Enhanced Heat Transfer in Composite Materials
A thesis presented to
the faculty of
Russ College of Engineering and Technology of Ohio University
In partial fulfillment
of the requirements for the degree
Master of Science
Sayali V. Pathak
August 2013
© 2013 Sayali V. Pathak. All Rights Reserved
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This thesis titled
Enhanced Heat Transfer in Composite Materials
by
SAYALI V. PATHAK
has been approved for
the Department of Mechanical Engineering
and Russ College of Engineering and Technology by
Khairul Alam
Moss Professor of Mechanical Engineering
Dennis Irwin
Dean, Russ College of Engineering and Technology
3
ABSTRACT
PATHAK, SAYALI V., M.S., August 2013, Mechanical Engineering
Enhanced Heat Transfer in Composite Materials
Director of Thesis: Khairul Alam
Many composite materials are composed of a matrix reinforced with fibers.
Carbon fiber composites are currently being used for high heat transfer
applications. Carbon fibers are known to have excellent thermal conductivities.
However, if the interface between the matrix and fibers has poor thermal
properties, it affects the overall thermal conductivity of the composite significantly.
The goal of this project is to quantify the thermal conductivity of the matrix-fiber
interface in a set of carbon fiber composites. This thesis describes two numerical
methods to determine the fiber-matrix interface heat transfer coefficient; the
numerical methods use the FLUENT solver in ANSYS. The resulting values from the
study compared well with results from an analytical equation.
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To my parents, Vijay Pathak and Sarita Pathak
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ACKNOWLEDGEMENTS
I would like to express my gratitude to my advisor, Dr. Khairul Alam, for his
excellent guidance throughout my master’s program. I thank him for his support,
and encouragement at every step of this project. Working under his guidance has
been a great learning experience. I also thank the project sponsors, Performance
Polymers Solutions Inc., for giving me the opportunity to work on this project.
I am thankful to my thesis committee members: Dr. Hajrudin Pasic, Dr. John
Cotton, and Dr. Annie Shen for their valuable advice. Their recommendations gave
me good ideas while working on the project as well as helped improve my thesis.
I would like to thank the Mechanical Engineering Department, Russ College
of Engineering, and Ohio University for supporting my graduate studies.
Randy Mulford, ME laboratory technician at Ohio University helped me with
the machining aspects of my experimentation. His prompt help made the timely
completion of experiments possible. I would like to thank him sincerely.
I also thank my family; my brother, Ankush Pathak, for his help with my
thesis document, my mother and father, Sarita Pathak and Vijay Pathak, for their
unwavering support throughout my education and life.
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TABLE OF CONTENTS
Abstract…………………………….………………………………………………………………………………...3
Dedication…..…………………..…………………………………………………………………………………...4
Acknowledgements…………………….………………………………………………………………………..5
List of Tables ............................................................................................................................................ 8
List of Figures ........................................................................................................................................... 9
Chapter 1:
Introduction ................................................................................................................ 11
1.1
Composite Materials ............................................................................................ 11
1.2
Thermal Applications of Composite Materials........................................... 16
1.3
Transverse Properties ......................................................................................... 17
1.4
Performance Polymers Solution Inc. (P2SI) ................................................ 18
1.5
Summary and Rationale of Proposed Research ........................................ 19
Chapter 2:
Carbon Fibers ............................................................................................................. 20
2.1
Anisotropy in Carbon Fibers Composites .................................................... 21
2.2
Anisotropic Micro-Structure of Carbon Fibers .......................................... 22
Chapter 3:
Matrix-Fiber Interface ............................................................................................. 29
3.1
Effect of Thermal Interface ................................................................................ 30
3.2
Causes of Imperfect Interfaces ......................................................................... 33
3.3
Theoretical and Analytical Studies ................................................................. 35
Chapter 4:
Numerical Model ....................................................................................................... 39
4.1
Geometry .................................................................................................................. 39
4.2
Named Selections .................................................................................................. 43
4.3
Meshing ..................................................................................................................... 44
4.4
Solver Set Up ........................................................................................................... 45
4.5
Post-Processing ...................................................................................................... 49
Chapter 5:
Comparison Of Two Numerical Models............................................................ 51
5.1
Variable Thicknesses of the Physical Wall................................................... 56
5.2
Simulation with Virtual Interface Layer ....................................................... 57
7
5.3
Physical Wall Vs. Virtual Interface Thickness ............................................ 59
Chapter 6:
Results And Discussion ........................................................................................... 61
6.1
Temperature Distribution.................................................................................. 61
6.2
Thermal Conductivity of Samples ................................................................... 62
6.3
Dispersion Factor .................................................................................................. 66
6.4
Analysis of Results Based On Fiber Characteristics ................................. 73
6.5
Theoretical Calculations ..................................................................................... 77
6.6
Discussion of h* Results ...................................................................................... 82
Chapter 7:
Summary And Conclusions ................................................................................... 84
References...………………………………………………….…………………………………………………...87
8
LIST OF TABLES
Table 2.1: Comparison of thermal conductivity of a composite having radial,
circumferential and isotropic fiber conductivity……………………………
Table 5.1: Variable interface thermal resistances and corresponding thermal
conductivities for various hicknesses...............................................................
Table 5.2: Thermal conductivities obtained after simulating for a thin wall
interface.........................................................................................................................
Table 5.3: Thermal conductivity with variable thickness of interface modeled
as a virtual wall..........................................................................................................
Table 5.4: Comparison of thermal conductivities with thin wall and virtual
wall models with variable interfacial resistance.........................................
Table 6.1: Thermal conductivities of samples obtained by simulations at
varying interfacial conductivity and by experiments...............................
Table 6.2: Matrix conductivities needed to match composite conductivity
with experimental conductivity…………………………………………………….
Table 6.3: Simulation conductivities of sample #1………………………………………...
Table 6.4: Experimental conductivities and corresponding values of
interfacial conductance (h*).................................................................................
Table 6.5: h* values obtained using various dispersion factors……………….……..
Table 6.6: Results grouped based on volume fraction………………………..………......
Table 6.7: Results grouped based on make of fibers………..……………………………..
Table 6.8: Results grouped based on fiber diameter……………………………….........
