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International Journal Of Recent Advances in Engineering & Technology (IJRAET)
ISSN (Online): 2347-2812, Volume-1, Issue -3, 2013
1
A PREDICTION OF EFFECTIVE THERMAL CONDUCTIVITY OF
POLYESTER COMPOSITES FILLED WITH MICRO-SIZED RICE
HUSK PARTICLES
1Kush Kumar Dewangan,
2Ankur Panwar,
3Neeraj Kumar Sharma
Gurgoan College of Engineering, Gurgoan , India
Email : [email protected],
Abstract—This paper describes the preparation of a new
class of polyester composites filled with rice husk, which is
an agro waste as well as a potential natural reinforcing
element. A numerical simulation of the heat-transfer
process within polyester matrix composites filled with
micro-sized rice husk particles is proposed in this paper. A
commercially available finite-element package ANSYS is
used for this numerical analysis. Three-dimensional
spheres-in-cube, cubes-in-cube and ellipsoids-in-cube
lattice array models are constructed to simulate the micro -
structure of composite materials with rice husk content
ranging from about 1 to 11 vol.% and the effective thermal
conductivities of the composites are estimated. Results
show that the effective thermal conductivity (Keff) of the
composites decreases with increase in the volume fraction
of the rice husk fillers. Finally, the simulations are
compared with measured Keff values obtained from other
established correlations such as Rule of Mixture (ROM),
Maxwell’s model and Lewis & Nielsen. This study reveals
that the incorporation of rice husk particles results in
reduction of thermal conductivity of polyester resin
thereby increasing its insulation capability. It is found that
with incorporation of about 11 vol. % of raw rice husk in
the polyester resin reduces its thermal conductivity by
approximately 15 %.
Index Terms—FEM Simulation, Polymer Composites, Rice
Husk Particles, Thermal Conductivity
I. INTRODUCTION
Recently, polymer based composite have more attracted
due to their outstanding properties such as, corrosion
resistance, low coefficient, wear resistance, thermal
conductivity etc. New generation electronic devices
require small size of packages and higher power along
with the high thermal conductivity of polymer
composite. Low thermal conductive polymer is also
demanded. Recently polymer is used as various
insulating purpose. That require new composite with
relatively low thermal conductivity. Thermal
conductivity of polymer may be improved or reduced by
using a low thermal conductivity or high thermal
conductive filler material.
Various type of Natural fillers such as jute, pineapple,
rice husk , hemp, curaua ,wood have been successfully
used to improve the mechanical properties of
thermoplastic composites [1-3]. The first approach on
the use of rice husk ash as a reinforcing filler in
polymers to produce composite is reported by Haxo and
Mehta [4]. Recently, Garcia et al. [5] have analyzed the
use of rice husk as reinforcing material for recovery of
tire rubber powder using the sintering method.
Mechanical properties and thermal stability of rice husk
ash filled epoxy foams, they used at 22.7 wt % of rice
husk ash, the particle average size of WRHA and BRHA
were 2.5 and 2.9 mm respectively was studied by Stefani
et al. [6] The mechanical properties of the epoxy foams
improve especially with the incorporation of WRHA.
Premalal et al. [7] studied on Comparison of the
mechanical properties of rice husk powder filled
polypropylene composites with talc filled polypropylene
composites; it was found that talc composites are easier
to process than RHP filled composites. However, talc
composites exhibit higher yield strength, Young‟s
modulus, flexural modulus and impact resistance. It was
found that RHP composites show higher elongation at
break compared to talc composites.
In recent study Griesinger, Hurler, and Pietralla [8]
found that the thermal conductivity of low-density PE
was seen to increase from 0.35 for an isotropic sample,
to the value of 50W/mK for a sample with an orientation
ratio of 50. This value of thermal conductivity is in the
range of thermal conductivity for steel. The thermal
conductivity of magnetite filled metal-oxide particle
International Journal Of Recent Advances in Engineering & Technology (IJRAET)
ISSN (Online): 2347-2812, Volume-1, Issue -3, 2013
2
filler (magnetite, Fe3O4) polymers increases from 0.22
to 0.93 W/(m K) for a filler content of 44 vol% of
magnetite, whereas the electrical resistivity decreases
more than seven orders of magnitude from an insulator
(0% of magnetite) to 10 kV m (47 vol% of magnetite).
