1-s2.0-s1290072910002590-main

8/12/2019 1-s2.0-S1290072910002590-main

1/9

Modeling thermal conductivity augmentation of nanouids using

diffusion neural networks

Mohammad M. Papari a,*, Fakhri Youse b, Jalil Moghadasi b, Hajir Karimi c, Antonio Campo d

a Department of Chemistry, Shiraz University of Technology, Shiraz 71555-313, Iranb Department of Chemistry, Shiraz University, Shiraz 71454, Iranc Department of Chemical Engineering, Yasouj University, Yasouj 75914-353, Irand Department of Mechanical Engineering, The University of Texas at San Antonio, San Antonio, TX 78239, USA

a r t i c l e i n f o

Article history:

Received 10 July 2010

Received in revised form

4 September 2010

Accepted 8 September 2010

Available online 14 October 2010

Keywords:

Carbon nanotube

Heat transfer

Nanouid

Thermal conductivity

a b s t r a c t

In the present investigation, neural network method is employed to estimate thermal conductivity of

nanouids consisting of multi-walled carbon nanotubes (MWCNTs) suspended in oil (a-oln), decene

(DE), distilled water (DW), ethylene glycol (EG) and also single-walled carbon nanotubes (SWCNTs) in

epoxy and poly methylmethacrylate (PMMA). The results obtained have been compared with other

theoretical models as well as experimental values. The predicted thermal conductivities are in good

agreement with the literature values.

2010 Elsevier Masson SAS. All rights reserved.

1. Introduction

Within the different classes of tubes made of organic or inor-

ganic materials, carbon nanotubes (CNTs) are extremely promising

for applications in materials and medicinal chemistry[1]. Since the

discovery of CNTs by Iijima[2], CNTs have received much attention

in theoretical and experimental studies on condensed matter

physics and material sciences. CNTs are fascinating materials which

combine the microscale (length) with the nanoscale (diameter)

dimensions.

It should be mentioned that the investigations dealing with the

thermal properties of nanotubes has not received as wide attention

as the modeling of the mechanical properties, electronic properties,

or even the modeling of ows through nanotubes. Because ofexistence of some impurities in synthesized nanotubes the

measurements of the thermal conductivity of nanotubes or any

other transport properties in different solvents are subject to some

degree of uncertainty.

The thermal conductivity of a material is one of its thermody-

namic response functions. The denition of thermal conductivity, k,

is based on the macroscopic equation of heat ow known as the

Fouriers law

Jq kVTx; t (1)

whereJqis the amount of heat owing through a unit surface per

unit time andT is the temperature of the material. Two contribu-

tions are made to the thermal conductivity, one is due to charge

carriers (electrons) and the other is due to lattice vibrations

(phonons).

The question of estimating the thermal properties of individual

SWCNTs and MWCNTs as well as of their bundles has been

addressed in a number of computational investigations. Therefore

a couple of predictive computational modeling techniques of carbon

nanotubeshave become the focalpoint of researchin computationalnano-science area. In general, from a theoretical point of view,

nanotubes can be modeled in terms of inter-atomic potentials

describing the various forces experienced by the carbon atoms in

thenanotubes andby thoseforeign atoms interacting with thetubes

such as uid particles. For example, by knowing the inter-atomic

potentials, computational toolsconsisting of the molecular dynamic

simulation (MD)[3e8]and the Monte Carlo (MC) [9]methods have

been employed to model the transport properties of CNTs. MD

simulation revealed that isolated SWCNTs had a very similar

thermal conductivity as those of a hypothetical isolated graphene

sheet with the same number of atoms at certain temperatures[10].* Corresponding author. Fax: 98 711 726 1288.

E-mail address:[email protected](M.M. Papari).

Contents lists available at ScienceDirect

International Journal of Thermal Sciences

j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c om / l o c a t e / i j t s

1290-0729/$ e see front matter 2010 Elsevier Masson SAS. All rights reserved.

doi:10.1016/j.ijthermalsci.2010.09.006

International Journal of Thermal Sciences 50 (2011) 44e52
mailto:[email protected]://www.sciencedirect.com/science/journal/12900729http://www.elsevier.com/locate/ijtshttp://dx.doi.org/10.1016/j.ijthermalsci.2010.09.006http://dx.doi.org/10.1016/j.ijthermalsci.2010.09.006http://dx.doi.org/10.1016/j.ijthermalsci.2010.09.006http://dx.doi.org/10.1016/j.ijthermalsci.2010.09.006http://dx.doi.org/10.1016/j.ijthermalsci.2010.09.006http://dx.doi.org/10.1016/j.ijthermalsci.2010.09.006http://www.elsevier.com/locate/ijtshttp://www.sciencedirect.com/science/journal/12900729mailto:[email protected]

8/12/2019 1-s2.0-S1290072910002590-main

2/9

In another study, the dependence of the thermal conductivity of

a nanotube on its structure, defects, diameter and chirality were

investigated[11,12].

