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Research Article

Numerical Investigation of Characteristic of

Anisotropic Thermal Conductivity of Natural Fiber

Bundle with Numbered Lumens

Guan-Yu Zheng

Department of Building Engineering, College of Civil Engineering, Tongji University, Shanghai 200092, China

Correspondence should be addressed to Guan-Yu Zheng; zheng guanyu@.com

Received July ; Accepted July ; Published August

Academic Editor: Song Cen

Copyright © Guan-Yu Zheng. is is an open access article distributed under the Creative Commons Attribution License,

which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Natural ber bundle like hemp ber bundle usually includes many small lumens embedded in solid region; thus, it can present lower

thermal conduction than that of conventional bers. In the paper, characteristic of anisotropic transverse thermal conductivity of

unidirectional natural hemp ber bundle was numerically studied to determine the dependence of overall thermal property of the

ber bundle on that of the solid region phase. In order to eciently predict its thermal property, the ber bundle was embedded into

an imaginary matrix to form a unit composite cell consisting of the matrix and the ber bundle. Equally, another unit composite cell

including an equivalent solid ber was established to present the homogenization of the ber bundle. Next, nite element thermal

analysis implemented by ABAQUS was conducted in the two established composite cells by applying proper thermal boundary

conditions along the boundary of unit cell, and inuences of the solid region phase and the equivalent solid ber on the composites

were investigated, respectively. Subsequently, an optional relationship of thermal conductivities of the natural ber bundle and the

solid region was obtained by curve tting technique. Finally, numerical results from the obtained tted curves were compared with

the analytic Hasselman-Johnson’s results and others to verify the present numerical model.

1. Introduction

Natural bers like kenaf ber [], hemp ber [], sisal ber

[], date palm ber [], wood ber [], and bamboo ber []

have unique advantages of low density, high specic prop-

erties, biodegradable nature, and low cost; thus, composites

lled with natural bers, such as natural ber reinforced

polymer/cement composites, are usually viewed as green

and environmentally friendly composites and have attracted

much attention of researchers for potential engineering

application. As one of inherent material properties of natural

bers, thermal property of natural bers is of great impor-

tance in natural ber reinforced composites, due to inherent

hollow microstructure of natural bers. Recent researches

have shown that natural bers consisting of cellulose or

lumens can present extremely lower thermal conduction than

conventional bers like glass bers and carbon bers [];

thus, natural bers reinforced composites can be considered

to be thermal insulator in such engineering as building and

furniture. In addition, it is viable to achieve the aim of light-

weight and proenvironment composite materials by consid-

ering natures of hollow microstructure and biodegradability

of natural bers.

In the past few years, many researchers investigated

thermal properties of natural bers and composites lled

with them. For example, El-Shekeil et al. experimentally

investigatedtheinuenceofbercontentonthemechanical

and thermal properties of kenaf ber reinforced thermo-

plastic polyurethane composites []. Liu et al. evaluated the

transverse thermal conductivity of Manila hemp ber in

solid region by the nite element method and analytical

Hasselman-Johnson’s model [].Also,theystudiedtheeect

ofthemicrostructureofnaturalberonthetransversether-

mal conductivity of unidirectional composite with abaca and

bamboo bers, by experiment and nite element simulation

[]. Behzad and Sain predicted the thermal conductivity for

hemp ber reinforced composites by experimental measure-

ment [], and subsequently they developed a nite element

Hindawi Publishing Corporation

Mathematical Problems in Engineering

Volume 2014, Article ID 506818, 8 pages

http://dx.doi.org/10.1155/2014/506818

Mathematical Problems in Engineering

50 𝜇m

(a)

Lumen

Solid region

(b)

F : (a) Cross-section morphology of the hemp ber bundle [] and (b) schematic version of the hemp ber bundle.

simulation procedure to predict the temperature prole and

the curing behavior of the hemp ber/thermoset composite

during the molding process []. Mangal experimentally

measured the thermal properties of pineapple leaf ber

reinforced composites []. Takagi and cooperators analyzed

thermalconductivityofPLA-bamboobercompositesusing

ahot-wiremethod[]. All these works mentioned above

have been benecial in understanding of thermal transfer

mechanism in the natural ber and design of natural ber

lled composites with desirable thermal properties.