Maxwell [9] proposed an equation to calculate the
effective thermal conductivity of Composites, when the
particles are well-dispersed and have no interaction one
with the others. Tsao [10] predicts an equation for the
thermal conductivity of a two-phase solid mixture by
using the additivity of conductance in parallel. Cheng
and Vachon [11] modified Tsao‟s model without
knowing the distribution of the discontinuous phase in
the continuous phase. A semi theoretical model by
Lewis and Nielsen [12] predicts the elastic module of
composites and then the thermal conductivity.
Veyret et al. [13] they proposed on a numerical
approach to predict the effective thermal conductivity of
granular of fibrous reinforced composite and compared
with theoretical and experimental results. Kumlutas¸ and
Tavman [14]they used a finite element model to obtain
the effective thermal conductivity of HDPE composites
as filler content up to 16% volume, as Shape of particle
sphere and cubes, further compared the effective
conductivity with empirical and experimental results.
Nayak et al. [15] they studied the thermal conductivity
of epoxy composites using a 3D finite and element
model of Sphere in cube in lattice array distribution.
Among all these study expect Nayak et al. [15] are used
organic filler (pine wood dust) which is eco-friendly and
bio-degradable. The objective of this paper is to develop
a high thermal insulating polymer composite, contenting
micro-sized Rice husk. Numerical approach was used to
determine the effective thermal conductivity of particle
composite in 3D models. FEM was performed to
understand the relationship between the effective
thermal conductivity of polymer composite at different
filler shape and different volume fraction. FEA is based
on steady-state, heat conduction. Further, the effective
thermal conductivity of Rice husk particles filled in
polymer composite is calculated numerically as a
function of filler concentration and the result obtained
from this investigation are compared with the other
theoretical and empirical models and also with
experimental results.
II. MATHEMATICAL ANALYSIS AND
THEORETICAL CONSIDERATION
In this section discussed several models for Thermal
conductivity of the two-phase system is predicted by
various methods, most of them are given more near to
experimental value.Among the models proposed the
thermal conductivity of composite has depend in the
dispersed region. In previous report, thermal conduction
model proposed by Maxell, Cheng and Vachon,
Ziebland, Lewis and Nielsen were discussed. Thermal
conduction model proposed by Agari and Uno was
discussed about the region of more than 30–vol %
content of filler particles.
First simplest model for two-component composite in
that material would be arranged in parallel or series with
respect to heat flow. This gives the effective thermal
conductivity of composite in either upper or lower
bounds. For parallel conduction model
(1- )c m fk ø k øk (1)
Where k: thermal conductivity, c: composite, m: matrix,
f: filler,ø :volume fraction of filler.
And for series conduction model:
1 (1- )
c m f
ø ø
k k k
(2)
The effective thermal conductivity of composite is given
by when considered geometric mean model:
1.f
mck k k
(3)
The correlations presented by Eqs. (1), (2) and (3) are
derived on the basis of the Rules-of- mixture. Tsao [10]
derived an equation relating the two-phase solid mixture
thermal conductivity to the conductivity of the
individual components and to two parameters which
describe the spatial distribution of the two phases. By
assuming a parabolic distribution of the discontinuous
phase in the continuous phase. Further the model of the
Tsao [10] is modified by the cheng and vachon [11] this
model that did not required data for additional
parameter. Assuming a parabolic distribution of the
discontinuous phase volume fraction cheng and vachon
[11] proposed a new model which is applicable for
spherical particle as well as foe fiber filled composite
mf kk
1
1 1
2
2
mmfm
mf
mfmmf
ck
B
kkBk
kkCBtan
kkBkkkCk
(4)
If mf kk
1 1
kc C k k k B k km m mf f
International Journal Of Recent Advances in Engineering & Technology (IJRAET)
ISSN (Online): 2347-2812, Volume-1, Issue -3, 2013
3
12
2
Bk B k k C k km m mf f B
InB kmk B k k C k km m mf f
(5)
Where for both equation
3 / 2B 4 2 / (3 )C
By the using potential theory. Maxwell [9] obtained an
exact solution for the conductivity of randomly
distributed and non-intersecting homogenous spheres in
a homogenous medium.
fmmf
fmmf
mckkkk
kkkkkk
2
22 (6)
Halpin [16] equation is modified by the Lewis and
Nielsen [12].to includes the effect of the shape of the
particles and the orientation or type of packing for a
two-phase system, and they have got:
B
ABkk mc
1
1 (7)
Akk
kkB
mf
mf
1 (8)
2
11
m
m (9)
The values of A and
m for many geometric shapes and
orientation are given in Tables I and II.
