roles of surfactants and particle shape in the enhanced thermal conductivity of tio2 nanofluids

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 nanopartic

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://aip.scitation.org/toc/adv/6/9

Published by the American Institute of Physics

Articles you may be interested in

Analysis and experimental study on the effect of a resonant tube on the performance of acoustic levitation devices

AIP Advances 6, 095302095302 (2016); 10.1063/1.4962546

Quantum spin Hall insulator phase in monolayer WTe2 by uniaxial strain

AIP Advances 6, 095005095005 (2016); 10.1063/1.4962662

Roles of surfactants and particle shape in the enhanced

Liu Yang,1,2, aXielei Chen,1Mengkai Xu,1and Kai Du1

1Key Laboratory of Energy Thermal Conversion and Control of Ministry of Education, School

of Energy and Environment, Southeast University, SiPaiLou 2#, 210096 Nanjing, China

2Jiangsu 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 estab-lished, few models for nanofluids containing surfactants or columnar nanoparticles are found This paper intends to consider the surfactants and particle shape effect in

respectively spherical and columnar TiO2nanofluids 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 differ-ent 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, stability1and thermal and transport characteristics2comparing

to the fluids containing millimeter or micrometer particles Nanofluids have been gradually used as advanced working fluids in industrial systems3such 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 Calcu-lating 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

base-fluid,7ultrasonic dispersion time,8storing time,9pH value,10,11surfactant12and temperature.13The third group is some microcosmic factors, such as surface charge state of nanoparticles,14Brownian motion,15the interfacial shell surrounding nanoparticles,16the dispersion stability,9and interactions

of nanoparticles,17the clustering of particles18etc Following the increase of researches on the influ-ence factors of thermal conductivity of nanofluids, more and more thermal conductivity models have been proposed via considering various parameters under various points of view

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 non-toxicity 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,10few 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

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

liquids suspended with small hard spherical particles, which had an expression as follows:

k eff = k f

k p + 2k f + 2φ(k p−k f)

k p + 2k f −φ(k p−k f) (1) Hamilton and Crosser20obtained a thermal conductivity model including the degree of sphericity

of the particles:

k eff = k f

k p + (n − 1)k f + (n − 1)φ(k p−k f)

where n is the empirical shape factor, which is given by n = 3/Ψ and Ψ is the sphericity.

an alternative formula for calculating thermal conductivity, which is expressed in the following form:

k eff = k f

k p + 2k f + 2φ(k p−k f)(1+ β)3

k p + 2k f −φ(k p−k f)(1+ β)3 (3) where β is the ratio of the nanolayer thickness to the original particle radius

Bruggeman22considered the effect of nanoparticle clustering and proposed a thermal

form:

k eff=1

4[(3φ − 1)k p + (2 − 3φ)k f]+k f

4

√

has an expression as follows:

an expression as follows:

Xue26proposed a model for calculating the thermal conductivity of nanotube nanofluids, which has an expression as follows:

k eff = k f

1 − φ+ 2φ k p

k p−k f lnk p +k f

2k f

1 − φ+ 2φ k f

k p−k f lnk p +k f

2k f

(8)

Jiang16considered the effect of interfacial layer and proposed a model for calculating the thermal conductivity of CNTs based nanofluids, which is expressed as follows:

k eff = k f

k pe + (n − 1)k f + (n − 1)φ(k pe−k f)

k pe = k f 2k p+ (β1 −1)(k p + k lr)

k lr=k p R(1 + t/R − k f /k p) ln(1+ t/R)

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 con-ductivity of nanofluids Leong27established 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 par-ticle 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:

k eff=(k p−k lr )φk lr(2 β

3

1−β3+ 1) + (k p + 2k lr) β13[φ β3(k lr−k f)+ k f]

β3

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 Hashimoto28presented 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:

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

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:

For nanofluids containing spherical TiO2nanoparticles with surfactant, as a result of TiO2is insuf-ferable, there is hardly any ionized ion of TiO2, the adsorption of surfactant to the TiO2nanoparticles

is supposed to abide by monolayer adsorption dispersion The mechanism of action of surfactants

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,17which was shown as following equation:

Thus, to sum up the above equations which we have just indicated, the present models for spherical TiO2nanofluids with and without surfactant are revealed in TableIrespectively

C Thermal conductivity model for columnar TiO2 nanofluids

deduced an equation in cylindrical coordinates using the same method, which was shown as follows:

k eff 2=(k p−k lr )φk lr( β

2

1−β2+ 1) + (k p + k lr) β2

1[φ β2(k lr−k f)+ k f]

β2

1(k p + k lr ) − (k p−k lr)φp( β2

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

keff=(k p−k lr )φk lr(2β3−β 3+1)+(k p +2k lr)β3[φ β3(k lr−k f)+k f]

β 3(k p +2k lr )−(k p−k lr)φp(β 3 +β 3 −1)

β = 1 + h/r

klr = 2.5k f

FIG 2 Simplification of a columnar TiO 2 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.2shows the simplification of a columnar TiO2nanoparticle into a cuboid geometry with an

equiv-alent 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.