As important llers of green composites, it is necessary to

establish comprehensive understanding of thermal properties

of natural bers or ber bundles. In this paper, the emphasis

is put on the study of thermal properties of the natural hemp

berbundle.Fromtheviewpointofcomposite,thenatural

hemp ber bundle can be viewed as composite material

with outstanding thermal properties, because, in the natural

ber bundle of interest, there are a large number of lumens

lled with air in transverse direction of it (see Figure (a)

for the cross-section morphology of the hemp ber bundle

[]). e thermal properties of the natural ber bundle vary

considerably depending on lumen volume and size and also

the thermal property of solid region phase, which encloses

thelumensinthenaturalberbundle.Figure (b) shows a

schematic illustration of the natural ber bundle consisting

of lumens and solid region. It is observed in Figure that

the large-scale ber bundle is lled with many small-scale

lumens in the solid region; thus, thermal properties of the

lumen and the solid region are important parameters of the

natural ber bundle. In practice, the lumen ller is lled

with air; thus, its thermal conductivity is usually specied

with very small value, for example, .W/(mK), which is

normally the thermal conductivity of air measured at the

standard atmosphere. erefore, the thermal conductivity of

the ber solid region more signicantly aects the whole

thermal performance of the ber bundle than that of lumen.

Here, the main purpose of this study is to investigate the

eect of material thermal property of the solid region phase

on the equivalent anisotropic thermal property of the natural

ber bundle by nite element simulation [,]ofcomposite

microstructure [] and then establish an optional interrela-

tionship between them by curve tting technique []togive

a rapid and highly accurate prediction of material thermal

properties for both of them.

2. Finite Element Model for

Anisotropic Natural Fiber Bundle

Reinforced Composites

In this paper, the natural ber bundle is assumed to be

embedded into a polymer matrix with constant thermal

conductivity to form a square representative volume element

(RVE) (or unit cell) (see Figure ), as was done by many

researchers in the analysis of heterogeneous materials [–

]. en, the nite element model of natural ber bundle

reinforced composites [,,]willbeestablishedto

investigate the inuence of solid region phase on the ber

bundle. e established unit cell consists of three dierent

regions, that is, matrix, solid region, and lumen. Each region

has isotropic thermal conductivity.

eoretically, the distribution of lumens in the natural

ber bundle will cause anisotropy of the composite under

consideration. erefore, in this study, the anisotropic ther-

mal conductivities of the composite will be investigated. In

steady-state heat conduction problem, the temperature eld

within the anisotropic representative volume element satises

the quasiharmonic dierential equation:

𝑥

+

𝑦

=0, ()

where the thermal conductivities 𝑥and 𝑦are piecewise

constant. Because the matrix, the solid region, and the lumen

are assumed to be locally isotropic and homogeneous, 𝑥=

𝑦=𝑚in the matrix, 𝑥=𝑦=𝑠in the solid region and

equal to 𝑙in the lumen, respectively.

Mathematical Problems in Engineering

Insulator

Insulator

T1T0

x

y

(a)

x

y

Insulator

Insulator

T

0

T

1

(b)

F : Schematic illustration of a square cell embeded with the natural hemp ber bundle. (a) Boundary conditions for 𝑒

𝑥.(b)Boundary

conditions for 𝑒

𝑦.

Besides, in the heat conduction system, the heat ux com-

ponents 𝑥and 𝑦are, respectively, dened by temperature

gradient:

𝑥=−𝑥

,

𝑦=−𝑦

.()

In the present composite computational model, the proper

thermal boundary conditions should be applied along the

boundary of the cell shown in Figure to construct a com-

plete composite heat transfer system being solved by nite

element technique [,,], which has been successfully

employed by many researchers for the analysis of eective

thermal properties of unidirectional ber composites [–

]. According to the work of Islam and Pramila [], the

prescribed temperature boundary conditions 1,0on the

vertical or horizontal boundaries of the cell can produce the

most accurate results up to a relatively high ber volume frac-

tion, and the remaining boundaries of the cell are assumed to

be insulating, as shown in Figure . It is assumed that 1>0;

thus, the average heat ux components 𝑥and 𝑦loaded on

the data collection face, for example, the le side face for the

case (a) and the bottom side face for the case (b), are positive.