Table I Value of A for various systems [12]
Type of dispersed
phase
Direction of
heat flow
A
Cubes Any 2
Spheres Any 1.5
Aggregates of spheres Any (2.5/ m) -1
Randomly oriented
rods Aspect ratio=2
Any 1.58
Randomly oriented
rods Aspect ratio=4
Any 2.08
Randomly oriented
rods Aspect ratio=6
Any 2.8
Randomly oriented
rods Aspect ratio=10
Any 4.93
Randomly oriented
rods Aspect ratio=15
Any 8.38
Uniaxially oriented
fibers
Parallel to
fibers
2L/D
Uniaxially oriented
fibers
Perpendicula
r to fibers
0.5
Table II Value of m for various systems [12]
Shape of
particle
Type of
packing m
Spheres Hexagonal
close
0.7405
Spheres Face centered
cubic
0.7405
Spheres Body centered
cubic
0.60
Spheres Simple cubic 0.524
Spheres Random close 0.637
Rods and fibers Uniaxial
hexagonal close
0.907
Rods and fibers Uniaxial simple
cubic
0.785
Rods and fibers Uniaxial
random
0.82
Rods and fibers Three
dimensional
random
0.52
III. EXPERIMENTAL STUDY
A. Composite Fabrication
Polyesteris used as matrix material in a liquid form, with
low thermal conductivity 0.345W/mk at room
temperature and density 1.35gm/cc. The organic bio-
based filler material rice husk is used in a powder form.
Particle approximately as shape like sphere and sized in
the range of 80-100 microns for rice husk. Solid density
of the rice husk is about 0.4 to 0.7 gm/cc and its thermal
conductivity 0.0359 w/mk. The dough (Polyester filled
with RHs) is then slowly decanted into the glass tubes,
coated beforehand with wax and uniform thin film of
silicone-releasing agent. The composites are cast by
conventional hand-lay-up technique in cup so as to get
plate
Specimens (dia. 40 mm, height 5 mm).
International Journal Of Recent Advances in Engineering & Technology (IJRAET)
ISSN (Online): 2347-2812, Volume-1, Issue -3, 2013
4
Table III Preparation of particulate filled composites by
hand- lay-up technique.
Sample Composition
1 Polyester +1.88vol%(0.765 wt% ) filler
2 Polyester +4.18 vol%(1.702wt% )filler
3 Polyester +6.54vol%(2.664 wt% )filler
4 Polyester +9.16vol%(3.731 wt% )filler
5 Polyester +11.30vol%(4.603wt% )filler
Composite samples of five different compositions, as
listed in Table 3are made. The mean particle size of rice
husk used in samples 1 (1.88vol.%), 2 (4.18 vol.%), 3
(6.54 vol.%),4(9.16 vol.%) and 5(11.3 vol.%) is 100
micron. The castings are left to cure at room temperature
for about 24 h after which the cup is broken and samples
are released. Specimens of suitable dimension are cut
using a diamond cutter for further physical
characterization and thermal conductivity test
B. Operating Principle of Unithermtm
Model 2022
The measurement of thermal conductivity of various
materials Unitherm™ Model 2022 is used. These
include polymers, ceramics, composites, glasses,
rubbers, some metals, and other materials oflow to
medium thermal conductivity. For this measurement a
relatively small test sample is required. In this test non-
solids, such as pastes or liquids, can be tested using
special containers. Thin films can also be tested
accurately using a multi-layer technique. The tests are in
accordance with ASTM E-1530 Standard.
The process for determine the measure effective thermal
conductivity of composite material is held under a
uniform compressive load between two polished
surfaces, each controlled at a different temperature. The
lower surface is part of a calibrated heat flow transducer.
The heat flows from the upper surface, through the
sample, to the lower surface, establishing an axial
temperature gradient in the stack. After reaching thermal
equilibrium, the temperature difference across the
sample is measured along with the output from the heat
flow transducer. These values and the sample thickness
are then used to calculate the thermal conductivity. The
temperature drop through the sample is measured with
temperature sensors in the highly conductive metal
surface layers on either side of the sample. For one-
dimensional heat conduction equation by Fourier‟s Low
at the steady state condition given as Eq. (10).