Table 6.9: Values of h* and ht* using 3 dispersion factors …….………………………
28
55
57
58
59
63
65
69
70
71
73
75
76
80
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LIST OF FIGURES
Figure 1.1: Typical cross section of a hybrid matrix composite transverse to
its unidirectional fibers……..……………………………………………………… 13
Figure 1.2 (a) & (b): Typical composite plies with cylindrical fibers bounded
by a matrix with various orientations of fibers..………………………... 14
Figure 1.2 ©: Hybrid matrix, containing conductive particles, reinforced by
cylindrical fibers………………………………………………………………………. 14
Figure 1.2 (d): Several plies, each having different orientations of fibers,
bonded into a single laminate……………………………….………………… 14
Figure 1.3: Anisotropic thermal conductivity (TC) in a composite with
carbon fiber reinforcement………………………………………………………. 15
Figure 2.1: (a):Layer of carbon atoms in an anisotropic structure of graphite
crystal……………………………………………………………………………………… 23
Figure 2.1: (b): Orientation of graphene planes in carbon fibers. ……………….. 24
Figure 2.2 (a): Transverse section of a carbon fiber with radially arranged
graphene planes............................................................................................... 25
Figure 2.2 (b): Carbon fiber cross section with concentric graphene planes. 25
Figure 2.3 (a): Strong conductivity along radial graphene planes can assist
heat flux in Y direction due to heat flow along the graphene
planes......................................................................................................................... 26
Figure 2.3 (b): Weak conductivity between graphene sheets may result in
poor conduction of heat in Y direction........................................................ 26
Figure 2.4: Model of a composite with minimal matrix……………………………….. 27
Figure 3.1 (a): Heat flux for Poor thermal interface.................................................... 31
Figure 3.1 (b): Heat flux for Good thermal interface. ………………………………… 32
Figure 3.2 (a): The conductivity of the composite compared to the
conductivity of the baseline model when thermal interface is
poor............................................................................................................................. 33
Figure 3.2 (b): The conductivity of the composite compared to the
conductivity of the baseline model with a good thermal
interface.................................................................................................................... 35
Figure 4.1: Unit cell model of a typical composite......................................................... 40
Figure 4.2: 2D Unit-cell model of a composite consisting of one carbon fiber
with radial arrangement of graphene planes……………………………. 41
Figure 4.3 (a) Temperature distribution in a fiber with strong radial
conductivity…………………………………………………………………….. 42
Figure 4.3 (b) Temperature distribution in a fiber with strong tangential
conductivity…………………………………………………………………….. 42
Figure 4.4: 5 layer model of a composite……………………………………...................... 43
Figure 4.5: Typical mesh of the composite model (RHS) used for the heat
transfer simulations………………………………………………………………… 45
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Figure 5.1: Interface of thickness Li added as a 3rd layer between matrix and
fiber............................................................................................................................. 52
Figure 5.2: Comparison of thermal conductivities obtained by different
models……................................................................................................................. 60
Figure 6.1: Temperature distribution in the 5 layer model of a composite....... 61
Figure 6.2: Variation of sample conductivity with increasing interfacial
thermal conductivity............................................................................................ 64
Figure 6.3: Comparison of h* values obtained using three different
dispersion factors.................................................................................................. 72
Figure 6.4: h* results obtained by dispersion factor of 5.2, grouped according
to volume fraction………………………………………….………………………… 74
Figure 6.5: h* results obtained by dispersion factor of 5.2, grouped according
to make of fiber……………………………………………………..…………………. 75
Figure 6.6: h* results obtained by dispersion factor of 5.2, grouped according
to fiber diameter………………………………………………………………………. 77
Figure 6.7 (a): Comparison of h* obtained from theoretical calculations and
simulations at dispersion factor based on hi,max=106 W/m2K........... 81
Figure 6.7 (b): Comparison of h* obtained from theoretical calculations and
simulations at dispersion factor based on hi,max=8x105 W/m2K.. 81
Figure 6.7 ©: Comparison of h* obtained from theoretical calculations and
simulations at dispersion factor based on hi,max=6x105 W/m2K.. 82
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CHAPTER 1: INTRODUCTION
1.1 Composite Materials
Composite materials are based on the concept that a combination of different
materials can attain properties that the constituent materials cannot attain
individually by themselves (Strong, 2008). This concept has been applied to create a
variety of composite materials having a wide range of desirable properties superior
to conventional materials.
The concept of composite materials has been known to mankind since as
early as 1500 B.C. when composites existed in various forms, for example mud walls
reinforced by bamboo or use of laminated metals in forging swords later in 1800
A.D (Kaw, 2006). Modern composites however, were discovered in the 20th century,
after the discovery of fiber glass in the 1930s, after which glass fiber reinforced
resins were used in aircrafts. The development and use of composite materials has
been increasing ever since. After the development of carbon, boron and aramid
fibers, composites were widely used in the structural parts of aircrafts especially
during World War II. After World War II, composites were introduced in
automobiles and have gained popularity in many other fields due to their superior
mechanical properties (Strong, 2008).
Typically, composite materials are “solid materials composed of two phases:
‘Binder’ or a ‘matrix’ and ‘reinforcements’ or ‘fillers’. The matrix surrounds the
reinforcements and holds them in place” (Chung, 2010; Strong, 2008). Composites
can also be defined as, “a structural material that consists of two or more
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constituents that are combined at a macroscopic level, are not soluble in each other,
and have complimentary properties” (Kaw, 2006; Strong, 2008). One constituent
phase is the matrix phase and the other is the reinforcement phase.
Any composite material has properties based upon the properties of matrixphase and reinforcement-phase. Other factors include fraction of the phases in the
composite and the quality of the interface between the two phases (Chung, 2010).
Since the composite is a combination of two or more materials that is expected to
produce better properties than each individual component (Campbell, 2010), the
materials of individual components can be selected in order to tailor the resultant
properties of the composite. A variety of properties are produced through various
combinations of materials, having various proportions, various morphologies etc.
(Campbell, 2010). Since they can be tailored to obtain a wide range of properties
that conventional materials cannot attain, composite materials have become
popular in the engineering field (Chung, 2010).
Each phase plays a complementary role in composite materials. The matrix
(or binder) binds the reinforcement-fibers together and gives shape to the
composite. It is generally the weaker element and cannot withstand external loads
upon the composites by itself. It shields and protects the fibers from the
environmental and handling damage. The matrix does not contribute much to the
strength of the composite; however, the external load is transferred by the matrix to
the fibers through their interfacial contact. The matrix phase is usually either made
of polymers, metals, ceramic, or a combination of more than one of the previously
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mentioned materials (Chung, 2010). A matrix made of more than one component
can be referred to as a hybrid matrix. A cross section of a typical composite with a
hybrid matrix reinforced with fibers with circular cross-section can be seen in
Figure 1.1.
Figure 1.1: Typical cross section of a hybrid matrix composite transverse to its
unidirectional fibers
Reinforcements are the stronger materials and dominate the mechanical
properties of the composite such as strength and stiffness. They can be
discontinuous (in the form of particles or whiskers), or continuous fibers. The fibers
may be oriented in a particular direction or in a random fashion within the
composite (Chung, 2010). Glass fibers, carbon fibers and aramid fibers are a few
examples of reinforcements widely used in composite materials. The composite
material samples used for experimentation in this project are in the form of
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unidirectional laminates with polymer (epoxy) as the matrix, reinforced by
continuous carbon fibers.
