |

Ag | Water | 96 | 1.7×10−5 vol.% | 20.8 | TWRC | [143] |

96 | 3.5×10−6 vol.% | 4 | TWRC | [143] | ||

DI | 5–25 | 0.5 vol.% | 16 | THW | [144] | |

Fe | EG | 20 | 4 vol.% | 38.8 | Theory | [145] |

10 | 0.55 vol.% | 18 | THW | [64] | ||

10 | 0.3 vol.% | 16.5 | THW | [28] | ||

50 | 2 vol.% | 15.5 | Theory | [145] | ||

Al | EG | 80 | 5 vol.% | 45 | THW | [127] |

SiC (sphere) | DI | 100 | 3 vol.% | 7.2 | THW | [146] |

DO | 30 | 0.8 vol.% | 7.36 | THW | [147] | |

SiO_{2} | Water | 40–50 | 3 vol.% | 38.2 | THW | [148] |

12 | 1 vol.% | 3.2 | THW | [132] | ||

12 | 1 vol.% | 3 | THW | [40] | ||

$MgO/Ti{O}_{2}$ | Water | 10–45 | 50:50, 80:20, 20:80, 60:40 and 40:60 wt% | 21.8 at 50 °C | THW | [149] |

Graphene nanoplatelet(GFnanopl) | Water | 0.025 to 0.1 wt% | 27 | THW | [150] | |

EG | 0.7–1.3 | 0.05vol.% | 86 | TSHW | [151] | |

Graphene Oxide | EG | 0.7–1.4 | 0.05vol.% | 61 | TSHW | [151] |

GFnanopl – SDBS | Water | 0.025, 0.05 and 0.1 wt.% | About 11.56 | Theory | [152] | |

GFnanopl – COOH | Water | 0.025, 0.05 and 0.1 wt.% | About 21.03 | Theory | [152] | |

$A{l}_{2}{O}_{3}-MWCNT$ | Water | 10–100 ($A{l}_{2}{O}_{3}$) | 0.01 vol.% | 10.85 for MWCNT (0:5) NFs | THW | [153] |

MWCNT | Solar glycol | 20–30 | 0.2, 0.4, and0.6 vol.% | 30.59 with MWCNT volume of 0.6% | THW | [154] |

Jatropha seed oil | Length × OD = 2.5–20 μm × 6–13nm | 0.2–0.8 wt.% | 6.76 | THW | [155] | |

Water | Ø40 | 0.49 vol.% | 80 | THW | [156] | |

Ø130 | 0.6 vol.% | 34 | THW | [157] | ||

Ø10–30 | 1 vol.% | 7 | THW | [132] | ||

Ø10–30 | 0.48 vol.% | 5 | THW | [158] | ||

HTO | Ø5–20 | 2 vol.% | 15 | THW | [159] | |

EG | Ø20–50 | 1 vol.% | 12.4 | THW | [160] | |

EO | Ø20–50 | 1 vol.% | 8.5 | THW | [160] | |

DWCNT | Water | Ø5 | 1 vol.% | 8 | THW | [157] |

SWCNT | Water | Ø1-2×5000-30,000 | 0.48 vol.% | 16.2 | THW | [157] |

Ø1-2×1000–3000 | 0.48 vol.% | 8.1 | THW | [158] | ||

EG | 100–600 | 0.21 vol.% | 15.5 | THW | [161] | |

C60-C70 fullerenes | Toluene | – | 0.378 vol.% | 0.816 | MSBD | [143] |

MO | 10 | 5 vol.% | 6 | THW | [132] |

_{eff}= Effective thermal conductivity; THW = Transient Hot-Wire; SSCB = Steady state “cut bar”; TSHW = Transient Short Hot-Wire; TPS = Transient Plane-Source; MSBD = Micronscale Beam Deflection Technique; TWRC = thermal-wave resonator cavity.

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**MDPI and ACS Style**

Gonçalves, I.; Souza, R.; Coutinho, G.; Miranda, J.; Moita, A.; Pereira, J.E.; Moreira, A.; Lima, R. Thermal Conductivity of Nanofluids: A Review on Prediction Models, Controversies and Challenges. *Appl. Sci.* **2021**, *11*, 2525.
https://doi.org/10.3390/app11062525

**AMA Style**

Gonçalves I, Souza R, Coutinho G, Miranda J, Moita A, Pereira JE, Moreira A, Lima R. Thermal Conductivity of Nanofluids: A Review on Prediction Models, Controversies and Challenges. *Applied Sciences*. 2021; 11(6):2525.
https://doi.org/10.3390/app11062525

**Chicago/Turabian Style**

Gonçalves, Inês, Reinaldo Souza, Gonçalo Coutinho, João Miranda, Ana Moita, José Eduardo Pereira, António Moreira, and Rui Lima. 2021. "Thermal Conductivity of Nanofluids: A Review on Prediction Models, Controversies and Challenges" *Applied Sciences* 11, no. 6: 2525.
https://doi.org/10.3390/app11062525