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Roles of surfactants and particle shape in the enhanced thermal conductivity of TiO2 nanofluids Liu Yang, Xielei Chen, Mengkai Xu, and Kai Du Citation: AIP Advances 6, 095104 (2016); doi: 10.1063/1.4962659 View online: http://dx.doi.org/10.1063/1.4962659 View Table of Contents: http://scitation.aip.org/content/aip/journal/adva/6/9?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Study of thermal conductivity enhancement of aqueous suspensions containing silver nanoparticles AIP Advances 5, 057103 (2015); 10.1063/1.4919808 Deterioration in effective thermal conductivity of aqueous magnetic nanofluids J. Appl. Phys. 116, 224904 (2014); 10.1063/1.4902441 Strong enhancement in thermal conductivity of ethylene glycol-based nanofluids by amorphous and crystalline Al2O3 nanoparticles Appl. Phys. Lett. 105, 063108 (2014); 10.1063/1.4893026 Role of surface charge, morphology, and adsorbed moieties on thermal conductivity enhancement of nanofluids Appl. Phys. Lett. 101, 173113 (2012); 10.1063/1.4764050 The limiting behavior of the thermal conductivity of nanoparticles and nanofluids J. Appl. Phys. 107, 114319 (2010); 10.1063/1.3354094

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AIP ADVANCES 6, 095104 (2016)

Roles of surfactants and particle shape in the enhanced thermal conductivity of TiO2 nanofluids Liu Yang,1,2,a Xielei Chen,1 Mengkai Xu,1 and Kai Du1 1 Key

Laboratory of Energy Thermal Conversion and Control of Ministry of Education, School of Energy and Environment, Southeast University, SiPaiLou 2#, 210096 Nanjing, China 2 Jiangsu Provincial Key Laboratory of Solar Energy Science and Technology, School of Energy and Environment, Southeast University, SiPaiLou 2#, 210096 Nanjing, China (Received 13 July 2016; accepted 18 August 2016; published online 8 September 2016)

Although several forms of thermal conductivity models for nanofluid have been established, few models for nanofluids containing surfactants or columnar nanoparticles are found. This paper intends to consider the surfactants and particle shape effect in the thermal conductivity of TiO2 nanofluids. The thermal conductivity models for respectively spherical and columnar TiO2 nanofluids are proposed by considering the influences of solvation nanolayer and the end effect of columnar nanoparticles. The thicknesses of the solvation nanolayers are defined by the surfactant molecular length and a few atomic distances for nanofluid with and without surfactant respectively. The end effect of the columnar nanoparticles is considered by analyzing the different thermal resistances and probability of the heat conduction for the selected small element in axial direction and radial direction. Finally, the present models and some other existing models were compared with some available experimental data and the comparison results show the present models achieve higher accuracy and precision for all the four kinds of applications. © 2016 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). [http://dx.doi.org/10.1063/1.4962659]

I. INTRODUCTION

Over the course of the past three decades, nanofluids have hit the scientific and industrial world because of their outstanding fluidity, stability1 and thermal and transport characteristics2 comparing to the fluids containing millimeter or micrometer particles. Nanofluids have been gradually used as advanced working fluids in industrial systems3 such as refrigeration system, solar energy system and cooling lubrication system etc. Since the excellent performance of nanofluids is generally attributed to the physical properties of fluids with addition of nanoparticles, the thermal conductivity of nanofluids should be an important issue in the experimental or theoretical investigations on nanofluids. Calculating of thermal conductivity of nanofluids is an important step in the designing or using nanofluids for the actual applications. Many researchers have investigated the peculiar influence factors on the thermal conductivity of nanofluids. The current influence factors on the thermal conductivity of nanofluids are divided into three groups. The first is the particles’ parameters, including the particles’ type, content,4 size5 and shape6 etc. The second is the fluid and environmental parameters, such as the type of basefluid,7 ultrasonic dispersion time,8 storing time,9 pH value,10,11 surfactant 12 and temperature.13 The third group is some microcosmic factors, such as surface charge state of nanoparticles,14 Brownian motion,15 the interfacial shell surrounding nanoparticles,16 the dispersion stability,9 and interactions of nanoparticles,17 the clustering of particles18 etc. Following the increase of researches on the influence factors of thermal conductivity of nanofluids, more and more thermal conductivity models have been proposed via considering various parameters under various points of view. a Corresponding author: Liu Yang.