Composites are typically in the form of plies and laminates. As shown in
Figure 1.2, when fibers are bound by a matrix in a single layer it is called a ply. When
a number of plies are in a lay-up stacked together, it is called a laminate.
(a)
(b)
(c)
Material Fiber
Laminate
Ply
(d)
Figure 1.2(a) & (b): Typical composite plies with cylindrical fibers bounded by a
matrix with various orientations of fibers. (c): Hybrid matrix,
containing conductive particles, reinforced by cylindrical fibers
(d): Several plies, each having different orientations of fibers,
bonded into a single laminate (SolidWorks 2013).
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In each ply, the fibers or tows are arranged parallel as shown in Figure 1.2
(a), (b) & (c). In a laminate, orientation of the fibers in all the plies could be parallel
or it could differ from ply to ply as is shown in Figure 1.2d (Campbell, 2010).
In the case of continuous fiber laminates, as properties of fibers are strongest
along their axial direction, fibers can be oriented in multiple directions in the plane
of the laminate to achieve strength in multiple directions. However, in the direction
perpendicular to the plane, properties are poor as none of the fibers are oriented in
that direction (Chung, 2010). This project focuses on enhancing thermal properties
of composite laminates in the perpendicular direction. Figure 1.3 shows the
directions in a composite along which the properties are good (x) or poor (y, z).
Y (Poor TC)
Z (Good TC)
X (Poor TC)
Figure 1.3: Anisotropic thermal conductivity (TC) in a composite with carbon fiber
reinforcement.
One of the most important advantages of composite materials is that they
tend to be much lighter in weight compared to conventional materials. Especially in
aircrafts and other automotive applications, replacing conventional metal alloys
with lighter composites will result in considerable mass reduction without
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compromising strength and hence, significant reduction in fuel requirements (Kaw,
2006).
A few characteristics of composite materials such as the cost of materials and
fabrication (Kaw, 2006), the lack of design rules and long development periods are
disadvantageous to the use of composite materials (Strong, 2008). Due to their
highly anisotropic geometry at the macroscopic as well as the microscopic level,
mechanical characterization of composites has proven difficult (Kaw, 2006). In the
case of thermal properties in a composite, simple rules of mixtures can determine
the effective thermal conductivity of a composite in a direction along the fiber axes,
but very few analytical models have been developed to predict the thermal
conductivity in the transverse direction (Islam & Pramila, 1999). Hence, finite
element analysis is a logical choice to study transverse heat transfer in composite
laminates.
1.2 Thermal Applications of Composite Materials
Polymers are currently being used in many applications as a replacement for
metals. However polymers are very poor thermal conductors. Polymer matrix
composites can be designed to be much more conductive than polymers, especially
with the use of carbon fibers. Carbon fiber composites, which are the subject of
analysis in this project, are being used due to their good thermal properties in heat
shields of missiles and rockets, brakes of automobiles where friction generates heat
that must be dissipated, housing of computers, motors and electrical control panels
where heat is generated and needs thermally conductive covers for high heat
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dissipation rates. Enhancing the thermal conductivities of polymer materials beyond
their current values becomes essential to meet the demands for dissipation of ever
increasing power generation rates per unit area (Strong, 2008).
1.3 Transverse Properties
The material properties of a composite depend on various factors: individual
properties of the components that form the composite, the volume fractions of both
components, and the interfacial bonding between the two components. The fibers
are the component that imparts strength to the composite. The matrix transfers the
external load on to the fiber through their interfacial bonding. It follows that the
strength of the composite as a whole depends on the integrity of the interfacial
bonding between matrix and fibers and how well the load is transferred (Kaw, 2006;
Strong, 2008). Similar to the mechanical strength of the composite, the quality of the
interface also affects the thermal conductivity of the composite, but significantly
only in transverse direction to fiber axes (Grujicic et al., 2006). In the longitudinal
direction of the fibers, the interface is not critical to longitudinal heat conduction
and since carbon fibers can have excellent longitudinal (axial) conductivity. In the
transverse direction, heat flux must cross the fiber-matrix interface repeatedly.
Therefore, the thermal characteristics of the interface have a major effect on the
transverse thermal conductivity. This study involves evaluation of the thermal
interface quality in carbon fiber reinforced epoxy composites in order to determine
the best way to enhance the thermal properties of the interface to increase
transverse thermal conductivity.
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The composite materials involved in this study are in the form of laminates of
an approximate thickness of 2 mm to 3 mm, manufactured by the project sponsors
Performance Polymers Solution Inc. (P2SI, Moraine, OH). They are made of the
epoxy, Epon 862/W, reinforced with pitch-carbon fibers. Epoxy is the most common
type of matrix used in advanced composite materials. It is available in multiple
grades which provide a wide range of properties to select from while tailoring
composites (Kaw, 2006). Epoxies have excellent shear strength which results in
stronger interfacial bonding with fibers but the strength of those bonds are highly
susceptible to poor surface quality and requires meticulous surface preparation
during manufacturing. Epoxies show good strength, stiffness and thermal stability
due to high distortion and degradation temperatures (Strong, 2008).
Carbon fibers, the reinforcement, are known to have excellent thermal
properties. Since their thermal properties and microstructure in addition to the
interface-quality are major factors contributing to the thermal conductivity in the
composite, they will be considered in the model developed in the study. The model
is used to study the effect of the interface thermal resistance on the bulk thermal
conductivity. Carbon fibers are discussed in further details in the next chapter.
1.4 Performance Polymers Solution Inc. (P2SI)
P2SI is a company founded in 2002 and is based in Moraine, Ohio. The
company offers prepregs, structural adhesives, fiber molding compounds, and
syntactic foams for light weight applications primarily in wind energy and
aerospace industries. They also provide composite materials for high temperature
19
and high thermal conductivity applications. P2SI provided Ohio University with
samples to be tested for their thermal conductivities and gave access to
experimental data from their specialized thermal characterization equipment. The
materials used in the composites along with the fiber volume fractions were
provided for simulations and further calculations.
1.5 Summary and Rationale of Proposed Research
This study is mainly focused on the properties of composites with
unidirectional fibers in the plane transverse to the fibers. In the transverse plane,
fibers are intermittently dispersed in the matrix phase. The heat flux in the
transverse plane travels alternately through matrix and fibers along its path.