1 2-T TQ KA
x
(10)
Where Q is the heat flux (W), K is the thermal
conductivity (W/m K), A is the cross-sectional area
(m2), T1-T2 is the difference in temperature (K), x is the
thickness of the sample (m).
The thermal resistance of a sample can be given as
xR
KA
(11)
where R is the resistance of the sample between hot and
cold surfaces (m2 K/W).Unitherm™ 2022 measures
thermal conductivity of solid material by determine the
thermal resistance between the upper and lower
surfaces. It‟s have a heat flux transducer that measures
the Q value and the temperature difference can be
obtained between the upper plate and lower plate. Thus
the thermal resistance can be calculated between the
upper and lower surfaces. Further, the thermal
conductivity of the samples can be calculated using Eq.
(11).
IV. NUMERICAL ANALYSIS
To predict the effective thermal conductivity of the
composite the finite element method (FEM) is more
efficient. Thermal analysis of heat transfer was carried
out using the finite element software of the program
ANSYS.The finite element method (FEM), originally
introduced by Turner et al. [17], is a powerful
computational technique for approximate solutions to a
variety of „„real-world” engineering The FEM is a
numerical procedure that can be used to obtain solutions
to a large class of engineering problems involving stress
analysis, heat transfer, fluid flow etc. ANSYS is general-
purpose finite- element modeling package for
numerically solving a wide variety of mechanical
problems that include static/dynamic, structural analysis
(both linear and nonlinear), heat transfer, and fluid
problems, as well as acoustic and electromagnetic
problems.
Numerical Modeling of the Problem
Basic step used for determine the thermal conductivity
of the composite the finite element method (ANSYS) is
modeling, material properties, element type, meshing
and boundary condition. Among these one of most
important step is element type and meshing that affect
the result very much. In Three-dimensional numerical
analysis solid98, solid87 and solid70 are used. Three-
dimensional numerical analysis was carried out for the
conductive heat transfer in the composite material. In
which matrix phase is taken as a cube and dispread
phase is as micro-sized particle in different shape like
sphere, cube and ellipsoid, which is in the form of lattice
array. Three dimensional-models have been used to
simulate the micro-structure of composite material for
various filler concentration1.41 vol.% to 11.3 vol% are
shown in figure (2).
In the numerical analysis of the heat conduction
problem, the temperatures at the nodes along the
International Journal Of Recent Advances in Engineering & Technology (IJRAET)
ISSN (Online): 2347-2812, Volume-1, Issue -3, 2013
5
surfaces ABCD is prescribed as T1 (=100˚C) and the
convective heat transfer coefficient of ambient is
prescribed as 2.5 W/m2 K at ambient temperature of
27˚C. The heat flow direction and the boundary
conditions are shown in Fig. 1. The other surfaces
(ABFE, AEDH, BCGF, and CDHG) parallel to the
direction of the heat flow are all assumed adiabatic. The
temperatures at the nodes in the interior region and on
the adiabatic boundaries are unknown. These
temperatures are obtained with the help of finite-element
program package ANSYS. In this analysis of the ideal
case it will be assumed that the composites are
macroscopically homogeneous, locally both the matrix
and filler are homogeneous and isotropic, the thermal
contact resistance between the filler and the matrix is
negligible, the composite lamina is free of voids and the
filler are arranged in a square periodic array/uniformly
distributed in matrix. Temperature profile obtains from
FEM for composites with cubes, spheres and ellipsoids
particle concentration of 11.3 vol% are shown in Fig 3-
5.