Because of stiffness of the sp2 bonding in CNTs, we expect that

CNTs exhibit a very high thermal conductivity [13]. Hence, it is

interesting to study the increase of thermal conductivity of

a nanouid which is obtained by dispersing small amount of CNTs

in traditional uids. Experimentally, during the last years the study

of the enhancement of the thermal conductivity of nanouids has

attained considerable interest [14e17]. Assael et al. [14] showed

that by adding 0.6 vol% suspension of MWCNTs to water, the

effective thermal conductivity of the resulting uid experiences

38% enhancement compared to the thermal conductivity of pure

water. Further, Choi et al.[16]found that 1 vol% of nanotubes in oil

leads to 2.5 fold increase in thermal conductivity of the uid.

In this paper, we have used neural network method to calculate

the effective thermal conductivities of suspension of multi-walled

CNTs in (a-oln) oil, decene (DE), distilled water (DW), ethylene

glycol (EG) and also single-walled CNTs in epoxy and poly meth-

ylmethacrylate (PMMA). In this respect, rst, we utilize a hybrid

model based upon the mega-trend-diffusion technique and neural

networks to estimate the expected domain range of data as hypo-

thetical. Second, we generate a number of virtual data points toreduce the error of estimatedfunctionwith respect to a small actual

dataset. Third, we build the robust prediction model for predicting

thermal conductivity of the aforementioned nanouids. Fourth, we

compare the obtained results with actual data as well as other

models showing good harmony with the literature values.

2. Literature review on the available theoretical approaches

for predicting the thermal conductivity of nanouids

As was mentioned earlier, isolated CNTs have anomalously high

thermal conductivity[13]. It can be expected that the CNTs based

suspensions can enhance the thermal conductivity and improve the

thermal performance of energy systems [16]. The thermal

conductivities of different particle-in liquid suspensions withspherical and nonspherical particles are of great interest in various

engineering applications because of their extra higher effective

thermal conductivities over those of base liquids at very low

particle volume concentrations of nano particles.

Several reasons are proposed for the anomalous increase of

thermal conductivity of nanouids. This enhancement may be

controlled by: 1) Brownian motion of particles [18e22], 2) the size

and concentration of nanoparticles [18,19,23], 3) the interfacial

conduction between the liquid and solid layer [24e30], 4) the

ordered layering of liquid molecules near the solid particles [24,25],

5) the existence of an interparticle potential [21], 6) convection in

the liquid because of the Brownian movement of the particles[21],

7) space distribution and axial effects of the CNTs in liquids [31], 8)

nanoparticle clustering [18,32,33], and 9) the particle shape [26,34].Further, a large surface-area-to-volume ratio can also explain the

anomalous enhancement in the thermal conductivity of nanouids

[23]. The latter effect varies inversely with the radius of the parti-

cles. Prasher et al.[21]claimed that the local convection caused by

the Brownian motion of the particles is the only acceptable

mechanism for explaining the observed enhancement. However,

previousstudies indicate that the heat transport in a CNTcomposite

is limited by the interfacial resistance and therefore, the thermal

conductivity of the composite would be much lower than the value

estimated from the intrinsic thermal conductivity of the nanotubes.

There are two ways to manage the interfacial resistance between

the nanotube and the base uid. They are: 1) chemical function-

alization on nanotubes [17,35] and 2) adding a dispersant to

nano

uid [14,15,36]. It is found that chemical modi

cation on

nanotubes can largely reduce the tube-matrix boundary resistance,

but at the same time decreases the intrinsic tube conductivity due

to truncating the CNTs to shorter segments[35,37].

Recently, it has been shown that the anomalousenhancement in

thermalconductivity is due to the effective heat transport through

the chainlike aggregates of nanoparticles and the thus Brownian

motion of nanoparticles is severely restricted after the formation of

chains and therefore has minor contribution in heat transport

within nanouid[38,39].

In addition, a couple of works on the role of temperature on the

thermal conductivity enhancement of nanouids were carried out

by some researchers [18,23]. Xuan et al. model reported a weak

dependency of heat transfer increasing on temperature [18]. Kumar

et al. [23] found a strong dependence of thermal conductivity

enhancement with temperature using moving particle model. The

temperature dependence was attributed to the variation of Brow-

nian motion velocity for the particles. However, this model cannot

be used for a large concentration of particles where inter-particle

interactions become important.