Therefore, when it passes from one phase to another, the interface between the two
phases obstructs heat conduction due to contact thermal resistance. Therefore, the
interface thermal resistance can have strong influence on the transverse thermal
conductivity of fiber reinforced composites. However, there are very few numerical
or analytical studies on calculation of interface resistance. Therefore, the goals of
this study are to calculate the interface thermal resistance of continuous carbon
fiber reinforced hybrid composites by a combination of numerical analysis and
experimental data. The numerical results will be compared with experimental data
to predict the interface thermal resistance. The results from this study can provide
insight on the dominant parameters that control the transverse thermal
conductivity of fiber reinforced composites.
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CHAPTER 2:
CARBON FIBERS
Carbon fibers are one of the most popular forms of reinforcement in
composite materials. They have excellent mechanical properties (Chung, 2010);
thermal conductivities that can be even better than most conductive metals like
copper (as high as 1000 W/mK), and a low coefficient of thermal expansion
(Campbell, 2010). Good and stable thermo-physical properties even under extreme
thermal environments combined with high stiffness render them an attractive
option for applications where heat dissipation is crucial, and where temperatures
are possibly greater than 2000°C. Examples of applications include aircraft and
other vehicle brakes or heat shields in missiles where conventional materials would
either undergo melting or a severe change in properties which would therefore
compromise their performance (Donnet & Bansal, 1984).
In this project, where the focus is on enhancing heat transfer properties of
composite materials, naturally carbon fibers are a good choice for reinforcement.
The composite laminates provided by P2SI are carbon fiber reinforced polymer
matrix composites.
There are several types of carbon fibers. They are classified based on the
precursor material used for their manufacturing, since the resultant properties are
dependent on the precursor (Huang, 2009). Two most common precursors used are
polyacrylonitrile (PAN) and mesophase pitch (Huang, 2009). Pitch based carbon
fibers have become popular due to their excellent thermal properties (Uemera,
2010). They have far better thermal conductivities (up to 1000 W/mK) compared to
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PAN carbon fibers (10 to 20 W/mK) (Campbell, 2010). For achieving the best
mechanical properties and weight reduction while retaining good thermal
properties, continuous pitch carbon fibers have been used (Donnet & Bansal, 1984).
Carbon fibers are highly anisotropic and their thermal properties are best in
the axial direction. In most carbon-fibers, transverse properties are typically about
1% of the axial properties. In carbon fibers where modulus in the axial direction is
145 msi, the transverse value is only 5 msi (Campbell, 2010). Anisotropic
microstructures of carbon fibers are responsible for the anisotropic nature of their
properties. Similarly, the anisotropy of the thermal conductivity of carbon fibers is
attributed to their anisotropic microstructure as discussed below.
2.1 Anisotropy in Carbon Fibers Composites
Since the fibers are strong in the axial direction and weaker in the transverse
direction, fiber axial orientation is usually selected to be in the direction of the load.
(Referring to Figure 1.3) To handle loads from multiple directions in plane x-z, plies
with varying orientation of carbon fibers in are combined into a laminate. However,
very often, the heat flux is perpendicular to the load (or fiber axes) and thermal
conductivity of the laminate in a direction perpendicular to fiber axes (y direction
shown in Figure 1.3 repeated below) remains poor. The transverse thermal
conductivity depends on the microstructure of the fiber in the transverse plane
(plane x-y). The laminates that are the samples to be studied in this project have a
similar arrangement of carbon fibers, which is why it is important to review the
microstructures of carbon fibers in order to study heat transfer in such laminates.
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Y (Poor TC)
Z (Good TC)
X (Poor TC)
Figure 1.3: Anisotropic thermal conductivity in a carbon fiber composite. As
graphene planes are aligned along fiber axis, conductivity in that
direction is good. Transverse to the fiber, conductivity is poor.
2.2 Anisotropic Micro-Structure of Carbon Fibers
Carbon fibers consist of graphite crystals. The unique properties of these
fibers are due to highly anisotropic graphite lattice structure. Several flat-sheets like
layers of carbon atoms, called “graphene layers”, are arranged in a stack to form a
graphite crystal as shown in Figure 2.1(a) (Fitzer & Manocha, 1998).
Figure 2.1(a): Layers of carbon atoms in an anisotropic structure of graphite crystal.
Graphene planes, formed by a layer of carbon atoms (Fitzer &
Manocha, 1998)
23
Figure 2.1(a) is a modification of the original figure from the book ‘Carbon
Reinforcements and Carbon/Carbon Composites’ (Fitzer & Manocha, 1998). Figure
2.1(b) depicts the orientation of graphene planes with respect to the fiber-axis. The
basal planes of each layer are bonded by strong covalent bonds in the plane of the
layer (or the crystallographic direction). But each layer of strongly bonded planes is
bonded to other such layers by weaker van der Waals forces. Therefore, material
properties have a high value along strong covalent bonds (in the plane of each layer)
and poor along weak van der Waal’s bonds (perpendicular to each layer) resulting
in the anisotropy. Therefore, properties of carbon fibers will depend on the
orientation of these bonds. The thermal conductivity along a graphene plane is
dominant compared to its value perpendicular to the planes (Fitzer & Manocha,
1998). It follows that the value of thermal conductivity of the composite in any
direction will depend on the orientation of graphene planes with respect to that
direction.
The carbon atoms form a nearly perfect graphite crystal structure, well
aligned along the fiber-axis giving a high axial thermal conductivity as shown in
Figure 2.1(b). Pitch fibers show highest degree of orientation of graphite structure
along fiber axis and therefore, good axial thermal conductivities compared to any
other types of carbon fibers (Campbell, 2010).
24
Transverse
direction/
Perpendicular to
fiber axis
Crystallographic
direction/ Parallel to
fiber axis
Figure 2.1(b): Orientation of graphene planes in carbon fibers. Strongly bonded
planes in a layer run along the fiber axis. Weak bonds between
layers are along transverse direction to the fiber axis (Strong, 2008).
The path of the heat flux and hence the transverse thermal conductivity also
depends on the arrangement of graphene planes in the x-y plane shown in Figure
2.2 (a) & (b). Looking at the circular cross section of pitch carbon fibers, the
graphene planes usually appear to be either fanning out radially from the center of
the fiber as shown in Figure 2.2(b) or arranged in concentric rings (like in an onion),
around the center of the fiber as shown in Figure 2.2(a) (Fitzer & Manocha, 1998).
25
Radially
fanned out
planes
Concentric
graphene planes
(a)
(b)
Figure 2.2(a): Transverse section of a carbon fiber with radially arranged graphene
planes.
(b): Carbon fiber cross section with concentric graphene planes
Figure 2.2 (a) & (b) are modifications of figures from the book ‘Carbon
Reinforcements and Carbon/Carbon Composites’ (Fitzer & Manocha, 1998).
