Modeling of thermal conductivity of nanofluids by modifying Maxwell’s equation using cell model approach

Nanofluid is an innovative heat transfer fluid with superior potential for enhancing the heat transfer performance of conventional fluids. Though many attempts have been made to investigate the abnormal high thermal conductivity of nanofluids, the existing models cannot precisely predict the same. An attempt has been made to develop a model for predicting the thermal conductivity of different types of nanofluids. The model presented here is derived based on the fact that thermal conductivity of nanofluids depends on thermal conductivity of particle and fluid as well as micro-convective heat transfer due to Brownian motion of nanoparticles. Novelty of the article lies in giving a unique equation which predicts thermal conductivity of nanofluids for different concentrations and particle sizes which also correctly predicts the trends observed in experimental data over a wide range of particle sizes, temperatures, and particle concentrations.     

A model for the thermal conductivity of nanofluids – the effect of interfacial layer

A model for predicting the effective thermal conductivity of nanofluids is proposed. It has been documented that the interfacial layer at the solid (particle)/liquid interface and particle size is one of the major mechanisms for enhancing the thermal conductivity of nanofluids. Comparing with other classical models, the proposed model takes into account some additional effects including volume fraction, thickness, thermal conductivity of the interfacial layer and particle size. The proposed model is found to be better than the existing models since the predicted effective thermal conductivity of different types of nanofluids are closer to the experimental results.     

Model for heat conduction in nanofluids

A comprehensive model has been proposed to account for the large enhancement of thermal conductivity in nanofluids and its strong temperature dependence, which the classical Maxwellian theory has been unable to explain. The dependence of thermal conductivity on particle size, concentration, and temperature has been taken care of simultaneously in our treatment. While the geometrical effect of an increase in surface area with a decrease in particle size, rationalized using a stationary particle model, accounts for the conductivity enhancement, a moving particle model developed from the Stokes-Einstein formula explains the temperature effect. Predictions from the combined model agree with the experimentally observed values of conductivity enhancement of nanofluids.     

Influence of the particle size distribution on the thermal conductivity of nanofluids

In a previous study, we have obtained an equation to predict the thermal conductivity of nanofluids containing nanoparticles with conductive interface. The model is maximal particle packing dependent. In this study, the maximal packing is obtained as a function of the particle size distribution, which is the Gamma distribution. The thermal conductivity enhancement depends on the averaged particle size. Discussion concerning the influence of the suspension pH on the particle packing is made. The proposed model is evaluated using number of sets from the published experimental data to the thermal conductivity enhancement for different nanofluids.     

The role of interfacial layers in the enhanced thermal conductivity of nanofluids: A renovated Hamilton–Crosser model

We previously developed a renovated Maxwell model for the effective thermal conductivity of nanofluids and determined that the solid/liquid interfacial layers play an important role in the enhanced thermal conductivity of nanofluids. However, this renovated Maxwell model is limited to suspensions with spherical particles. Here, we extend the Hamilton--Crosser model for suspensions of nonspherical particles to include the effect of a solid/liquid interface. The solid/liquid interface is described as a confocal ellipsoid with a solid particle. The new model for the three-phase suspensions is mathematically expressed in terms of the equivalent thermal conductivity and equivalent volume fraction of anisotropic complex ellipsoids, as well as an empirical shape factor. With a generalized empirical shape factor, the renovated Hamilton--Crosser model correctly predicts the magnitude of the thermal conductivity of nanotube-in-oil nanofluids. At present, this new model is not able to predict the nonlinear behavior of the nanofluid thermal conductivity.     

Brownian dynamic simulation for the prediction of effective thermal conductivity of nanofluid

Nanofluid is a colloidal solution of nanosized solid particles in liquids. Nanofluids show anomalously high thermal conductivity in comparison to the base fluid, a fact that has drawn the interest of lots of research groups. Thermal conductivity of nanofluids depends on factors such as the nature of base fluid and nanoparticle, particle concentration, temperature of the fluid and size of the particles. Also, the nanofluids show significant change in properties such as viscosity and specific heat in comparison to the base fluid. Hence, a theoretical model becomes important in order to optimize the nanofluid dispersion (with respect to particle size, volume fraction, temperature, etc.) for its performance. As molecular dynamic simulation is computationally expensive, here the technique of Brownian dynamic simulation coupled with the Green Kubo model has been used in order to compute the thermal conductivity of nanofluids. The simulations were performed for different concentration ranging from 0.5 to 3 vol%, particle size ranging from 15 to 150 nm and temperature ranging from 290 to 320 K. The results were compared with the available experimental data, and they were found to be in close agreement. The model also brings to light important physical aspect like the role of Brownian motion in the thermal conductivity enhancement of nanofluids.     