Fig.1Boundary conditions
Fig.2 Geometric model of Rice husk particle (cubes) in
polyester (cube) at 11.3vol%
Fig.3Temperature profile for composite with cube
particle concentration of 11.3 vol%
Fig. 4.Temperature profile for composite with sphere
particle concentration of 11.3vol%
Fig.5 Temperature profile for composite with ellipsoid
(aspect ratio1:0.7) particle concentration of 11.3 vol%
International Journal Of Recent Advances in Engineering & Technology (IJRAET)
ISSN (Online): 2347-2812, Volume-1, Issue -3, 2013
6
Table IV Effective Thermal conductivity values of composites obtained from different FEM models
Sample Filler content
(vol. %)
Effective thermal conductivity of the composite (W/mK)
FEM model
cubes-in cube
FEM model Ellipsoids -in
cube Aspect ratio (1:0.7)
FEM model
Spheres -in-cube
Experimental
value
1 0 0.345 0.345 0.345 0.345
2 1.8 0.336 0.331 0.331 0.334
3 4.18 0.327 0.324 0.316 0.325
4 6.54 0.317 0.313 0.309 0.316
5 9.16 0.308 0.304 0.300 0.306
6 11.3 0.297 0.293 0.296 0.299
V. RESULT AND DISCUSSION
In real materials, particle filled polyester composites
contain countless filler particles, because of the small
micron-scale filler. Note that the size of individual
particles remains unchanged. The distance between
particles will also decrease as the volume fraction of
fillers increases, Models with different volume fraction
content different no of filler particle ranging from 36
to 216. Basic step used for determine the thermal
conductivity of the composite the finite element
method (ANSYS) is modeling, selection of material
properties, element type, meshing and boundary
condition. Among these one of most important step
areselection of element type and meshing result. As
different filler shape like cubes, spheres and ellipsoids
effects on the effective thermal conductivity was
estimate by Finite element method are in table IV
shown in fig.6 with comparing experimental results.
The percentage errors associated with each particle
shape for individual composites are given as Table V.
It is further noted that while the particle cube give
more near to experimental value.
In percent study, a numerical and empirical approach
is used to obtain the effective Thermal conductivity of
the polyester composite material filled with micro-
sized RHs particle. The values of effective thermal
conductivities of the particulate filled polyester
composites with varied proportions of micro-sized
RHs particle obtained using Maxwell‟s correlation,
Rules-of-mixture model, Lewis and Nielsen (A=3
Fig. 6Thermal conductivity of polyester composites as
a function of filler content with different filler shape
International Journal Of Recent Advances in Engineering & Technology (IJRAET)
ISSN (Online): 2347-2812, Volume-1, Issue -3, 2013
7
Table V Percentage errors of keff obtain from different
Sampl
e
Particulat
e content
Vo1%
Percentag
e error
with
respected
to
measured
value
FEM
model
cubes–in-
cube
FEM
model
Ellipsoids
–in-cube
Aspect
ratio
(1:0.7)
FEM
model
Spheres
–
in-cube
1 1.88 0.59 0.89 0.89
2 4.18 0.61 0.30 1.84
3 6.54 0.31 0.94 2.25
4 9.16 0.65 0.65 1.96
5 11.3 0.66 2.00 0.67
Filler shape with respected to the measured value
and m=0.637), Cheng and Vachon model and those
obtained from FEM analysis are presented in Table VI
and fig. 7. It presents a comparison among the results
obtained using these models with regard to the values
of effective conductivity obtained experimentally. It is
further noted that while the FEM and Maxwell‟s
model overestimate the value of thermal conductivity
the Rules-of-mixture and Lewis & Nielsen model
underestimates the value with respect to the
experimental value.
It leads to a conclusion that for a particulate filled
composite of this kind the FEM model as cube particle
shape can very well be used for predictive purpose in
determining the effective thermal conductivity for a
wide range of particle concentration of polyester
composite.
Table VI Thermal conductivity values of composites obtained from different methods
VI. CONCLUSION Rice husk reinforced polyester composites were
successfully prepared. Effective thermal conductivities
of the particulate filled polyester composites with
micro-sized RHs particle is increases with as the filler
content. From several experimental and computational
findings presented above, following conclusions can
be drawn:
Sample Particulat
e content
(vol. %)
Effective thermal conductivity of the composites (W/mK)
ROM model Lewis and
Nielsen
model
Maxwell‟s
model
FEM model
cubes–in-cube
Experimental
value
1 0 0.345 0.345 0.345 0.345 0.345
2 1.88 0.296 0.331 0.339 0.336 0.334
3 4.18 0.253 0.324 0.334 0.327 0.325
4 6.54 0.220 0.313 0.327 0.317 0.316
5 9.16 0.192 0.304 0.316 0.308 0.306
6 11.3 0.174 0.293 0.308 0.297 0.299
International Journal Of Recent Advances in Engineering & Technology (IJRAET)
ISSN (Online): 2347-2812, Volume-1, Issue -3, 2013
8
(a)Finite element method (FEM) can be gainfully
employed for determination of effective thermal
conductivity of these composites with different amount
of Rice husk content.
(b)The values of effective thermal conductivity
obtained for various composite models from FEM are
in reasonable agreement with the experimental values
for a wide range of filler contents from 1.88vol% to
11.3 vol. %.