It has been established that the thermal conductivity in a carbon fiber will be
dominant along graphene planes and poor otherwise. Since both structures are
radially symmetric, thermal conductivity of a single fiber studied for any one
direction (x or y) holds for all directions in the x-y plane. In the following discussion
y-direction will be considered.
In a finite element study of heat transfer through composites with pitch
carbon fibers (Grujicic et al., 2006), it was shown that the interfacial thermal
resistance significantly reduces the thermal conductivity in the transverse direction.
The microstructure of the fiber is such that the graphene planes are fanned out
radially from the center of the fiber, resulting in heat flux vectors following the same
path (Grujicic et al., 2006). The model by Grujicic et al. consisted of a cuboid
26
composite structure consisting of four fibers of diameter 14 µm and the cube having
dimensions to correspond with the volume fraction of the actual composite. Heat
transfer in the longitudinal as well as transverse direction to the fibers was
analyzed. Figure 2.3 shows the effect of the graphene plane orientation on the heat
transfer is in the y-direction.
Radial orientation
q
Concentric ring
q
q
q
(a)
(b)
Figure 2.3(a): Strong conductivity along radial graphene planes can assist heat flux
in Y direction due to heat flow along the graphene planes.
(b): Weak conductivity between graphene sheets may result in poor
conduction of heat in Y direction.
The radially oriented graphene planes will have strong radial conductivity. It
will cause the transfer of heat in the desired y -direction by transferring heat flux
along the radial graphene planes, as shown in Figure 2.3(a). Therefore, a better
value of conductivity in the y-direction can be expected. On the other hand, in the
circumferentially arranged concentric graphene planes shown in Figure 2.2(b), the
27
tangential conductivity will be strong, however; much of the desired heat flux must
pass from one graphene plane to another as shown in Figure 2.3(b); where the
bonding forces are weak, therefore the conduction of heat may be comparatively
poor.
A composite model was set up to evaluate the differences in the composites
made with radial arrangement vs. concentric arrangement of graphene planes. The
square model is shown below in Figure 2.4.
Matrix
Fiber
Figure 2.4: Model of a composite with minimal matrix showing the line of symmetry
in the middle
The axial fiber conductivity used in this model is 200 W/mK and matrix
conductivity is 0.2 W/mK. In order to reduce the resistance effect of the low
conductivity matrix, the matrix volume in the model was minimized by making each
exactly the same length as the fiber diameter (taken to be 10 µm in this model). In
this way, the volume of the model would mostly fiber and the transverse
28
conductivity would be highly sensitive to any change in the fiber conductivity. The
details of the typical composite model as implemented in the ANSYS-FLUENT
software are described in detail in the numerical model chapter (Chapter 4).
The results of comparison between radial fiber conductivity composite and
circumferential fiber conductivity composites are given in Table 2.1. These results
are obtained using the value of 200W/mK for longitudinal conductivity of fiber. It
can be seen that the radial arrangement produces better transverse composite
conductivity than the concentric arrangement of graphene planes. The details of the
general numerical model used for simulations are described in another chapter.
Table 2.1: Comparison of thermal conductivities of a composite having radial,
circumferential and isotropic fiber conductivity.
Type of fiber conduction
Radial Concentric Isotropic
Conductivity of
composite
W/mK
7.84
6.09
17.24
The overall conductivity obtained using isotropic fiber conductivity is much less
compared to the longitudinal fiber conductivity (200 W/mK) because of the low
matrix conductivity. This demonstrates the strong effect of the matrix conductivity
on the conductivity of the composite. The isotropic fiber model can be used to
compare the numerical model with published theoretical analysis which assumes
isotropic fibers in the matrix. The comparison is discussed in Chapter 5.
29
CHAPTER 3:
MATRIX-FIBER INTERFACE
As discussed earlier, there are two phases of materials in a composite: matrix
and fibers (reinforcement). The matrix is the continuous phase while the fibers are
dispersed in the matrix. The surface of every fiber in contact with the matrix phase
is called the matrix-fiber interface. Since fibers are typically cylindrical, the matrixfiber interface is a two-dimensional cylindrical planar surface between matrix and
fibers. Its properties dramatically affect the mechanical as well as thermal
properties of the composite (Nan et al., 1997). The thermal properties of such an
interface between the carbon fibers and epoxy matrix are the focus of this study.
The results of this study will be important for applications where heat dissipation is
crucial. For example, in electronic packaging, a large load of 4x105 W/m2 of heat flux
is to be dissipated by a small 5 mm x 5 mm chip while maintaining a temperature as
low as a 100 °C (Dunn & Taya, 1993). In such cases, it is desirable to use a material
with enhanced thermal conductivities. In order to achieve excellent thermal
conductivities to match such demands, it is important to consider not only the effect
of macroscopic factors on the thermal behavior of the material but also the
microscopic geometrical factors like interfacial imperfections (Dunn & Taya, 1993).
Since the fibers are typically longitudinal and continuous within a single ply,
they appear as a discontinuous phase in the direction perpendicular to their lengths.
Therefore, composites are anisotropic with regards to their material properties.
Heat propagation in composites in the direction along fibers depends primarily on
the thermal conductivity of the fibers which have high conductivity. When it comes
30
to heat transfer in a direction transverse to the fiber length, heat flux travels across
the interface between the matrix and fibers. Hence, effect of this interface on the
heat transfer in composites is important (Macedo & Ferreira, 2003).
3.1 Effect of Thermal Interface
On a microscopic level, the interface obstructs the heat flux which crosses
over between matrix and fibers and insulates fibers from the matrix (Duschlbauer,
Bohm, & Pettermann, 2003) in the form of a contact thermal resistance. On a
macroscopic level it has a negative effect on the bulk thermal conductivity of the
composite (Duschlbauer et al., 2003). The contact thermal resistance results in a
discontinuity in the temperature across the interface between matrix and fiber, and
has been termed as ‘Kapitza resistance’ (Dunn & Taya, 1993). In a carbon fiber
reinforced epoxy composite, a good thermal interface is essential to utilize the
thermal conductivity of fibers to the fullest extent. In a study regarding modeling the
interfacial thermal resistance and studying its effects on the overall thermal
conductivity of the composite (Islam & Pramila, 1999) it was observed that
increasing the interfacial resistance by 1/10th reduced the transverse conductivity
by approximately 50 %. Therefore, the effect of the interface resistance on
conductivities of composites cannot be ignored.
A preliminary simulation at Center for Advanced Materials Processing of
Ohio University was carried out by (Taposh, 2010) to study the effect of interfacial
thermal resistance on the thermal conductivity of a composite. Figure 3.1(a) and (b)
(Taposh, 2010) are from an internal report on this project. Please note that the
31
images were taken from the report and cannot be modified in order to magnify the
temperature and flux indicator legends. However, the colors of the contours can be
used to evaluate the relative quality of the heat flux at any location in the model.