It is of interest to understand the mechanism of heat transfer

and to model it. In this respect, several studies have been initiated

for the development of such model accounting for all above-

mentioned effects in the model for the prediction of the thermalconductivity of nanouids containing nanosphere particles

[23,40e42] or carbonnanotubes [43]. But most of the proposed

models suffer from the fact that they fail to model the inherent

nature of the ow and energy transport process inside the nano-

uids. To understand the enhancement mechanism, Xuan and Yao

employed the lattice Boltzmann model [44]. According to this

model, there are several forces acting on each nanoparticle which

can be expressed as the vector sum of the Brownian force, the

interaction potential, the drag force, the buoyancy force and

the gravitational force. Also, the interactions potential between the

nearest neighbor nanoparticles are taken into account. It was

concluded that the Brownian motion is a dominant factor of

affecting the random displacement and the aggregation of the

nanoparticles. While the suspended nanoparticles are in motion inthe base uid, aggregation and clustering maybe occur. The cluster

formation depends on the internal potentials between the base

liquid and nanoparticles, as well as the internal potentials among

the particles. Therefore, the resulting increase of thermal conduc-

tivity is very small compared to the experimental. A corresponding

model for nanouids containing nanotubes was proposed by Nan

et al. [43]. The predicted thermal conductivity was greater than

those obtained experimentally. There are also some correlations for

estimating thermal conductivity enhancement [19,21,25,31,45e47].

Among all models cited we have compared our results with

those coming from Xues models[25,31,47]as well as from exper-

imental data. These models are presented in the following section.

It should be added at this point that the computer-based

simulation of thermal conductivity of CNTs-based nanouids hasalso become the focal points of research in computational nano-

science[20,36,44].

2.1. Xue models

In this chapter we review Xues models. It has been shown that

the measured enhancement of thermal conductivity for 1 vol%

nanotubes in oil is 160%[16], while the enhancements predicted by

the older models[40,41]are not more than 10%. That is to say, that

wecannot understand the effective thermal conductivity of nano-

uids based on the old models. As we know, all the old theoretical

models depend only on the thermal conductivity of the solid and

the liquid and their relative volume fraction, noton the particle size

and the interface between the particles and the

uid. In fact,

M.M. Papari et al. / International Journal of Thermal Sciences 50 (2011) 44e52 45

8/12/2019 1-s2.0-S1290072910002590-main

3/9

a recent actual study has revealed that molecules of normal liquids

close to a solid surface organize into layered structures much like

a solid[21]. Furthermore, there is evidence that such an organized

solid-like structure of a liquid at the surface is a governing factor in

heat conduction from a solid wall to an adjacent liquid[48].In this

regard, Choi and coworkers postulate that this organized solid/

liquid interfacial shell makes the transport of energy across the

interface effective[16]. Xue[25](hereafter referred as Xue-2003)

considered the above-mentioned postulates and based on Maxwell

theory and average polarization theory he formulated a new model

for calculating the effective thermal conductivity of nanouids.

Because the interfacial shells existed between the nanoparticles

and the liquidmatrix, both the interfacial shell and the nanoparticle

were considered as a complex nanoparticle. So, the nanouid

systemwas regarded as the complex nanoparticles dispersed in the

uid. Ifke is the effective thermal conductivity of the nanouid; kc is

the thermal conductivity of the complex nanoparticles and km is

the thermal conductivity of the uid, the relation for the effective

thermal conductivity of the complex nanoparticleeuid system is

given by:

9

1

n

lkekm

2kekm

n

l kekc;x

keB2; x

kc; x ke

4 kekc;y

2ke

1 B2; x

kc; y ke

0 2

where n and n/l arethe volumefraction of the nanoparticles and the

complex nanoparticles, respectively. Other parameters in the above

equation are represented by the following relations:

l abc

atbtct (3)

B2;y B2;z 1 B2; x

2 (4)

B2;x abc

2

ZN0

duua2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiua2

ub2

uc2

r (5)

kc; j k1

1 B2; j

k1 B2 ;jk2

1 B2; j

lk2k1

1 B2; j

k1B2; j k2B2; jlk2k1 (6)

In these expressions, it is assumed that the complex nanoparticle is

composed of an elliptical nanoparticle of thermal conductivity k2with half radii of (a,b,c) and an elliptical shell of thermal conduc-

tivity k1 with thickness t. Based on Maxwell theory and average

polarization theory he considered the interface effect between thesolid particles and the base uid. The author concluded that the

interfacial shell about the nanotube or nanoparticle has a major

contribution in thermal conductivity of the nanouid. As the

thickness of this interfacial shell becomes larger, the thermal

conductivity of the nanouid increases.

Xue [31] (hereafter we named Xue-2005) developed a new

model for the effective thermal conductivity of CNTs-based

composites relying on Maxwell theory considering the space

distribution and axial effects of the CNTs. Considering the very large

axial ratio and the space distribution of the CNTs, Xue built a new

model for the effective thermal conductivity of CNTs-based

composites. Based on Maxwell theory, two formulae of calculating

the effective thermal conductivity of CNTs-based composites were

generated. The model shows that the axial ratio and the space

distribution of the CNTs can signicantly affect the effective

thermal conductivity of CNTs-based composites so that the

dispersion of a very small amount of CNTs can result in a remark-

able enhancement in the effective thermal conductivity of the

composites.