Asaresult,basedonFourier’slawinheattransfer(see()),

the eective thermal conductivities of the composite can be

given as [,]

𝑒

𝑥=𝑥

1−0,

𝑒

𝑦=𝑦

1−0,()

where is the side length of the square cell,

𝑥=1

𝐿

0𝑥0,>0 ()

for the case in Figure (a),and

𝑦=1

𝐿

0𝑦(,0)>0 ()

for the case in Figure (b). e integrals in ()and()canbe

evaluated by trapezoidal numerical integration.

In the practical computation, the side length of the square

cell is set to be , which is a normalized length. If the volume

fraction of the ber bundle to the cell is assumed to be a

moderate value of %, the normalized radius of the ber

bundle is .. Furthermore, if the volume fraction of

lumen to the ber bundle keeps constant, that is, .%,

which is the experimental result [], the normalized radius

ofthelumencanbeevaluatedby0.0492/,whereis

the number of lumens in the ber bundle. For example, if

thenumberoflumensis,whichisclosetotheactual

disperse of lumens in the ber bundle (see Figure ), then

the normalized radius of each lumen is .. Besides, the

specied temperature boundry conditions along the two ver-

tical edges of the unit cell are set to be and , respectively.

Moreover, the thermal conductivities of matrix, solid region,

and lumen are, respectively, normalized with the reference

value . W/(mK), which is the thermal conductivity of

lumen, in the nite element procedure below. In the paper,

the symbols 𝑚,,and𝑙, respectively, indicate the thermal

condctivities of matrix, solid region, and lumen.

3. Numerical Results and Discussions

3.1. Convergence Investigation. Generally, the nite element

(FE) solution will be more accurate as the model is subdivided

into smaller elements. e only sure way to know if we

have suciently dense mesh is to make several models with

dierent grids of elements and check the convergence of the

solution. In order to investigate the convergence of the FE

Mathematical Problems in Engineering

T : Summary of the numerical test for the solution mesh size independence.

Approximated element

size Number of elements Number of nodes Ave rage he at ux

component 𝑥Deviation (%)

Size . . .

Size . . .

Size . . .

Size . . Reference

(a) (b)

F : Finite element model of the composite with natural ber bundle including lumens. (a) Computational domain. (b)

Computational mesh.

solution, the composite model in Figure (a) is studied by

ABAQUS and the FE size is changed from very coarse to

very ne. e element type employed in ABAQUS is DCD.

In each of FE size levels, the average horizontal heat ux

component 𝑥at the le wall of the square unit cell is

calculated. Tab l e gives a summary of the output of these

size levels indicating the number of elements and nodes used

in the computational domain corresponding to each element

size. In this table, Size stands for very coarse elements and

Size means very ne elements. e table also indicates

the deviation between the average heat ux component at

thelewallofthesquareunitcellcalculatedusingvarious

element sizes and that calculated using the nest element

size of . e summarized results in Table indicate that the

maximum deviation between the solution using the nest

element of that corresponds to elements and the

coarsest element of that corresponds to elements is

.%. is reects clearly that the numerical solution

obtained via this FE simulation is mesh size independence.

Additionally, looking for high accuracy, the authors decided

to use a ne element size of in the following computation.

3.2. Anisotropy Investigation. It is known that the distribution

mode of lumen may cause anisotropy of both ber bundle

and composite. To investigate this eect, let us consider the

composite model involving polymer matrix, solid region, and

lumens, as displayed in Figure (a),inwhichlumens

are regularly distributed in the ber bundle to approximate

the real distribution of lumens in the practical natural ber

T : Anisotropic thermal conductivities of the composite for

various thermal conductivities of the solid region phase.

𝑠/𝑙𝑒

𝑥/𝑙𝑒

𝑦/𝑙

. .

. .

. .

. .

..

bundle (see Figure ). Figure (b) presents the computational

mesh of element Size . Results in Tab l e display the change

of anisotropic thermal conductivities of the composites for

various thermal conductivities of the solid region phase

in the natural ber bundle. It is obvious that the thermal

conductivities of the composite along two directions are

extremely similar, so it is concluded that the approximated

practical distribution of lumen in the ber bundle causes

thecompositeandberbundleisotropy.esimilarcon-

clusion was drawn by Liu et al., who predicted that the

anisotropy of the composite became smaller with the number

of lumens increasing []. erefore, it is reasonable to assume

theisotropyofthecompositeandtheberbundleinthe

following analysis.