From the results of simulations done using ALGOR; it was observed that with
a poor quality interface i.e. high thermal resistance at the interface, the fibers were
blocked out of the transverse heat flux path, as shown in Figure 3.1(a). Therefore,
fibers with good thermal conductivity will show excellent conduction in the axial
direction but may not fully participate in transverse heat conduction due to being
insulated from the matrix. Whereas, in the case of good thermal interface with low
resistance, the fiber can significantly contribute to the heat flow in the transverse
direction (z direction) as shown in Figure 3.1(b) resulting in good overall thermal
conductivity of the composite (Taposh, 2010).
Figure 3.1(a): Heat flux for Poor interface between (Taposh, 2010). Blue and green
colors indicate poor heat transfer; orange and red indicate good heat
transfer. Heat transfer along fiber axis is good and across it is poor.
(b) Heat flux for Good interface (Taposh, 2010). Since entire field shows
colors close to red, it indicates good and even heat transfer
throughout the composite along all directions.
32
The effect of thermal interface between matrix and reinforcements on the
effective conductivity of composites was also studied in comparison with the
baseline composite conductivity where the matrix and fibers have the same thermal
conductivity (Taposh, 2010). This study showed that the rise in the composite
conductivity relative to the baseline conductivity with increasing fiber or matrix
conductivity is significantly steeper in the case of good thermal interface. For
example, as shown in Figure 3.2 (a) & (b), for a poor thermal interface, even if the
composite conductivity is high, it does not exceed the baseline value more than 8
times. However, for a good thermal interface, the composite conductivity exceeds
the baseline value by over 20 times.
9
Kcomposite / Kbaseline
8
7
6
5
Km1
Kmatrix=1
4
Km5
Kmatrix=5
3
Km10
K
matrix=1
2
1
0
1
5
10
50
100
Fiber conductivity, Kf W/mK
Figure 3.2(a): The conductivity of the composite compared to the conductivity of the
baseline model (Kmatrix =Kf=1) when thermal interface is poor
(Taposh, 2010).
33
Kcomposite / Kbaseline
25
20
15
Km=1
Km1
10
Km=5
Km5
Km=10
Km10
5
0
1
5
10
50
100
Fiber conductivity, Kf W/mK
Figure 3.2(b): The conductivity of the composite compared to the conductivity of the
baseline model (Kmatrix=Kf=1) with a good thermal interface (Taposh,
2010).
It shows that even if better conductive fibers and matrix will raise the bulk
thermal conductivity, having a good thermal interface will give an even steeper
raise.
3.2 Causes of Imperfect Interfaces
Properties of the interface are dependent on surface characteristics of the
interfacial region, contact pressure, and compatibility between fibers and the matrix
(Macedo & Ferreira, 2003; Wan et al., 2011). Difference in morphologies of the
surfaces interfacing at the contact is mainly responsible for imperfect interfaces
which obstructs the heat flux between fiber and matrix and is termed as thermal
interface resistance (Duschlbauer et al., 2003). This contact thermal resistance may
be a result of an interfacial gap occurring rather unintentionally between matrix and
fiber during manufacturing; the bonding between them might be perfect initially,
34
but may have deteriorated after exposure to varying temperatures (Park, Lee, &
Kim, 2008). In that case, the difference in their coefficients of thermal expansion
causes interfacial de-bonding because the temperature at which the bond is formed
changes after cooling over time and the two materials undergo different amount of
deformation due to the different coefficients of thermal expansion resulting in slip
at the interface (Chung, 2010; Dunn & Taya, 1993). Poor mechanical bonding
between the matrix and fiber surfaces, imperfect chemical adherence or presence of
impurities between them also results in poor thermal interfaces (Duschlbauer et al.,
2003; Park et al., 2008). Experimental as well as model results (Grujicic et al., 2006)
have shown that the de-bonding or de-cohesion of fiber-matrix interface
significantly reduces the transverse thermal conductivity of composites. Even if the
contact is nearly perfect, scattering of phonons (principle carriers of energy) may be
inevitable (Duschlbauer et al., 2003). Also, air pockets form at the joining of two
surfaces and it reduces heat conduction and hampers the overall composite
conductivity (Chung, 2010).
There are a few methods to enhance the interface in order to improve
matrix-fiber bonding as well as improve the thermal conduction through the
interface. Usually, finishes or coupling agents are used to enhance the interfacial
bonding (Strong, 2008). Also, a thermal fluid/grease/paste or a molten solder are
some remedies for topographical unconformities and poor thermal interfaces
(Chung, 2010).
35
3.3 Theoretical and Analytical Studies
Even though the concept of interfacial thermal resistance had been known
(Nan et al., 1997), it was ignored due to its demand for complex mathematical
treatment and not because it was insignificant (Gu & Tao, 1988). An initial study
performed regarding heat transport in composites was carried under the
assumption of perfect interfaces (Benveniste, 1987). However, knowledge of
thermal properties of the interfaces can be used to tailor the thermal properties of
composites in the future to achieve high thermal conductivities and currently makes
for an active field of study (Nan et al., 1997).
A few studies have been conducted in the attempt of quantifying the
interfacial thermal resistance and studying its effect on the overall conductivity of
the composite. A study by Nan et al. (1997) uses the concept of the interfacial
resistance as a ‘Kaptiza resistance’. The Kapitza resistance is associated with the
Kapitza radius, which is defined as its span from the center of the fiber (Nan et al.,
1997). The following equation gives the effective conductivity of a composite after
taking the interfacial resistance into account.
Transverse thermal conductivity in a composite laminate with aligned
continuous fibers is given by-
(
)
(
)
(
)
(
)
3.3
36
Where
is the fiber conductivity
is the matrix conductivity
is the fiber volume fraction
{
3.4
3.5
is the length of fiber
is the radius of fiber
3.6
Where,
is the interfacial heat transfer coefficient (W/m2K)
The range of the Kapitza radius is:
interface. When
. When
, it represents a perfect
, there is discontinuity across the interface in the
temperature profile. The interfacial thermal resistance with a non-zero thickness
can have a dramatic effect on the composite overall conductivity (Nan et al., 1997).
The equation for effective conductivity of a composite considering the effect
of interfacial thermal resistance based on the multiple-scattering approach (Nan et
al., 1997) can be used to reverse calculate the interfacial resistance when the
effective conductivity is known from experiments. This relation will be used to
support the results obtained from the numerical model.
37
Macedo and Ferreira (2003) used the following equation for the calculation of
interfacial thermal resistance ( ): -
3.7
Where,
is the interfacial heat transfer coefficient (W/m2K)
is the density
is the specific heat
is the phonon propagation velocity = √
is the probability of transmission of phonons
The above equation is a convenient way to predict the interfacial
conductivity. However, since the shear modulus at interfaces in the samples
provided is unknown, the above equation cannot be used for this project.