Xue assumed thatkeis the effective thermal conductivity of the

CNTs-based composites,kcis the thermal conductivity of the CNTs

and km is the thermal conductivity of the main media. By consid-

ering two kinds of distribution functions, the following relations for

the effective thermal conductivities were developed:

ke km1 f 4f=p

ffiffiffiffiffiffiffiffiffiffiffiffiffikc=km

p arctg

p=4


p 1 f 4f=p

ffiffiffiffiffiffiffiffiffiffiffiffiffikm=kc

p arctg

p=4


p (7)wheref is the total volume fraction of the CNT and

ke km

1 f 2f kc

kckmln

kckm2km

1 f 2f km

kckmln

kckm2km

(8)

Xue concludedthat the distribution state andlarge ratio of the CNTs

play an important role in the thermal conductivity of nanouids.

However, the large discrepancy between the calculated values

using eqs.(7) and (8) and the experimental measurements of the

thermal conductivity of CNTs-oil composite was noticeable.

Recently, Xue [47] (hereafter we name Xue-2006) proposed

a novel approach for predicting the effective thermal conductivity

for carbon nanotube composites by incorporating the thermal

resistance and the average polarization theory in his previous

works[25,31].In this new method, the combined effect of carbon

nanotube length, diameter, concentration, interface and matrix on

the thermal conductivity of the carbon nanotube composite have

been considered simultaneously:

91 fkekm2kekm

f" kekc33

ke 0:14 dL

kc33ke

4 kekc11

2ke12

kc11ke

#

0 9

whereRkis thermal resistance, L and d are length and diameter of

carbon nanotube,fis the carbon nanotube volume fraction, kcand

km are the thermal conductivities of carbon nanotube and the

matrix, respectively. Also, the transverse and longitudinal equiva-

lent thermal conductivities kc11 and kc

33 are obtained from the

following equations:

kc

11

kc

1 2Rkkcd

; kc

33

kc

1 2RkkcL(10)

It has been shown that the effective thermal conductivity increases

rapidly with increasing nanotube length, but the effective thermal

conductivity changes very little when the diameter changes. Also, it

is shown that the effective thermal conductivity increases rapidly

with increasing thermal conductivity of CNTs, whereas the effective

thermal conductivity decreases rapidly with increasing thermal

conductivity of the matrix. It was also found that the large interface

thermal resistance between the CNTs and the base uid causes

a signicant degradation in the thermal conductivity. This thermal

resistance can be attributed to the weak atomic bonding at the

particle-matrix interface [49]. It should be mentioned that this

model is valid in the whole range of carbon nanotube

concentration.

M.M. Papari et al. / International Journal of Thermal Sciences 50 (2011) 44e5246

8/12/2019 1-s2.0-S1290072910002590-main

4/9

2.2. A renovated HamiltoneCrosser model

Yu and Choi [26]developed a method to modify the Maxwell

equation [50] for spherical particle-in-liquid suspensions with each

spherical particle surrounded by an ordered liquid layer. They

reported that the ordered liquid layer plays an important role in

enhancing the thermal properties of nanouids. Choi et al. [16] cited

that multi-walled carbon nanotubes enhanced the thermal

conductivity of the base uid by as much as 160% at a concentration

of 1 vol%; the highest enhancement reported for a nanouid.

Because nanotubes are non-spherical, the renovated Maxwell

model [50] cannot be used to predict the effective thermal

conductivity of nanotube-based nanouids. In contrast, Hamilton

and Crosser[34]showed that when the particle-to-liquid conduc-

tivity ratio of a suspension is above 100, the particle shape has

a substantial effect on the effective thermal conductivity of the

suspension. For nanotube-in-liquid suspensions, the ratio of the

thermal conductivities of the two phases would be of the order of

10,000. Although the renovated Maxwell model is limited to

suspensions with spherical particles, an important feature of the

HamiltoneCrosser model [34] is that it can predict the effective

thermal conductivity of suspensions owing non-spherical particles.

However, the Hamiltone

Crosser model does not include the effectof the nanolayer. Yu and Choi, in 2004, extended the Hamil-

toneCrosser model [26] to ellipsoidal particle-in-liquid suspensions

with each ellipsoidal particle surrounded by an interfacial layer.

