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Numerical Investigation of Characteristic of Anisotropic Thermal Conductivity of Natural Fiber Bundle with Numbered Lumens

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Natural fiber bundle like hemp fiber bundle usually includes many small lumens embedded in solid region; thus, it can present lower thermal conduction than that of conventional fibers. In the paper, characteristic of anisotropic transverse thermal conductivity of unidirectional natural hemp fiber bundle was numerically studied to determine the dependence of overall thermal property of the fiber bundle on that of the solid region phase. In order to efficiently predict its thermal property, the fiber bundle was embedded into an imaginary matrix to form a unit composite cell consisting of the matrix and the fiber bundle. Equally, another unit composite cell including an equivalent solid fiber was established to present the homogenization of the fiber bundle. Next, finite 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 influences of the solid region phase and the equivalent solid fiber on the composites were investigated, respectively. Subsequently, an optional relationship of thermal conductivities of the natural fiber bundle and the solid region was obtained by curve fitting technique. Finally, numerical results from the obtained fitted curves were compared with the analytic Hasselman-Johnson’s results and others to verify the present numerical model.
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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 eciently 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 inuences 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 specic 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
investigatedtheinuenceofbercontentonthemechanical
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-Johnsons model [].Also,theystudiedtheeect
ofthemicrostructureofnaturalberonthetransversether-
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 prole 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-bamboobercompositesusing
ahot-wiremethod[]. All these works mentioned above
have been benecial 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
thelumensinthenaturalberbundle.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 specied
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 signicantly aects the whole
thermal performance of the ber bundle than that of lumen.
Here, the main purpose of this study is to investigate the
eect 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 inuence of solid region phase on the ber
bundle. e established unit cell consists of three dierent
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 satises
the quasiharmonic dierential 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, dened 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 eective
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,basedonFourierslawinheattransfer(see()),
the eective 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
0𝑥0,>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
specied 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 suciently dense mesh is to make several models with
dierent 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 DCD.
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
thelewallofthesquareunitcellcalculatedusingvarious
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 reects 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 eect, let us consider the
composite model involving polymer matrix, solid region, and
lumens, as displayed in Figure (a),inwhichlumens
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
thecompositeandberbundleisotropy.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
theisotropyofthecompositeandtheberbundleinthe
following analysis.
3.3. Eect of the ermal Conductivity of the Solid Region in
the Natural Fiber Bundle. To estimate the eect 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 eective 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-
agevalueofitcanrstlybeobtainedonthedata-collection
surface for each specic value of the normalized thermal
conductivity of the solid region. en, the normalized eec-
tive thermal property of the composite can be evaluated by
(). e variation of the eective thermal conductivity of the
composite is given in Figure ,fromwhichitisobservedthat
the simulated eective 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=−0.008432𝑠
𝑙2+0.5236𝑠
𝑙+5.673. ()
(ii) Cubic polynomial tting:
𝑒
𝑙=0.0008027𝑠
𝑙30.02168𝑠
𝑙2
+0.5847𝑠
𝑙+5.604. ()
3.4. Eect 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 eect
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 DCD with  nodes are generated
by ABAQUS to discretize the computational domain (see
Figure (b)).
It is assumed that the normalized thermal conductivity of
thesolidberchangesintheinterval[,]; thus, the eective
thermalconductivityofthecompositecanbeevaluatedby
nite element simulation for any specic value of thermal
conductivity of the equivalent solid ber. e variation of the
eective 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
𝑙30.02104
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 eective 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
=−0.008432𝑠
𝑙2+0.5236𝑠
𝑙+0.372 ()
or
0.0003747
𝑙30.02104
𝑙2+0.9218
𝑙
=0.0008027𝑠
𝑙30.02168𝑠
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 eective 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
betweenthesolidregionandberbundle,thetheoretical
Hasselman-Johnsons 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-Johnsonsmodelcanbeemployedtoinvestigate
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−11+V𝑙
21V𝑙
+1+V𝑙2
41V𝑙2
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 specied 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 signicantly 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,theberbundleisassumedtobeembeddedintoa
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 eect 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 inuence 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 veried 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
oftheberbundleorthesolidregioninthepractice,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 conict of interests regarding the
publication of this paper.
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