Methods to compute effective properties of composites are based on known
or assumed properties of the interface. In the case of this project, the thermal
interface properties as well as the composite conductivity are unknown. The
experimental value can be combined with thermal analysis by ANSYS (FLUENT) to
quantify the interface thermal properties at Ohio University. In a previous study
(Islam & Pramila, 1999), numerical analysis was performed to predict the effective
‘through-thickness’ (transverse to fiber axes) thermal conductivity in ideal
composites as well as composites with imperfect fiber-matrix interfaces to prove
38
the applicability of numerical analysis in doing so. The obtained results matched
with the experimental results and it was concluded that it is an effective method for
thermal conductivity prediction (Islam & Pramila, 1999).
The model, considering interface resistance and simulation set up, is detailed
in following chapters. It is to be noted that in all the following chapters, the term
‘interfacial resistance’ refers to the contact thermal resistance (Ri) offered by the
interface between matrix and fiber.
39
CHAPTER 4:
NUMERICAL MODEL
This chapter details the model of composite materials created in ANSYSWorkbench software and to be simulated using the FLUENT solver. The composite
material samples used for experimental measurement of their thermal
conductivities are in the form of unidirectional laminates. Each laminate is an epoxy
based carbon fiber composite. Each sample was modeled using the Design modeler
feature in ANSYS-workbench; their respective meshes were generated and solved
using the steady state solver in FLUENT. Heat conduction through these composites
is simulated to predict the matrix-fiber interface thermal properties based on
experimental values.
To reduce the computational domain, effort, and time; a square shaped unit
cell was modeled which is a representative unit of the composite i.e. consisting of
only one fiber surrounded by a matrix but having the same volume fraction and
material properties as the sample being modeled. Following are the details of the
model and solver set up.
4.1 Geometry
To study the transverse plane heat conduction problem, a 2D model is
created in the Design Modeler of ANSYS Workbench. The 2D model appears similar
to the cross section of a composite laminate with matrix and circular fiber cross
sections. This unit cell consists of a square section of the polymer with a circular
section at its center corresponding to the fiber and therefore having the same
diameter as the fiber, as shown in Figure 4.1 (Islam & Pramila, 1999). Since the
40
fibers in samples being modeled are all assumed to be parallel to each other, this
model can be repeated to represent the whole composite.
Figure 4.1: Unit cell model of a typical composite
The dimensions of the square unit cell are determined based on the volume
fraction of each composite. Therefore, the dimension of each side is s and it will vary
in each model representing a different composite with a different volume fraction.
The following equation gives the computation of cell dimensions based on volume
fraction.
4.1
Where,
is the fiber diameter, and is the length of each side of the unit cell. In the
simulation of transverse heat conduction, heat flux is established between one pair
of opposite sides of the unit cell in order to compute transverse thermal
conductivity. The unit cell is shown in Figure 4.2.
41
2
Q (W/m
q
)
2
Q (W/m
q )
Figure 4.2: 2D Unit-cell model of a composite consisting of one carbon fiber with
radial arrangement of graphene planes, and surrounding matrix. A
temperature difference (T1 – T2) applied across transverse direction y
resulting in heat flux Q.
Figures 4.3(a) and 4.3(b) show the results of simulations with a single unit
cell. This model was used to compare radial and circumferential graphene plane
arrangements as mentioned in Chapter 2. Although these results do not fulfill the
main purpose of the project, they support the study of different types of
microstructures in carbon fiber and its effect on their thermal conductivity. They
confirm that radial arrangement gives better conductivity compared to
circumferential. The results for a radially oriented graphene planes are shown in
Figure 4.3(a), while the results with circumferentially (onion ring structure)
oriented graphene planes is shown in Figure 4.3(b).
42
Figure 4.3(a): Temperature distribution in a fiber with strong radial conductivity
Figure 4.3(b): Temperature distribution in a fiber with strong tangential
conductivity
It should be noted that a 5 layer model with five fibers is used instead of a unit cell
model, to eliminate the end effect. It consists of 5 layers of the unit cell stacked along
y direction, as shown in Figure 4.4. All results shown in Chapter6 are based on the 5
unit cell model.
43
Figure 4.4: 5 layer model of a composite
Taking the advantage of geometrical symmetry to further reduce
computational effort, only the right (or left) half of the 5 layers is modeled and the
results are mirrored about the axis of symmetry indicated in Figure 4.4 by the Line
of symmetry.
4.2 Named Selections
After creating the geometry and surfaces of the model, it is important to
create named selections in order to facilitate the assignment of material properties
and boundary conditions to specific zones of the model in the solver set up. The
parts of the model in this case that will have assigned material properties or will be
subject to boundary conditions are as follows-
44
1. Matrix: The matrix part of the model is assigned with properties of Epon 862/W
which is the epoxy matrix used in all samples.
2. Fibers: All the fiber cross sections are assigned with the properties of the
particular fiber used in the sample being modeled.
3. Matrix-Fiber Interface: All the interfaces shared between the matrix and fibers
will be created as a named selection to be assigned with the desired thermal
resistance in terms of thickness (1µm) and thermal conductivity.
4. Top side: The top surface of the laminate being modeled which will appear to be
a line segment in the 2D model will be named as ‘Top side’ and will be subject to
temperature constraint (
).
5. Bottom side: Similar to the top surface, the bottom side will be subject to a
temperature boundary condition (
).
6. RHS: The RHS i.e. the right hand of the rectangular model
7. Symmetry: The left hand side of the created model, which in actuality is a line of
symmetry between the right half and the left half of the model, is named as
Symmetry. This is the axis about which the geometry and the pictorial results are
mirrored.
4.3 Meshing
The model created in the Design modeler is imported in ANSYS mesh
developer for meshing. Mesh settings were selected to generate a quadrilateral
dominant mesh with approximate side-length of a cell at 2 x 10-7 µm. The quality of
a mesh can be judged by the orthogonal quality of the elements and their skewness.
45
Since the dimensions of the model varies depending on the volume fraction and
fiber diameter of each sample, the number of elements and nodes, the orthogonal
quality, and skewness varies for each sample. The mesh shown in Figure 4.5 is for a
sample with 60% fiber volume fraction and 10 µm fiber diameter, and consists of
17928 nodes and 17671 elements. It depicts a typical mesh for any sample. An
orthogonal quality at 0.8 (ideal at 1) and maximum skewness at 0.1 (ideal at 0) was
ensured in each mesh which indicates a good quality mesh ensuring maximum
accuracy in the solution.