To calculate the effective thermal conductivity of the complex

ellipsoid-in-liquid suspension containing monosized anisotropic

ellipsoids of the equivalent thermal conductivitykpj(j a,b and c)

and the equivalent volume concentration fe, it is assumed that the

thermal conductivity of the base liquid,k1is isotropic. Based on the

average theoryand this assumption, by rewriting and averaging the

HamiltoneCrosser equation[26]along the semi-axis directions of

the complex ellipsoids, one obtains the effective thermal conduc-

tivity ke of the suspension of monosized ellipsoidal particles of

semiaxes a,b, andc. That is

ke

1

nfeA

1 feA

k1 (11)

n 3Ja

(12)

J 2et

1 e2t

1=6et

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 e2t

p arcsinet

(13)

et

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

b2 t

a2 t

s

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

c2 t

a2 t

s (14)

A 1

3 Xj a;b;ckpjk1

kpj n 1k1(15)

fe

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia2 t

b2 t

c2 t

rabc

f (16)

kpj

1

kpkskprdj;0 dj;t ksrdj;0 dj;t r

ks (17)

r

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia2 t

b2 t

c2 t

rabc

(18)

dj;n

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia2 n

b2 n

c2 n

r2

ZN0

dwj2 n w

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia2 n w

b2 n w

c2 n w

r(19)

where fis the volume concentration of the solid ellipsoids without

surrounding layers,fe is the volume concentration of the complex

ellipsoids, a is an empirical parameter and J is the sphericity

dened as the ratioof the surface area of a sphere to a volume equal

to that of the particle, n is the empirical shape factor, kpand ksare

the thermal conductivities of the solid ellipsoid and its surrounding

layer. However, this model fails to predict the nonlinear behavior of

the nanouid thermal conductivity.

3. Conventional articial neural networks

Although articial neural networks (ANNs) is widely utilized toextract management knowledge from acquired data, but neural

information processing models mainly assume that the data are

compatible and the learning data for training a neural network are

sufcient. For cases in which the data are insufcient, it is impos-

sible to recognize a nonlinear system. In other words, there exist

a non-negligible error between the real function and the estimated

function from a trained neural network[51]. Moreover using basic

neural networks with small data points (actual points), it is difcult

to guarantee that a good predictive model will be obtained in the

complete actual domain. In most cases, only good predictions are

achieved in regions in the vicinity of actual points contained in the

training dataset [52]. However, a sufcient training data set is

dened as a data set which provides enough information to

a learning method to obtain stable training accuracy.The ANNs supplies a non-linear function mapping of a set of

input variables into the corresponding network output variables,

without the requirement of having to specify the actual mathe-

matics form of the relation between the input and output variables.

The multilayer percepetron (MLP) is a type of feed forward neural

network that has been used commonly for the approximate func-

tions [53]. The ANN technique has been applied successfully in

various elds of modeling and prediction in many engineering

systems, mathematics, medicine, economics, metrology and many

others. It has become increasingly popular in during last decade.

The advantage of ANN compared to conceptual models are its high

speed, simplicity and large capacity which reduce engineering

attempt. The ANNs are discussed in details in the literature, and

therefore only a few prominent features of it are given here todescribe the general nature of the network. MLP neural networks

consist of multiple layers of simple activation units called neurons

that are arranged in such a way that each neuron in one layer is

connected with each neuron in the next by weighted connections

(seeFig. 1). Neurons are arranged in layers that make up the global

architecture. MLP networks are comprised of one input layer, at

least one hidden layer and an output layer. The number of neurons

in the input layer is dened by the problem to be solved. The input

layer receives the data. The output layer delivers the response

corresponding to the property values. The hidden layer processes

and organizes the information received from the input layer and

delivers it to the output layer. The number of neurons in the hidden

layer, which to some extent play the role of intermediate variables,

may be considered as an which might be either output from other


8/12/2019 1-s2.0-S1290072910002590-main

5/9

neurons or input from external sources. Each neuron thus has

a series of weighted inputs, wijwhich might be either output from

other neurons or input from external sources. Each neuron calcu-

lates a sum of the weighted inputs and transforms it by thefollowing transfer function:

nj 1

1 exp x (20)

wherenj is the output of the j-th neuron andx is given by:

x Xni 1

wijpibj (21)

wherewijrepresents the weights applied to the connections from

thei-th neurons in the previous layer to the j-th neurons, piis the

output from the i-th neuron and b is a bias term. MLP networks

operate in the supervised learning mode. In the training algorithm,

commonly back propagation error called BP, training data are given

to the networks and the network iteratively adjusts connection

weightsw and biases b (starting from initial random values) until

the network predicted values match the actual values satisfactorily.

In the BP training algorithm, this adjustment is carried out by

comparing the actual value tij and the predicted values aij of the

network by means of calculation of the total sumof the square error

(SSE) for the n data of the training dataset,

SSE Xni 1

tijaij

2(22)

3.1. Detailed steps in the ANN modeling procedure

In this work, the actual data set was provided by the Refs. [14,21]

and[54e57]. The number of actual data points is 43 in total and

presented in the sequence ofFigs. 4e9.