2158-3226/2016/6(9)/095104/12

6, 095104-1

© Author(s) 2016

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AIP Advances 6, 095104 (2016)

TiO2 nanofluids are thought one of the closest kinds to the practical energy application due to their comprehensive properties, such as the sensational dispersivity, chemical stability and nontoxicity. And the thermal conductivity definition is an important step for the further application. However, the existing models are not comprehensive. The addition of surfactants can distinctly affect the dispersion and thermodynamic properties of nanofluids,10 few thermal conductivity models have included the effect of surfactants. Besides, most of the current thermal conductivity models are proposed for spherical or nano-tubes nanofluids, none was specifically aimed at for nanofluids containing columnar nanoparticles. Therefore, there is a great need of thermal conductivity models for nanofluids containing surfactants or columnar nanoparticles. In this work, four thermal conductivity models for respectively spherical and columnar TiO2 nanofluids are proposed by considering the influences of solvation nanolayer and the end effect of columnar nanoparticles. It is expected that this study brings some supplementary ideas that can be helpful for the study of the thermal conductivity of some special shaped nanofluids. II. THERMAL CONDUCTIVITY MODELS A. Existing models

Prior to 1891, Maxwell19 reported a treatise which proposed a thermal conductivity model of liquids suspended with small hard spherical particles, which had an expression as follows: keff = kf

kp + 2kf + 2φ(kp − kf ) kp + 2kf − φ(kp − kf )

(1)

Hamilton and Crosser 20 obtained a thermal conductivity model including the degree of sphericity of the particles: keff = kf

kp + (n − 1)kf + (n − 1)φ(kp − kf ) kp + (n − 1)kf − φ(kp − kf )

(2)

where n is the empirical shape factor, which is given by n = 3/Ψ and Ψ is the sphericity. Yu and Choi21 took the interfacial nanolayer into account and upgraded Maxwell’s model into an alternative formula for calculating thermal conductivity, which is expressed in the following form: keff = kf

kp + 2kf + 2φ(kp − kf )(1 + β)3 kp + 2kf − φ(kp − kf )(1 + β)3

(3)

where β is the ratio of the nanolayer thickness to the original particle radius. Bruggeman22 considered the effect of nanoparticle clustering and proposed a thermal conductivity model of homogeneous spherical nanofluid, which is introduced by Murshed23 as the following form: kf √ 1 keff = [(3φ − 1)kp + (2 − 3φ)kf ] + ∆ 4 4

(4)

∆ = (3φ − 1)2 (kp /kf )2 + (2 − 3φ)2 + 2(2 + 9φ − 9φ2 )(kp /kf )

(5)

Bhattacharya24 considered Brownian motion and proposed a thermal conductivity model, which has an expression as follows: keff = φkp + (1 − φ)kf

(6)

Timofeeva25 used effective medium theory to propose a thermal conductivity model, which has an expression as follows: keff = kf (1 + 3φ)

(7)

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AIP Advances 6, 095104 (2016)

Xue26 proposed a model for calculating the thermal conductivity of nanotube nanofluids, which has an expression as follows: keff = kf

k

kp +kf 2kf

k

kp +kf 2kf

p ln 1 − φ + 2φ kp −k f f 1 − φ + 2φ kp −k ln f

(8)

Jiang16 considered the effect of interfacial layer and proposed a model for calculating the thermal conductivity of CNTs based nanofluids, which is expressed as follows: keff = kf

kpe + (n − 1)kf + (n − 1)φ(kpe − kf ) kpe + (n − 1)kf − φ(kpe − kf )

kpe = kf

klr =

2kp + ( β1 3 − 1)(kp + klr ) 2klr + ( β1 3 − 1)(kp + klr )

kp R(1 + t/R − kf /kp ) ln(1 + t/R) tkf ln[(1 + t/R)kp /kf ]

(9)

(10)

(11)

According to many research results, the addition of surfactants has a great effect on the dispersion situation and thermal behavior of nanofluids, but few thermal conductivity models have considered the influence of surfactants. Moreover, although many thermal conductivity models for spherical and nano-tubes nanofluids have been proposed, there is no available thermal conductivity model specifically targeted at columnar nanoparticle based nanofluids. For this reason, there is a great need of thermal conductivity model for nanofluids including surfactants or columnar nanoparticles. B. Thermal conductivity model for spherical TiO2 nanofluids