3.3. Eect of the ermal Conductivity of the Solid Region in

the Natural Fiber Bundle. To estimate the eect of the solid

region on the composite, it is assumed that the normalized

Mathematical Problems in Engineering

10.5

10

9.5

9

8.5

8

7.5

7

6.5

612345678910

ke/kl

ks/kl

FEM

Quadratic polynomial

Cubic polynomial

F : Variation of the eective thermal conductivity of the

composite against the solid region.

thermal conductivity of the solid region changes in the inter-

val [,]. By nite element computation, the distribution of

horizontal heat ux component and the corresponding aver-

agevalueofitcanrstlybeobtainedonthedata-collection

surface for each specic value of the normalized thermal

conductivity of the solid region. en, the normalized eec-

tive thermal property of the composite can be evaluated by

(). e variation of the eective thermal conductivity of the

composite is given in Figure ,fromwhichitisobservedthat

the simulated eective thermal conductivity of composite

increases with the increasing thermal conductivity of solid

region, as we expect. Simultaneously, it is found that the

variation shown in Figure shows slight nonlinearity, instead

of linearity. us, to describe the nonlinear variation shown

in Figure , the following quadratic and cubic polynomial

expressions from curve tting technology are, respectively,

employed.

(i) Quadratic polynomial tting:

𝑒

𝑙=−0.008432𝑠

𝑙2+0.5236𝑠

𝑙+5.673. ()

(ii) Cubic polynomial tting:

𝑒

𝑙=0.0008027𝑠

𝑙3−0.02168𝑠

𝑙2

+0.5847𝑠

𝑙+5.604. ()

3.4. Eect of the ermal Conductivity of the Homogenized

Fiber Bundle. In this section, the composite model shown in

Figure (a) is taken into consideration to investigate the eect

of the homogenized ber bundle on the composite. In the

model, the homogenized ber bundle is represented by a solid

ber with the same size. Also, the same thermal boundary

conditionsasthoseusedinthecompositemodelaboveare

applied along the outer boundaries of the cell. To conduct the

nite element analysis, a total of quadratic quadrilateral

elements of type DCD with nodes are generated

by ABAQUS to discretize the computational domain (see

Figure (b)).

It is assumed that the normalized thermal conductivity of

thesolidberchangesintheinterval[,]; thus, the eective

thermalconductivityofthecompositecanbeevaluatedby

nite element simulation for any specic value of thermal

conductivity of the equivalent solid ber. e variation of the

eective thermal conductivity of the composite against the

equivalent solid ber is displayed in Figure , which clearly

showsthatthesimulatedeectivethermalconductivityof

the composite nonlinearly increases with the increasing value

of the thermal conductivity of the equivalent solid ber. To

accurately capture the nonlinear variation shown in Figure ,

the following quadratic and cubic polynomial curves are,

respectively, employed by means of curve tting technology.

(i) Quadratic polynomial tting:

𝑒

𝑙=−0.01486

𝑙2+0.8932

𝑙+5.301. ()

(ii) Cubic polynomial tting:

𝑒

𝑙=0.0003747

𝑙3−0.02104

𝑙2

+0.9218

𝑙+5.269. ()

3.5. Optional Interrelationship between the ermal Conduc-

tivity of the Solid Region and the Fiber Bundle. Finally, the

equivalence of the two composite models, respectively, shown

in Figures (a) and (a) requires that the two composite

models should have same eective thermal conductivities.

erefore, combining ()–(), we have an optional interrela-

tionship between the thermal conductivity of the solid region

and the ber bundle; that is,

−0.01486

𝑙2+0.8932

𝑙

=−0.008432𝑠

𝑙2+0.5236𝑠

𝑙+0.372 ()

or

0.0003747

𝑙3−0.02104

𝑙2+0.9218

𝑙

=0.0008027𝑠

𝑙3−0.02168𝑠

𝑙2

+0.5847𝑠

𝑙+0.335,

()

Mathematical Problems in Engineering

(a) (b)

F : Unit square cell embeded with a solid ber to represent the homogenized ber bundle. (a) Computational domain. (b)

Computational mesh.