According to available data, there are two input variables cor-

responding to thermal conductivity ofuid (kf), volume fraction of

CNTs (f) and thermal conductivity ratio (ke/kf) of carbon nanotube

(ke/kf) as output variable. As explained previously, the available

experimental data points (43 data points) do not provide sufcient

information for training robust prediction neural network. There-

fore, the research of Li et al. [58]was referred to, and the mega-

trend-diffusion and estimation of domain range techniques were

used to create virtual data (adding a number of virtual data points)

to construct the predicted model. Because the purpose of this

research is to apply the procedure proposed by Li et al. to construct

a prediction model, the detailsof the derivation of the methodology

are omitted here. In order to ll all the data gaps and estimate the

data trend, the mega-trend-diffusion method assumes that the data

selected are located within a certain range and have possibility

indexes determined by a common membership function (MF). The

numbers of data which are smaller or greater than the average of

the data are considered the skewness of the MF, which is used to

estimate the trend of the population. When the data collected are

insufcient to build a reliable predicting model, the procedure

systematically generates some virtual data between the estimated

lower and upper extremes (L and U) of the range to ll the data

gaps. The computation of L and U mainly employs the statistical

diffusion techniques provided in the procedure. The procedure

further calculates the MF values for each virtual data as its impor-

tance level.

The Li et al. [57] showed that if the number of actual data

(experimental data) chosen for generation of virtual data increases

the average error rate of trained net decreases. So in this study rst

all available actual data were used for generating a virtual data then

the ANN was trained using generated data. Finally the actual data

were used for testing the ANN. There are a total of four steps in theprocedure which was implemented with 43 actual data:

Step 1 Select all actual data as the training data of ANN.

Step 2 Use eqs. (23) and (24) to calculate the estimated lower

bound (L) and upper bound (U),

L UsetSkL

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 bs2x=NL ln4Lq (23)

U UsetSkU

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 bs2x=NU ln4Uq : (24)

In these equations:

hset bs2x=n (25)Uset min max=2; (26)

wherebs2x Pni1xix2=n 1 is the data set variance; n is thedataset size; andUset, the core of the dataset so that

SkL NL=NLNU and SkU NU=NLNU (27)

The skewness (Sk) characterizes the degree of asymmetry of

a distribution around dataset core. SkU indicates a distribution

fraction with an asymmetric tail extending greater than the core

value.SkL indicates a distribution fraction with an asymmetric tail

extending less than core value.NLand NUrepresent the number of

data smaller and greater than Uset, respectively.Since a common function to diffuse the data set was utilized, the

values of the membership function for these two data range limits,

the left end L and the right end U, approach zero. Consequently, in

eqs.(23), and (24), a very small number, 1020 was set for4(L) and

4(U) to be the value of the membership function, respectively. The

lower (L) and upper (U) limits are calculated using equations(23)

and (24)for each variable (i.e. thermal conductivity of uid (kf),

volume fraction of CNTs (f) as inputs and thermal conductivity ratio

(ke/kf) of carbon nanotube (ke/kf) as output variable) of the ANN.

The results are listed inTable 1.

Step 3 generate a virtual data between L and U for each variable.

Step 4 for each variable, the corresponding value of the member-

ship function (MF) using eq. (28)is calculated. The thermal

Fig. 1. Topology of conventional P-S-1 MLP.

M.M. Papari et al. / International Journal of Thermal Sciences 50 (2011) 44e5248

8/12/2019 1-s2.0-S1290072910002590-main

6/9

conductivity ratio is the output for the ANN, and there is no

need to calculate its MF value. The numbers randomly

selected for the variables and the corresponding value of MF

are summarized inTable 2.

MF

( NvLUsetL

Nv < UsetUNv

UUsetNv > Uset

(28)

whereNvis a virtual number.

Because we repeat this procedure 90 times foreach variable, this

gives rise to 90 virtual data for the training the MLP network.Table

2contains the data generated for each variable.

The conventional topology of MLP can be shown as a P-S-1

network appeared in Fig. 1. In this topology, P represent the two

input variables corresponding to the thermal conductivity of theuid (kf) and volume fraction of CNTs (f); S is the neuron in hidden

layer and 1 is the output variable corresponding to the thermal

conductivity ratio (ke/kf).

In sum, the hybrid model that was employed in this work

comprises both the mega-trend diffusion technique along with the

neural network. This hybrid model comprises an input layer of four

neurons corresponding to the thermal conductivity of the uid (kf),

the volume fraction of CNTs (f) and theircorresponding values of MF,

plusa variable numberof neuronsin thehidden layer anda neuron in

the output layer corresponding to the thermal conductivity ratio

(ke/kf). This topology is illustrated inFig. 2. The output layer neuron

has a linear transfer function, while the hidden layer neurons have

a sigmoid transfer function. The neural network was trained

according to scaled conjugate gradient back propagation (train scg)algorithm available in the neural network toolbox of MATLAB.