The effect of particle size on the thermal conductivity of alumina nanofluids

We present new data for the thermal conductivity enhancement in seven nanofluids containing 8–282 nm diameter alumina nanoparticles in water or ethylene glycol. Our results show that the thermal conductivity enhancement in these nanofluids decreases as the particle size decreases below about 50 nm. This finding is consistent with a decrease in the thermal conductivity of alumina nanoparticles with decreasing particle size, which can be attributed to phonon scattering at the solid–liquid interface. The limiting value of the enhancement for nanofluids containing large particles is greater than that predicted by the Maxwell equation, but is predicted well by the volume fraction weighted geometric mean of the bulk thermal conductivities of the solid and liquid. This observation was used to develop a simple relationship for the thermal conductivity of alumina nanofluids in both water and ethylene glycol.     

Dispersion stability and thermal conductivity of propylene glycol-based nanofluids

The dispersion stability and thermal conductivity of propylene glycol-based nanofluids containing Al₂O₃ and TiO₂ nanoparticles were studied in the temperature range of 20–80 °C. Nanofluids with different concentrations of nanoparticles were formulated by the two-step method and no dispersant was used. In contrast to the common belief, the average particle size of nanofluids was observed to decrease with increasing temperature, and nanofluids showed an excellent stability over the temperature range of interest. Thermal conductivity enhancement for both studied nanofluids was a nonlinear function of concentration and was temperature independent. Theoretical analyses were also performed using existing models compared with experimental results. The model based on the aggregation theory appears to be the best.     

Particle agglomeration and properties of nanofluids

The present study demonstrates the importance of actual agglomerated particle size in the nanofluid and its effect on the fluid properties. The current work deals with 5 to 100 nm nanoparticles dispersed in fluids that resulted in 200 to 800 nm agglomerates. Particle size distributions for a range of nanofluids are measured by dynamic light scattering (DLS). Wet scanning electron microscopy method is used to visualize agglomerated particles in the dispersed state and to confirm particle size measurements by DLS. Our results show that a combination of base fluid chemistry and nanoparticle type is very important to create stable nanofluids. Several nanofluids resulted in stable state without any stabilizers, but in the long term had agglomerations of 250 % over a 2 month period. The effects of agglomeration on the thermal and rheological properties are presented for several types of nanoparticle and base fluid chemistries. Despite using nanodiamond particles with high thermal conductivity and a very sensitive laser flash thermal conductivity measurement technique, no anomalous increases of thermal conductivity was measured. The thermal conductivity increases of nanofluid with the particle concentration are as those predicted by Maxwell and Bruggeman models. The level of agglomeration of nanoparticles hardly influenced the thermal conductivity of the nanofluid. The viscosity of nanofluids increased strongly as the concentration of particle is increased; it displays shear thinning and is a strong function of the level of agglomeration. The viscosity increase is significantly above of that predicted by the Einstein model even for very small concentration of nanoparticles.     

Predicting the thermal conductivity enhancement of nanofluids using computational intelligence

Nanofluids, composed of nanoparticles in base liquids, have drawn increasing attention in heat transfer applications due to their anomalously increased thermal conductivity. Pertinent parameters, including the base liquid thermal conductivity, particle thermal conductivity, particle size, particle volume fraction, and temperature, have been shown to have significant but complex effects on thermal performance of nanofluids, which is commonly characterized by the thermal conductivity enhancement, E%. In this work, machine learning is used to develop the Gaussian process regression model to find statistical correlations between E% and aforementioned physical parameters among various types of nanofluids. Nearly 300 nanofluid samples, dispersions of metal and ceramic nanoparticles in water, ethylene glycol, and transformer oil, are explored for this purpose. The modeling approach demonstrates a high degree of accuracy and stability, contributing to efficient and low-cost estimations of E%.     

Application of SAXS to the study of particle-size-dependent thermal conductivity in silica nanofluids

Knowledge of the size and distribution of nanoparticles in solution is critical to understanding the observed enhancements in thermal conductivity and heat transfer of nanofluids. We have applied small-angle X-ray scattering (SAXS) to the characterization of SiO₂ nanoparticles (10–30 nm) uniformly dispersed in a water-based fluid using the Advanced Photon Source at Argonne National Laboratory. Size distributions for the suspended nanoparticles were derived by fitting experimental data to an established model. Thermal conductivity of the SiO₂ nanofluids was also measured, and the relation between the average particle size and the thermal conductivity enhancement was established. The experimental data contradict models based on fluid interfacial layers or Brownian motion but support the concept of thermal resistance at the liquid–particle interface.     