The interfacial nanolayer surrounding the particles is actual some kind of solvation nanolayer formed by the liquid or surfactant molecules. The effect of solvation nanolayers outside the solid particles has been considered in some current models because the thickness of the solvation nanolayer is not negligibly small for nanometer scaled particles and it will evidently affect the thermal conductivity of nanofluids. Leong27 established thermal conductivity model of nanofluids based on the analytic solution of the Laplace equation of two-dimensional steady heat conduction with boundary conditions which including the effects of the solvation nanolayer. The physical model of a single particle surrounded with a solvation nanolayer in the basefluid is as shown in Fig. 1. And the expression of Leong’s model can be shown as follows: keff =

(kp − klr )φklr (2 β13 − β 3 + 1) + (kp + 2klr ) β13 [φ β 3 (klr − kf ) + kf ] β13 (kp + 2klr ) − (kp − klr )φp ( β13 + β 3 − 1)

(12)

β = 1 + h/r

(13)

β1 = 1 + h/(2r)

(14)

FIG. 1. Sketch of a particle with interfacial layer in a fluid medium27 (Reproduced with permission from J. Nanopart. Res. 8, 245 (2006). Copyright 2006 Springer).

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AIP Advances 6, 095104 (2016)

The model herein presented for the thermal conductivity of TiO2 nanofluids was also based on Leong’s model. For nanofluids containing spherical TiO2 nanoparticles without surfactant, the solvation nanolayer is supposed to be generated by the solvation effect and its thickness is defined by a few atomic distances. Hashimoto28 presented a calculating formula for the thickness of solvation shell by considering the electron density distribution at the interface of particle and liquid, which has an expression as follows: √ (15) h = 2πσ where h is the thickness of the solvation shell, σ is a parameter characterizing the diffuseness of the interfacial boundary and the general value is about 0.2–0.8 nm. In the present model, the value of σ is 0.4 nm, and it can be calculated via Eq. (15) that the solvation shell thickness is 1 nm for the prediction of the thermal conductivity of nanofluids. In Murshed29 and Leong’s studies,27 after the layer thickness was set as 1 nm, the thermal conductivity of the interfacial nanolayer was set as 2 to 3 times that of base fluid for many kinds of nanofluids. And they found that this definition could fit most of the experimental data in their study and the other literatures. And there are also several researches30–32 that considered k lr to be several times, such as 2 or 3 times of k f depending on the type and the size of the nanoparticles. Therefore, in this paper for all the present models, the thermal conductivity of the interfacial nanolayer herein is set certainly as an in-between equation: klr = 2.5kf

(16)

For nanofluids containing spherical TiO2 nanoparticles with surfactant, as a result of TiO2 is insufferable, there is hardly any ionized ion of TiO2 , the adsorption of surfactant to the TiO2 nanoparticles is supposed to abide by monolayer adsorption dispersion. The mechanism of action of surfactants to the nanoparticles for monolayer adsorption has been discussed in our previous study.33 When surfactant is added into the nanofluids, the solvation nanolayer is a kind of adsorption layer which is thought to consist of surfactant and liquid molecules, and hence the region of solvation shell is related to the surfactant molecule length. Therefore, when assuming that the surfactant molecules are fully extended and the thickness of the solvation nanolayer can be defined as the surfactant molecule length,17 which was shown as following equation: h=l

(17)

Thus, to sum up the above equations which we have just indicated, the present models for spherical TiO2 nanofluids with and without surfactant are revealed in Table I respectively C. Thermal conductivity model for columnar TiO2 nanofluids

A thermal conductivity model for nanotube based nanofluids was built by Murshed.32 They deduced an equation in cylindrical coordinates using the same method, which was shown as follows: keff 2 =

(kp − klr )φklr ( β12 − β 2 + 1) + (kp + klr ) β12 [φ β 2 (klr − kf ) + kf ]

(18)

β12 (kp + klr ) − (kp − klr )φp ( β12 + β 2 − 1)

For columnar TiO2 nanofluids, the length to diameter ratio of columnar nanoparticle is not of such a big size as that of nanotube, hence the end effect of columnar nanofluids should be taken TABLE I. Thermal conductivity model for spherical TiO2 nanofluids. keff =