The 90 virtual data obtained in Step 4 were assigned to the

training of MLP, while the actual data were utilized to test the

model.

In the rst step of the training procedure, all data points were

scaled to the range of [1, 1]. For the training step, the number of

neurons in the hidden layer plays an important role in the network

optimization. Therefore, in order to optimize the network, to ach-

ieve generalization of the model and avoid over-tting, we started

with 1 neuron in the hidden layer and gradually increased the

number neurons until no signicant improvement in he perfor-

mance of the net was observed. For this study, the mean square

error (MSE) was chosen as a measure of the performance of the

networks. The network model with eight neurons in the hiddenlayer, maximum epoch 300, momentum constant 0.8 and learning

rate 0.01. The MSE for this conguration was 0.14%. After deter-

mining the optimal topology of the network, the actual data and

the corresponding MF values areused as testing data for the trained

MLP to calculate the absolute average error. The average absolute

deviation (AAD) is dened as:

AAD

PNi 1

j

Xexp:i

Xcalc:i

.X

expi

j 100

N (29)

where N is the number of data.

4. Results and discussion

As already mentioned, in this study articial neural network

(ANN) scheme has been employed to compute the effective thermal

conductivity of nanouids. The selected nanouids were the multi-

walled CNTs suspended in oil (a-oln), decene (DE), distilled water

(DW), ethylene glycol (EG) and also single-walled CNTs suspended in

epoxy and poly methylmethacrylate (PMMA). Several improved

models cited in the previous sections[25,26,31,47]capable of calcu-

lating the therrmal conductivity of CNTs-based composites were

selected to be compared against our results. In these models, the size

of the particles and interface between CNTs and uid (interface

effect), and the concentration of CNTs are considered. It should be

remembered that most of the old theoretical models[40,41]dependonly on the thermal conductivity of particles, the thermal conduc-

tivity of the base liquid and the particle volume fraction.

In the present calculations, the thermal conductivity of CNTs is

taken as 600e3000 W/m K, and the thermal conductivity of oil,

decene, distilled water, ethylene glycol, epoxy and poly methyl-

methacrylate are taken as 0.1448, 0.14, 0.60, 0.25, 0.198 and 0.21 W/

m K, respectively.

As already stated, Xue-2003 [25] developed a new model for

predicting the effective thermal conductivity of nanouids. This

model is based on the average polarization theory and takes into

account the effect of the interface between the solid particles and

the base uid. Xue-2003 claims that the model predictions are in

good agreement with the actual data for nanotube-in-oil nano-

uids and exhibits nonlinearity with the nanotube loadings.We selected the following parameters to predict the thermal

conductivity of the aforementioned nanouids: a 50,000 nm,

b c 12.5 nm, kl 0.1448 W/m K, kp 2000 W/m K, interfacial

layer thickness 3 nm, ks 5 W/m K, and depolarization factor

along a axis 0.0062. We performed the calculations of eqs.

(2)e(6)using the above parameters for MWCNTS in oil suspension

Table 2

The random numbers selected between L and U for each factor.

kf f ke/kf

Nv 0.1455 0.1993 0.2513 0.2713 0.1295 0.3333 1.2867 1.0767 1.0337

MF 0.0238 0.2580 0.4841 0.1088 0.0519 0.1336 0.3675 0.983 0.0432

Fig. 2. The topology of 2P-S-1 MLP for the small data set training.

Table 1

The lower and upper limits of virtual data for each variable.

Limit kf f ke/kf

L 0.14 0.0 1

U 0.6 4.99 2.56


8/12/2019 1-s2.0-S1290072910002590-main

7/9

8/12/2019 1-s2.0-S1290072910002590-main

8/9

30 108,15.9 107,13.9 107, 20 10-8, 25 10-8 and 20 10-7

Km2W-1 for CNT/oil, CNT/DE, CNT/EG, CNT/DW, CNT/epoxy and

CNT/PMMA, respectively.We carried out articial neural network calculations, explained

in the previous section, for all uids and compared the results

obtained against those calculated by the above-mentioned models.

For verication of the validity of our calculations and the models,

the predicted values of the thermal conductivities have been

compared with the experimental measurements[16,17,56].

Figs. 4e7 compare the thermal conductivity ratio (ke/kf) of

a multi-walled carbon nanotube (MWCNT) in oil, decene, distilled

water, and ethylene glycol, respectively, as a function of the volume

fraction of nanotubes using the articial neural network with those

values obtained by models and experiments. As it may be observed

inFigs. 4e7, the calculated thermal conductivities using the ANN

method agree well with both the experimental values [16,17,56]

and other models[26,31,47].

Further, Figs. 8 and 9 illustrate the comparison between the

calculated thermal conductivity of single-walled carbon nanotube

(SWCNT) in epoxy and PMMA uids using the ANN method and

models. Shown for comparison, the experimental values of thermal

conductivities of aforementioned composites [14,52,57e59] have

also been demonstrated.