The thermal conductivity of alumina nanofluids in water, ethylene glycol, and ethylene glycol + water mixtures

We present new data on the thermal conductivity of nanofluids consisting of alumina nanoparticles dispersed in water, ethylene glycol, and ethylene glycol + water mixtures. We also demonstrate that our previously published model is able to describe the temperature, particle size, and particle volume fraction dependence of these nanofluids without any adjustable parameters, irrespective of the base fluid used (water, ethylene glycol, or water + ethylene glycol mixtures). Furthermore, we demonstrate how the model may be used to check the consistency of literature data on all alumina nanofluids.     

Novel nanofluids based on mesoporous silica for enhanced heat transfer

Nanofluids, which are liquids with engineered nanometer-sized particles suspensions, have drawn remarkable attraction from the researchers because of their enormous potential to enhance the efficiency in heat-transfer fluids. In the present study, water-based calcined mesoporous silica nanofluids were prepared and characterized. The commercial mesoporous silica (MPSiO₂) nanoparticles were dispersed in deionized water by means of pH adjustment and ultrasonic agitation. MPSiO₂ nanoparticles were observed to have an average particle size of 350 ± 100 nm by SEM analysis. The concentration of MPSiO₂ was varied between 1 and 6 wt%. The physicochemical properties of nanofluids were characterized using various techniques, such as particle size analyzer, zeta-potential meter, TEM, and FT-IR. The thermal conductivity was measured by Transient Plane Source (TPS) method, and nanofluids showed a higher thermal conductivity than the base liquid for all the tested concentrations.     

Investigation of thermal conductivity and rheological properties of vegetable oil based hybrid nanofluids containing Cu–Zn hybrid nanoparticles

Vegetable oils (Ground nut) are being investigated to serve as a possible substitute for non-biodegradable mineral oils, which are currently being used as metal-cutting fluids in machining processes. In this study, thermophysical properties of hybrid nanofluids (vegetable oil) to be used as metalworking cutting fluids are investigated. In-situ synthesis of copper (Cu) and zinc (Zn) combined hybrid particles is performed by mechanical alloying with compositions of 50:50, 75:25, and 25:75 by weight. Characterizations of the synthesized powder were carried out using X-ray diffraction, a particle size analyzer, FE-SEM, and TEM. Hybrid nanofluids with all the three combinations of hybrid nanoparticles were prepared by dispersing them into a base fluid (vegetable oil). The thermophysical properties, such as thermal conductivity and viscosity, were studied for various volume concentrations and at a range of temperatures. Experimental results have shown enhancement in thermal conductivity in all cases and also an increase in viscosity. The enhancement in viscosity is similar in all three combinations, as the particle size and shape are almost identical. The enhancement in thermal conductivity is higher in Cu–Zn (50:50), resulting in better enhancement in thermal conductivity due to the Brownian motion of the particles and higher thermal conductivity of the nanoparticles incorporated.     

水基ZnO纳米流体电导和热导性能研究

利用水热法生成了形状规则、粒径均匀的球形ZnO纳米颗粒, 并超声分散于水中, 制备得到稳定的水基ZnO纳米流体. 实验测量水基ZnO纳米流体在体积分数和温度变化时的电导率, 并测试室温下水基ZnO纳米流体在不同体积分数下的热导率. 实验结果表明, ZnO纳米颗粒的添加较大地提高了基液(纯水)的热导率和电导率, 水基ZnO纳米流体的电导率随纳米颗粒体积分数增加呈非线性增加关系, 而电导率随温度变化呈现出拟线性关系; 纳米流体的热导率与纳米颗粒体积分数增加呈近似线性增加关系. 本文在经典Maxwell热导模型和布朗动力学理论的基础上, 同时考虑了吸附层、团聚体和布朗运动等因素对热导率的影响, 提出了热导率修正模型.将修正模型预测值与实验值对比, 结果表明修正模型可以较为准确地计算出纳米流体的热导率. 关键词： 水热法 电导率 热导率 热导模型     

Temperature Dependence of Thermal Conductivity of Nanofluids

Mechanism of thermal conductivity of nanofluids is analysed and calculated, including Brownian motion effects, particle agglomeration and viscosity, together influenced by temperature. The results show that only Brown- Jan motion as reported is not enough to describe the temperature dependence of the thermal conductivity of nanofluids. The change of particle agglomeration and viscosity with temperature are also important factors. As temperature increases, the reduction of the particle surface energy would decrease the agglomeration of nanopartieles, and the reduction of viscosity would improve the Brownish motion. The results egree well with the experimental data reported.     