TABLE II Thermal conductivity model for columnar TiO 2 nanofluids.

keff = k eff 2 2πr 2πrH2+2πrH + k eff 2 R R X Z 2πr

2

2πr2+2πrH keff 2=(k p−k lr )φk lr(β 2 −β 2+1)+(k p +k lr)β 2 [φ β 2(k lr−k f)+k f]

β 2(k p +k lr )−(k p−k lr)φp(β2+β 2 −1)

β = 1 + h/r

klr = 2.5k f

RZ= H

k p πr2+k f (H2 −πr 2 )

RX=H−π r/2

k f H2 +k p 2rH +k πr/2 f (H−2r)H

Based on the series and parallel thermal resistance analysis, the thermal resistance in z-axis can

be calculated by following equation:

1

R z =k p πr2

H +k f (H2−πr2)

k p πr2+ k f (H2−πr2) (20) Accordingly, the thermal resistance in X-axis can be calculated by following equation:

R X=H − πr/2

k f H2 + πr/2

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:

k z

k X=R X

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:

k eff = k eff 2

2πrH 2πr2+ 2πrH + k eff 2

R X

R Z

2πr2

For the thermal conductivity of columnar TiO2nanofluids without surfactant, the modified size

of nanofluids can be obtained by following equation:

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

nanofluids with and without surfactant are revealed in TableII, respectively

III APPLICATIONS

Some available experimental results in recent researches on the thermal conductivity for both spherical and columnar TiO2nanofluids with and without surfactant were cited to verify the practica-bility of the present models And the references are divides into following four groups for spherical

A Applications of the present model to spherical TiO 2 nanofluids without surfactant

Reddy35prepared three types of spherical TiO2 nanofluids with different base fluids including

between experimental and theoretically determined thermal conductivity of spherical TiO2

It can be observed that Maxwell model, Bruggman’s model and Timofeeva’s model distinctly

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 con-taining such a low volume loading of TiO2nanoparticles 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 TiO2nanoparticle and the base fluids

It can be observed that the present model shows better precision than other models for the thermal conductivity of TiO2nanofluids 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

(b) 40%:60% EG/water based nanofluids (c) and 50%:50% EG/water based nanofluids, (d) the relatively errors of a, b and c.

FIG 5 Comparisons (a) and relative errors (b) of various models with the data in Chen’s experiment 34 for spherical TiO 2

nanofluids (red lines fit the red points, black lines fit the black points).

It seems that when the content of nanoparticles is very low (<1%), the thermal conductivity

of TiO2 nanofluids is significantly higher than the theoretically determined data of the referred classical models, which may be as a result of the nanofluids with better dispersion stability at lower concentrations can achieve better performance of heat transport.36 This circumstance can also be observed at Chen’s experimental research,34in which the thermal conductivity of TiO2 nanofluids

is also obviously larger than the theoretically determined data of classical models Fig.5(a)gives the comparisons of models with experimental data for spherical TiO2nanofluids with water and EG

as basefluids, respectively And Fig 5(b)gives the relative errors of those various models It can

be observed that the classical models over-estimate or under-estimate the thermal conductivity of TiO2nanofluids especially at a little higher concentration For instance, for EG based nanofluids, the maximum relative errors of those classical models are about 8% While the present models shows

nanofluids since it can be observed from Fig.5(b)that the maximum relative error of the present model is lower than 2%

B Applications of the present model to spherical TiO 2 nanofluids with surfactant

Murshed23dispersed TiO2nanoparticles in spherical shapes of 15 nm in deionized water by using surfactant CTAB as dispersant and measured the thermal conductivity used a transient hot-wire appa-ratus Fig.6(a)shows the comparisons of models with experimental data for the thermal conductivity

of spherical TiO2nanofluids with CTAB as dispersant The calculated value of the length of CTAB

nanofluids containing surfactant.

molecular is 2.6 nm by commercially-available software in the present model It can be observed that when added CTAB as dispersant, the thermal conductivity are greatly increased, which may be also as a result of the improvement of stability of nanofluids by surfactant It can be found from the relative errors of the models as shown in Fig.6(b)that the maximum enhancement is determined to

be about 13%-15% higher than the calculated values of Maxwell model, Yu and Choi model, Tim-ofeeva model and Bruggeman model While Bhattacharya’s model well fitted the experimental data within the volume loading range of 2%, but it over estimates the thermal conductivity of nanofluids

at higher volume loadings (>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 TiO2nanofluids 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 experiment34 and the theoretically

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

H-C model, Xue’s model, Murshed model and the present model show better precision on the thermal

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

Murshed25also 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 TiO2nanofluids The results also showed that the enhancement is significantly higher than that estimated by conventional

thermal conductivity of columnar TiO2nanofluids 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

nanofluids.