(c)The values of thermal conductivity obtained from
FEM (cubes-in-cube arrangement) are found to be
more accurate (close to experimental values) than the
calculated values from existing theoretical models
such as Rule of mixture, Maxwell‟s equation, Lewis
and Nielsen‟s model and FEM spheres-in-cube model
and Ellipsoid-in-cube model(aspect ratio 1:0.7 ).
(d)Incorporation of micro-sized Rice-Husk results in
significant improvement of thermal insulation for
polyester resin .With addition of 1.88 vol. % of Rice
Husk the thermal insulation of polyester improves by
about 3.18 % and with addition of 11.3 vol. % of Rice
Husk the improvement is about 13.33 %.
(e) With increased thermal insulation, this new class of
Rice husk filled polyester composites can be used for
applications such as electronic packages,
encapsulations, die (chip) attachments, thermal grease,
thermal interface material and electrical cable
insulation.
REFERENCE [1] T. A. Bullions, D. Hoffman, R. A. Gillespie, J.
Price-O‟Brien and A. C. Loos, “Contributions
of Feather Fibers and Various Cellulose Fibers
to the Mechanical Properties of Polypropylene
Matrix Composites,” Composite Science
Technology, 66, pp. 102, 2006.
[2] M. D. H. Beg, K. L. Pickering, “Reprocessing
of wood fibre reinforced polypropylene
composites. Part I: Effects on physical and
mechanical properties,” Composites Part A:
Applied Science and Manufacturing,39(7), pp.
1091-1100, 2008.
[3] H.E.Haxo and P.K.Mehta,“Ground rice hull ash
as filler for rubber,” Rubber Chem.
Technology, 48, pp.271–88,1975.
[4] D.Gracia, J.Lopez, R.Balart, R.A.Ruseckaite
and P.M. Stefani,“Composites based on
sintering rice husk-waste tire rubber mixtures,”
Materials and Design, 28, pp. 234-38,2007.
[5] P.M.Stefani and J.Jime´mez, “Thermal
Degradation of Rice Husk and Other Biomass
Materials,” International Perspectives on
Chemistry and Biochemistry Research, Nova
Sci. Pubs.,2005.
[6] Hattotuwa G.B. Premalal, H. IsmailA.
Baharin,“Comparison of the mechanical
properties of rice husk powder filled
polypropylene composites with talc filled
polypropylene composites,” Polymer
Testing,21(7),pp.833–9,2002.
[7] A. Griesinger, W. Hurler, and M. Pietralla,“A
Photothermal Method with StepHeating for
Measuring the Thermal Diffusivity of
Anisotropic Solids,” International Journalof
Heat and Mass Transfer, 40(13), pp. 3049–
3058,1997.
[8] J.C.Maxwell, “A Treatise on Electricity and
Magnetism,” 3rd
edn, Ch.9. Dover Inc. New
York, NY.,1954.
[9] T.N.G.Tsao,“Thermal Conductivity of Two
Phase Materials,” Industrial andEngineering
Chemistry, 53(5), pp. 395–397, 1961.
[10] S. C.Cheng, and R. I.Vachon, “The Prediction
of the Thermal Conductivityof Two and Three
Phase Solid Heterogeneous Mixtures,” Int. J.
HeatMass Transfer, 12(3), pp. 249–264,1969.
[11] T. Lewis, and L. Nielsen, “Dynamic
Mechanical Properties of Particulate-Filled
Polymers,” J. Appl. Polym. Sci., 14(6), pp.
1449–1471, 1970.
[12] D.Veyret, S.Cioulachtjian, L.Tadrist and J.
Pantaloni, “Effective ThermalConductivity of a
Composite Material: A Numerical Approach,”
Transactions of theASME- Journal of Heat
Transfer, 115, pp. 866–871,1993.
[13] RajlakshmiNayak, Aloksatapathy, “A study on
thermal conductivity of particulate reinforced
epoxy composites,”Composite Material
Science,48, pp. 576-581, 2010
[14] J.C.Halpin,“Stiffness and expansion estimates
for oriented short fiber composites,” Journal of
Composite Materials, 3, pp. 732–4,1969
[15] M.J. Turner, R.W. Clough, H.C. Martin, L.J.
Topp, “Stiffness and deflection analysis of
complex structures,”J. Aeronaut. Sci., 23,
pp.805–823,1956.