12345678910

13

12

11

10

9

8

7

6

ke/kl

FEM

Quadratic polynomial

Cubic polynomial

k/kl

F : Variation of the eective thermal conductivity of the

composite against the homogenized ber bundle.

from which the variation of 𝑠in terms of is plotted in

Figure .

To verify the obtained relation of thermal conductivity

betweenthesolidregionandberbundle,thetheoretical

Hasselman-Johnson’s model derived from the interface inter-

action between the circular matrix and circular inclusions

embedded in the matrix [] is taken as reference for the pur-

pose of comparison. Here, an analytical expression from the

Hasselman-Johnson’smodelcanbeemployedtoinvestigate

the thermal conductivity of the solid region 𝑠with respect

to that of the ber bundle ;thatis,

ks/kl

1234567

14

12

10

8

6

4

2

0

Hasselman-Johnson’s model

Results of cubic tting curve

Results of quadratic tting curve

k/kl

F : Approximated relation of thermal conductivities of the

natural ber bundle against the ber solid region.

𝑠

𝑙=

𝑙−11+V𝑙

21−V𝑙

+1+V𝑙2

41−V𝑙2

𝑙−12+

𝑙,()

where V𝑙represents the volume fraction of lumen to the ber

bundle.

Mathematical Problems in Engineering

According to the experiment data in [], the practical

volume content of the lumen to the ber bundle is about

.%; thus, the substitution of V𝑙=30.87%into() yields

𝑠

𝑙=0.9465

𝑙−1+0.8960

𝑙−12+

𝑙

()

from which one can get the variational curve of 𝑠in terms

of ,asshowninFigure forthepurposeofcomparison.

Specially, if is taken to be . W/(mK) [,], the thermal

conductivity of the solid region 𝑠is calculated by ()as

. W/(mK).

In Figure , it is observed that there is good agreement

between the numerical results from either quadratic or cubic

curves and the theoretical result of Hasselman-Johnson’s

model for the case of moderate change of . For example,

for the case of = 0.115W/(mK), the thermal conduc-

tivity of the solid region 𝑠is calculated as . W/(mK)

for the quadratic approximation and . W/(mK) for

the cubic approximation, which has relative derivation of

.% and .% of the theoretical solution .W/(mK),

respectively. erefore, both quadratic and cubic relations of

thermal conductivity of the solid region and the ber bundle

can be used to evaluate thermal properties of the natural ber

bundle or the solid region in the bundle. Also, in contrast to

the analytical expression (), it can be seen from ()and()

that either 𝑠or is given, and one can easily determine

another.ismeansthatwecandoinverseprediction

conveniently by a specied material thermal conductivity of

the solid region. is is an advantage of the optional relation

presented in the paper over the analytical solution.

Besides, it is obvious in Figure that the existence of

lumen signicantly weakens the capacity of heat transmission

in the ber bundle. As a result, the thermal conductivity of the

ber bundle is greatly less than that of 𝑠.

4. Conclusion

In this paper, D computational composite model of the nat-

ural ber bundle including numbers of lumens is developed.

Due to the geometrical limitation of the ber bundle, it is not

convenient to directly apply thermal boundary conditions to

it to perform nite element analysis of composite. To treat

this,theberbundleisassumedtobeembeddedintoa

matrix with known thermal conductivity to construct unit

composite cell, which is numerically analyzed by applying

proper thermal boundary conditions along the cell boundary.

By means of the developed nite element computational

composite model, the eect of the solid region in the

bundle on the overall thermal property of the composite is

studied. Simultaneously, a homogenized composite model

is constructed, in which the ber bundle is replaced by

an equivalent solid ber to investigate the inuence of the

homogenized ber bundle. By comparing the two composite

models developed in this study, an optional interrelationship

between thermal conductivities of the solid region and the

homogenized ber bundle was obtained by curve tting

technique. Finally, the present computational composite

model is veried and numerical experiments show that either

quadratic or cubic predictions can produce almost similar

results for the solid region in the ber bundle, in contrast

to the theoretical Hasselman-Johnson’s model and other

numerical results. Moreover, the direct or inverse predictions

can be easily performed to evaluate the thermal conductivity

oftheberbundleorthesolidregioninthepractice,ifoneof

them is given. More importantly, the present computational

method can be easily extended for the prediction of thermal

property of other natural ber bundles with various lumen

patterns.

Conflict of Interests

e authors declare no conict of interests regarding the

publication of this paper.

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