(kp −klr )φklr (2β13 −β 3 +1)+(kp +2klr )β13 [φ β 3 (klr −kf )+kf ] β13 (kp +2klr )−(kp −klr )φp (β13 +β 3 −1)

β = 1 + h/r β1 = 1 + h/(2r) klr = 2.5kf

Without surfactants √ h = 2πσ

with surfactants h=l

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AIP Advances 6, 095104 (2016)

FIG. 2. Simplification of a columnar TiO2 nanoparticle into a cuboid with an equivalent cross-sectional area.

into account. As a result of the columnar nanoparticle is not absolute regular but generally with sharp corners,34 in order to quantitatively investigate the end effect of columnar nanoparticles on the thermal conductivity of nanofluids, the cross-section of the columnar nanoparticle is simplified. Fig.2 shows the simplification of a columnar TiO2 nanoparticle into a cuboid geometry with an equivalent cross-sectional area. Assuming the radius and the height of the columnar nanoparticles are r and H, respectively, the cross-section of the columnar nanoparticle is simplified as an equivalent area of rectangle with 2r nm in length and πr/2 nm in width. After the simplification of the cross-section of the columnar nanoparticle, a small cube element with H nm on each edge is set to analyze the thermal resistance of the cube in x-axis (axial direction) and z-axis (radial direction) respectively. The front and vertical views of the cube are shown in Fig. 3(a) and (b), respectively.

FIG. 3. Front view (a) and vertical view (b) of the cube element selected for analyzing the thermal resistances in different directions.

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AIP Advances 6, 095104 (2016)

TABLE II. Thermal conductivity model for columnar TiO2 nanofluids. keff = keff 2 keff 2 =

2πrH 2πr 2 +2πrH

R

+ keff 2 RX

2πr 2

Z 2πr 2 +2πrH

(kp −klr )φklr (β12 −β 2 +1)+(kp +klr )β12 [φ β 2 (klr −kf )+kf ] β12 (kp +klr )−(kp −klr )φ p (β12 +β 2 −1)

β = 1 + h/r β1 = 1 + h/(2r) klr = 2.5kf H RZ = 2 2 RX =

kp πr +kf (H −πr 2 ) H−πr/2 πr/2 + kp 2rH+k kf H 2 f (H−2r)H

Without surfactants √ h = 2πσ

with surfactants h=l

Based on the series and parallel thermal resistance analysis, the thermal resistance in z-axis can be calculated by following equation: 1 kp πr 2 kf (H 2 − πr 2 ) = + Rz H H

(19)

H + kf (H 2 − πr 2 )

(20)

Rz =

kp

πr 2

Accordingly, the thermal resistance in X-axis can be calculated by following equation: RX =

H − πr/2 πr/2 + kp 2rH + kf (H − 2r)H kf H 2

(21)

Because the cube is of the same size in X-axis and Z-axis, the ratio of the thermal conductivity of the cube in X-axis and Z-axis can be calculated by following equation: kz R X = (22) kX RZ Assuming that the probability of the direction of the heat conduction is directly proportional to the surface area of the columnar nanoparticle in X-axis and Z-axis direction, the thermal conductivity of the columnar nanofluids can be modified as following equation: 2πrH RX 2πr 2 + k (23) eff 2 RZ 2πr 2 + 2πrH 2πr 2 + 2πrH For the thermal conductivity of columnar TiO2 nanofluids without surfactant, the modified size of nanofluids can be obtained by following equation: √ (24) h = 2πσ keff = keff 2

For the thermal conductivity of columnar TiO2 nanofluids with surfactant, the modified size of nanofluids can be obtained by following equation: h=l

(25)

As a consequence of the above processes of illation, the present models for columnar TiO2 nanofluids with and without surfactant are revealed in Table II, respectively III. APPLICATIONS

Some available experimental results in recent researches on the thermal conductivity for both spherical and columnar TiO2 nanofluids with and without surfactant were cited to verify the practicability of the present models. And the references are divides into following four groups for spherical and columnar TiO2 nanofluids with and without surfactant, respectively.