In order that the validity of the results obtained are checked

further, the AAD of thermal conductivity ratios from the literature

values in the entire range of volume fraction for whole of nano-

uids has been tabulated in Table 3. As the numbers in Table 3

attest, the thermal conductivity ratio computed from the Mega-

Trend-Diffusion neural network method stands over other models.

FromTable 3, it is immediately veriable that the DNN-MLP model

possesses a high ability to predict the thermal conductivity with an

absolute average error of 3.26%. There is a good agreement between

the predicted and the experimental values of the thermal

conductivity ratio (r 0.991). Herein, the standard deviation in the

relative errors was 2.3%, meaning that the dispersion around theaverage was very small.

It should be mentioned that examining articial neural network

approach to predict other transport properties, say viscosity, of

nanouids remains for future work.

Acknowledgements

The authors thank computer facility provided by Shiraz

University of Technology and Shiraz University.

References

[1] C.P. Poole, F.J. Owens, Introduction to Nanothechnology. Wiley-Interscience,

Hoboken, New Jersey, 2003.[2] S. Iijima, Helical microtubules of graphitic carbon, Nature 354 (1991) 56e58.[3] R.E. Tuzun, D.W. Noid, B.G. Sumpter, R.C. Merkle, Dynamics ofuidow inside

carbon nanotubes, Nanotechnology 7 (1996) 241e246.[4] R.E. Tuzun, D.W. Noid, B.G. Sumpter, R.C. Merkle, Dynamics of He/C60 ow

inside carbon nanotubes, Nanotechnology 8 (1997) 112e118.[5] D. Qian, W.K. Liu, R.S. Ruoff, Mechanics of C60 in nanotubes, Journal of

Physical Chemistry B 105 (2001) 10753e10758.[6] V.P. Sokhan, D. Nicholson, N. Quirke, Fluid ow in nanopores: accurate

boundary conditions for carbon nanotubes, Journal of Chemical Physics 117(2002) 8531e8539.

[7] S. Supple, N. Quirke, Rapid imbibition ofuids in carbon nanotubes, PhysicalReview Letters 90 (2003) 214501e2145014.

[8] J. Li, L. Porter, S. Yip, Atomistic modeling ofnite-temperature properties ofcrystallineb-SiC II. Thermal conductivity and effects of point defects, Journalof Nuclear Materials 255 (1998) 139e152.

[9] Q. Wang, J.K. Johnson, Molecular simulation of hydrogen adsorption in single-walled carbon nanotubes and idealized carbon slit pores, Journal of Chemical

Physics 110 (1999) 577e

586.

1

1.05

1.1

1.15

1.2

0 0.2 0.4 0.6 0.8 1 1.2

CNT volume Fraction %

Thermalcon

ductivityratio

Liu et al (2005)

Xie et al (2003)

Assael et al (2006)

Xue (2005) model

Xue (2006) model

Yu and Choi model

+ Neural network

Fig. 7. Comparison of the thermal conductivity ratio of the MWCNTs-EG suspensions

obtained from the Diffusion-Trend-Neural network with experiment and with some

models.

1

1.5

2

2.5

3

3.5

4

4.5

5

0 0.5 1 1.5 2 2.5

CNT volume Fraction %

Thermalconductivityratio

Biercuk et al (2002)

Assael et al (2003)

Bryning et al (2005)

Xue (2005) model

Xue (2006) model

Yu and Choi model

+ Neural network

Fig. 8. Comparison of the thermal conductivity ratio of the SWCNTs-epoxy suspen-

sions obtained from the Diffusion-Trend-Neural network with experiment and with

some models.

1

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

0 2 4 6 8CNT volume Fraction %

Thermalconductivityratio

Du. F et al (2006)

Xue (2005) model

Xue (2006) model

Yu and Choi model

+ Neural network

Fig. 9. Comparison between actual data of SWCNTs-PMMA suspensions and the

calculated values from mentioned models and Diffusion-Trend-Neural network.

Table 3

The AAD% of the calculated effective thermal conductivity ratio using Xue-2005,

Xue-2006, Yu and Choi models, as well as ANN method from the experimental data.

System Xue-2005 Xue-2006 Yu and Choi 2004 Neural network

MWCNT/Oil 4.75 7.89 4.52 2.79MWCNT/DE 0.38 0.62 0.81 2.50

MWCNT/DW 7.22 4.89 5.22 3.64

MWCNT/EG 1.24 1.15 1.79 1.86

SWCNT/Epoxy 15.86 5.47 3.32 2.48

SWCNT/PMMA 15.74 16.90 15.29 6.31

Overall 7.53 6.15 5.16 3.26


8/12/2019 1-s2.0-S1290072910002590-main

9/9

1-s2.0-s1290072910002590-main

Documents