A thermal conductivity model for nanofluids including effect of the temperature-dependent interfacial layer

The interfacial layer of nanoparticles has been recently shown to have an effect on the thermal conductivity of nanofluids. There is, however, still no thermal conductivity model that includes the effects of temperature and nanoparticle size variations on the thickness and consequently on the thermal conductivity of the interfacial layer. In the present work, the stationary model developed by Leong et al. (J Nanopart Res 8:245–254, 2006) is initially modified to include the thermal dispersion effect due to the Brownian motion of nanoparticles. This model is called the ‘Leong et al.’s dynamic model’. However, the Leong et al.’s dynamic model over-predicts the thermal conductivity of nanofluids in the case of the flowing fluid. This suggests that the enhancement in the thermal conductivity of the flowing nanofluids due to the increase in temperature does not come from the thermal dispersion effect. It is more likely that the enhancement in heat transfer of the flowing nanofluids comes from the temperature-dependent interfacial layer effect. Therefore, the Leong et al.’s stationary model is again modified to include the effect of temperature variation on the thermal conductivity of the interfacial layer for different sizes of nanoparticles. This present model is then evaluated and compared with the other thermal conductivity models for the turbulent convective heat transfer in nanofluids along a uniformly heated tube. The results show that the present model is more general than the other models in the sense that it can predict both the temperature and the volume fraction dependence of the thermal conductivity of nanofluids for both non-flowing and flowing fluids. Also, it is found to be more accurate than the other models due to the inclusion of the effect of the temperature-dependent interfacial layer. In conclusion, the present model can accurately predict the changes in thermal conductivity of nanofluids due to the changes in volume fraction and temperature for various nanoparticle sizes.     

Simulation of the thermal conductivity of a nanofluid with small particles by molecular dynamics methods

The thermal conductivity of nanoliquids has been simulated by molecular dynamics method. We consider nanofluids based on argon with aluminum and zinc particles with sizes of 1–4 nm. The volume concentration of nanoparticles is varied from 1 to 5%. The dependence of the thermal conductivity on the volume concentration of nanoparticles has been analyzed. It has been shown that the thermal conductivity of a nanofluid cannot be described by classical theories. In particular, it depends on the particle size and increases with it. However, it has been established that the thermal conductivity of nanofluids with small particles can even be lower than that of the carrier fluid. The behavior of the correlation functions responsible for the thermal conductivity has been studied systematically, and the reason for the increase in the thermal conductivity of nanofluid has been explained qualitatively.     

Thermal conductance of nanofluids: is the controversy over?

Over the last decade nanofluids (colloidal suspensions of solid nanoparticles) sparked excitement as well as controversy. In particular, a number of researches reported dramatic increases of thermal conductivity with small nanoparticle loading, while others showed moderate increases consistent with the effective medium theories on well-dispersed conductive spheres. Accordingly, the mechanism of thermal conductivity enhancement is a hotly debated topic. We present a critical analysis of the experimental data in terms of the potential mechanisms and show that, by accounting for linear particle aggregation, the well established effective medium theories for composite materials are capable of explaining the vast majority of the reported data without resorting to novel mechanisms such as Brownian motion induced nanoconvection, liquid layering at the interface, or near-field radiation. However, particle aggregation required to significantly enhance thermal conductivity, also increases fluid viscosity rendering the benefit of nanofluids to flow based cooling applications questionable.     

Physics and mechanics of heat exchange processes in nanofluid flows

Nanofluids present a new type of dispersed fluids consisting of a carrier fluid and solid nanoparticles. Unusual properties of nanofluids, particularly high thermal conductivity, make them eminently suitable for many thermophysical applications, e.g., for cooling of equipment, designing of new heat energy transportation and production systems and so on. This requires a systematic study of heat exchange properties of nanofluids. The present paper contains the measurement results for the heat transfer coefficient of the laminar and turbulent flow of nanofluids on the basis of distilled water with silica, alumina and copper oxide particles in a minichannel with circular cross section. The maximum volume concentration of particles did not exceed 2%. The dependence of the heat transfer coefficient on the concentration and size of nanoparticles was studied. It is shown that the use of nanofluids allows a significant increase in the heat transfer coefficient as compared to that for water. However, the obtained result strongly depends on the regime of flow. The excess of the heat transfer coefficient in the laminar flow is only due to an increase in the thermal conductivity coefficient of nanofluid, while in the turbulent flow the obtained effect is due to the ratio between the viscosity and thermal conductivity of nanofluid. The viscosity and thermal conductivity of nanofluids depend on the volume concentration of nanoparticles as well as on their size and material and are not described by classical theories. That is why the literature data are diverse and contradictory; they do not actually take into account the influence of the mentioned factors (size and material of nanoparticles). It has been shown experimentally and by a molecular dynamics method that the nanofluid viscosity increases while the thermal conductivity decreases with the decreasing dispersed particle size. It is found experimentally for the first time that the nanofluid viscosity coefficient depends on the particle material. The higher is the density of particles, the higher is the thermal conductivity coefficient of nanofluid.