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A. Applications of the present model to spherical TiO2 nanofluids without surfactant

Reddy35 prepared three types of spherical TiO2 nanofluids with different base fluids including water, 40%:60% and 50%:50% ethylene glycol (EG)/water. Fig. 4 (a)-(c) shows the comparisons between experimental and theoretically determined thermal conductivity of spherical TiO2 nanofluids with water, 40%:60% EG/water and 50%:50% EG/water as basefluid at room temperature (30◦ C). It can be observed that Maxwell model, Bruggman’s model and Timofeeva’s model distinctly underestimate the thermal conductivity of spherical TiO2 nanofluid. This is presumably because those models have not taken into account the particle size effect and solvation nanolayer effect. However, although Yu and Choi’s model has including the influences of particle size and solvation nanolayer, the model data still under-estimate the effective thermal conductivity ratio for the nanofluid containing such a low volume loading of TiO2 nanoparticles. On the contrary, it can be observed that Bhattacharya’s model distinctly over-estimates the thermal conductivity of TiO2 nanofluids. This observation is might because that Bhattacharya’s model primarily considered the particle type and loading effects. And the predictions exceed the experimental values as a result of the big disparity in the thermal conductivity between TiO2 nanoparticle and the base fluids. It can be observed that the present model shows better precision than other models for the thermal conductivity of TiO2 nanofluids with all kinds of base fluids in their experiments. For water or 50%:50% EG/water based nanofluids, the present model shows a very little over prediction on the thermal conductivity of nanofluids when the volume fraction of nanoparticles approximate to 1%. For the 40%:60% EG/water based nanofluids, the present model under-estimate the thermal conductivity of nanofluids, which is the same with other models. However, it can be observed from Fig. 4 (d) that the maximum deviation of present model for the thermal conductivity of nanofluid with all kinds of base fluids is only about 1% which is much smaller than that of other models.

FIG. 4. Comparisons of models with data in Reddy’s experiment 35 of spherical TiO2 nanofluids, (a) water based nanofluids, (b) 40%:60% EG/water based nanofluids (c) and 50%:50% EG/water based nanofluids, (d) the relatively errors of a, b and c.

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FIG. 5. Comparisons (a) and relative errors (b) of various models with the data in Chen’s experiment34 for spherical TiO2 nanofluids (red lines fit the red points, black lines fit the black points).

It seems that when the content of nanoparticles is very low (2 vol%). It can also be seen from Fig. 6(b) that the present model shows better precision than all other models for the thermal conductivity of TiO2 nanofluids with CTAB as dispersant. C. Applications of the present model to columnar TiO2 nanofluids without surfactant

Fig. 7 (a) shows the comparisons between data in Chen’s experiment 34 and the theoretically determined values of various models for columnar TiO2 based nanofluids with pure water and pure EG as base fluids, respectively. It can be observed that the experimental data for columnar TiO2 /water and columnar TiO2 /EG nanofluids are distinctly larger than the theoretically determined values of Timoffeeva model and much lower than that of Bhattacharya model. All the other models including HC model, Xue’s model, Murshed model and the present model show better precision on the thermal conductivity of columnar TiO2 nanofluids because it can be observed from Fig.7 (b) that all the relatively errors are within 5%. However, the present model considering the influence of shape, solvation nanolayer and the end effect of the columnar nanoparticles still shows the best estimation precision among all the reference models for both water and EG based nanofluids since the maximum relatively error is within 2%. As a result of the thermal resistance in axial direction are lower than that in radial direction, the theoretically determined values of present model considering the end effect of the columnar nanoparticles is larger than that of other models for infinite length cylinder (nanotube) nanofluids, for instance, Murshed model. D. Applications of the present model to columnar TiO2 nanofluids with surfactant

Murshed25 also dispersed columnar TiO2 nanoparticles in deionized water by using surfactant CTAB as dispersant and measured the thermal conductivity. It was found that besides the particle loading, particle size and shape could also affect the thermal conductivity of TiO2 nanofluids. The results also showed that the enhancement is significantly higher than that estimated by conventional existing models. Fig. 8(a) shows comparisons between experimental and theoretically determined thermal conductivity of columnar TiO2 nanofluids with CTAB as dispersant. It can be observed that for nanofluids containing surfactant, all the models shows poorer precision of prediction of thermal conductivity than that in earlier application as already stated in this paper. Fig. 8(b) shows the relative errors of the present model and other reference models. It can be observed that the maximum

FIG. 7. Comparisons (a) and relative errors (b) of various models with the data in Chen’s experiment 34 for columnar TiO2 nanofluids.

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AIP Advances 6, 095104 (2016)

FIG. 8. Comparisons (a) and relative errors (b) of various models with the data in Murshed’s experiment 25 for columnar TiO2 nanofluids with surfactant.

relative error of all the models take place at the lower volume fraction (1%-2%) of nanoparticles except for Bhattacharya’s model. Assael37 attributed the reason for high enhancement of thermal conductivity of nanofluids when adding surfactant to the surface modification of particles by the dispersant. As a result of the surfactant can improve the dispersion stability of nanofluids, the influence of surfactant should be considered. It can be observed from Fig. 8(b) that the invigoration effect of nanoparticles on the thermal conductivity of fluids is more obviously, which may be as a result of the stability of nanofluids is better in lower concentration with surfactant. Bhattacharya’s model shows good agreement with the experimental data when the volume fraction of nanoparticles is within 2%. However, due to there is a huge difference between the theoretically determined values of Bhattacharya’s model and the experimental data when the volume fraction of nanoparticles exceeds 2%, the relative errors of Bhattacharya’s model is not shown completely in Fig. 8 (b). And this form of treatment is also employed in Fig. 4-7. It seems that the Bhattacharya’s model is easily to over-estimate the thermal conductivity for the nanofluids containing high thermal conductivity nanoparticles. Generally, although the maximum relative error is about 8%, the present model shows better precision of prediction on the thermal conductivity of TiO2 nanoparticles than other classical models whose maximum relative error exceed 13%. It can also be found that the theoretically determined values of present model when considering the end effect of the columnar nanoparticles and the addition of surfactant will be enlarged and closer to the experimental data. And the models for infinite length cylinder (nanotube) nanofluids can be upgraded by this consideration because the precision of prediction can be distinctly improved, for instance Murshed’s model which is built for the thermal conductivity of nanotube nanofluids. The present model was limited for TiO2 nanofluids since it is hard to validate the present model in different applications especially for containing surfactant and columnar nanoparticles for other kinds of nanofluids. It was found that TiO2 nanofluids are qualified for the four kinds of application and hence this paper selects TiO2 nanofluids as research subject. There are still many experimental papers on TiO2 nanofluids are available in the literature. But most of them are focused on spherical TiO2 nanofluids without surfactant. viz. the first application case of the present model. The experimental data on this case is distributed in a wide range. Our model for spherical TiO2 nanofluids without surfactant can fit many experiment data, and also many data that are not suitable for our model. However, the main innovation in our paper is the models for nanofluids containing surfactant and columnar particles. The experimental data in above conditions are relative less and the present models have shown better predicting accuracy on those two circumstances. Through the above analysis on the four kinds of applications, it can be concluded that the present models can upgraded Leong’s model and murshed’s model in new areas according to the different shape of nanoparticles and ingredients of the solvation nanolayer. And the present models show good applications for spherical and columnar TiO2 nanofluids with and without surfactant so as to

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AIP Advances 6, 095104 (2016)

extend the scope of application. However, due to the restrictions of quantity of experimental study on the thermal conductivity of rod-like TiO2 nanofluids or nanofluids containing surfactant, there are only few experimental results for the verification of present models on those special kinds of nanofluids. Therefore, the present models can be further improved or modified after more literatures that containing the experimental results of the thermal conductivity of those kinds of nanofluids are found. IV. CONCLUSIONS

This paper proposed thermal conductivity models for respectively spherical and columnar TiO2 nanofluids by considering the influences of the solvation nanolayer and the end effect at the top and bottom side of columnar nanoparticles. The thickness of the solvation nanolayer is defined by a few atomic distances for the nanofluids without surfactant, and defined by the surfactant molecule length for nanofluid containing surfactant, respectively. The end effect at the top and bottom side of the columnar nanoparticles is considered by analyzing the different thermal resistances and probability of the direction of the heat conduction for the selected small element in axial direction and radial direction. The present models and some other existing models were compared with some experimental data for both spherical and columnar TiO2 nanofluids with and without surfactant. The comparison results show that the present models gained better accuracies and precisions for the four kinds of applications. And the present models are recommended to be employed for estimating the thermal conductivity of TiO2 nanofluids in the following three circumstances: 1) in low volume concentrations (

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