Figure 4.5: Typical mesh of the composite model (RHS) used for heat transfer
simulation
4.4 Solver Set Up
Once the meshed model of the composite sample was imported into the
FLUENT solver, appropriate solver settings, as described below in this section, were
selected to simulate heat transfer in the model. Since the imported geometry was
46
2D, the 2D setting is selected by default. For increased accuracy, the double
precision option was chosen. In general, double precision calculations require more
computational effort and RAM than single precision calculations. However, since the
model was a simple geometry, selecting the double precision option achieved better
accuracy without significant increase in solving time.
Heat transfer in these composite models was simulated using the steadystate solver in a 2D planar setting. The Energy model was activated which includes
the equations for heat transfer in the solver.
4.4.1
Materials
The materials of the composites involved in this study are not available in the
FLUENT database. Therefore, the materials epoxy and fiber materials specific for
each sample model were created in the preprocessing set-up and then assigned to
their respective ‘named selections’ or zones as discussed in the beginning of this
chapter. Material properties are assigned to the appropriate zones prior to solving.
Following are the materials and the zones they are assigned to:
4.4.1.1 Matrix
The matrix is considered to have an isotropic conductivity. The matrix used
in every sample is the same: Epon 862/W. The thermal conductivity of this polymer
is 0.2 W/mK.
4.4.1.2 Fibers
Each sample has a different fiber material of which conductivities are known.
As discussed earlier, pitch fibers have anisotropic conductivity. For a fiber with
47
radially arranged graphene planes, the tangential conductivity is 1% of the radial
conductivity; while radial conductivity is the same as the longitudinal conductivity
of the fiber. This type of anisotropy in the fiber material properties can be specified
by selecting the cylindrical-orthotropic type of thermal conductivity in FLUENT.
However, the cylindrical-orthotropic conductivity accommodates specification for
only one center as a reference point for the radial and tangential components. Since
the centers of all fibers have different co-ordinates, the same material definition
cannot be used to assign properties to all the fibers. Therefore, each fiber cross
section in the same composite model is technically assigned a different material
although all the attributes remain the same except the co-ordinates of the reference
point.
4.4.1.3 Matrix-Fiber Interface
The thermal resistance of the matrix-fiber interface is to be evaluated based
on simulations. But this property is required for input in the solver set up.
Therefore, the numerical solution was repeated with various values of the matrixfiber interface. By comparing the numerical results and the experimental data, the
correct value for the interface is determined. The simulations were repeated with 9
different values over a range of conductivities.
The thermal resistance ( ) between fiber and matrix was imparted by
defining their interface as a wall of thickness 1 µm ( ) and a variable thermal
conductivity ( ).
(4.1)
48
Since the matrix-fiber interface thermal conductivity was to be varied, it was input
in the form of an ‘input parameter’ instead of a constant value. In this case, thermal
conductivity of the interface is gradually increased from 0.0001 W/mK to 10,000
W/mK in 9 steps.
As an alternative approach, the interface resistance can also be given in
terms of convective heat transfer coefficient “ ” which is dependent on
&
&
With the variation of
.
(4.2)
through a range from 0.0001 W/mK to 10000 W/mK,
varies from 102 to 1010 W/m2K.
4.4.2
Boundary Conditions
Following are the boundary conditions for the name selections discussed
previouslyTop side: The top side is constrained at temperature
.
Bottom side: The bottom side is defined as a wall similar to the top side and set to a
temperature of
.
Right hand side: The RHS is by default an insulating wall with zero thickness.
Symmetry: The named selection Symmetry is automatically construed as the line of
symmetry after the model being imported in to the FLUENT solver after meshing.
Matrix-fiber interface:
The most important boundary type in these simulations is the interface
between matrix and fibers, the two solid zones, across which heat transfer occurs. It
can be modeled in two ways: (i) by adding a 3rd layer between matrix and fiber
49
having a certain thickness and conductivity, or (ii) by coupling the matrix wall with
the fiber wall and specifying the thickness and conductivity of the coupled wall. In
the coupled wall approach, a virtual layer at the interface is used to model the
interface; and is assigned a thickness and a thermal conductivity. The comparison of
results obtained using these two methods are discussed in detail in a later chapter.
4.5 Post-Processing
After solving with the above settings and boundary conditions, the main
results to be viewed in this case are the temperature contours in the composite, and
heat flux across the thickness of the laminate which is further used for calculation of
thermal conductivity of the composite.
4.5.1
Calculation of Thermal Conductivity
There are 16 samples provided by P2SI. Each sample is simulated with 9
different interfacial conductivities. The results obtained from all 9 simulations are
used to calculate the thermal conductivity of the composite (which will result in 9
possible thermal conductivities). The heat flux
is induced as a result of the
temperature difference applied across the composite. This value was used to
calculate the overall thermal conductivity of the composite in the following manner(4.3)
(4.4)
Where,
q is the heat flux (W/m2)
50
= Heat rate obtained as result from simulation (W)
is the thermal conductivity of the composite (W/mK)
is the cross-sectional area across the flux (m2)
is the temperature difference across thickness of laminate
(K)
51
CHAPTER 5:
COMPARISON OF TWO NUMERICAL MODELS
The tasks in this project are focused towards obtaining values of interfacial
thermal resistance ( ) in the given samples of composite materials. This can be
achieved through comparison of experimental values with simulation results
obtained by modeling corresponding composites using ANSYS-FLUENT and
simulating them under appropriate boundary conditions to obtain their overall
thermal conductivities.
As mentioned earlier, the interface can be modeled in two ways, in order to
impart thermal resistance at the contact of matrix and reinforcement.
1. One way is to insert a physical, solid geometrical wall between the matrix
and fiber such that its thickness and assigned material conductivity
produces the thermal resistance to be modeled.
2. Another way available in FLUENT is to set up a virtual wall at the interface,
giving values of thickness and conductivity to impart thermal resistance.
The advantage is that the virtual wall makes absolutely no geometrical
changes in the basic dimensions of the model.
In the first approach, the interface where the matrix and fiber surfaces meet
can be imagined as a wall of a certain thickness made of a certain material. This
approach raises the question of the effect of this wall-thickness on the model. To
investigate this, the thickness of the wall is varied and its effect on the overall
conductivity is studied. However, it is essential for the comparison of two models
that their interface resistance matches regardless of wall-thickness. This issue can
52
be well addressed by controlling the material thermal conductivity of the wall. The
value
depends on two factors: the thickness of interface ( ) and the thermal
conductivity of interface material ( ). Hence, for a particular value of
, even if
thickness of the interface wall ( ) varies from case to case, the thermal conductivity
of the interface ( ) is adjusted such that the equivalent thermal resistance
remains the same. Figure 5.1 shows the interface between matrix and fiber added as
a 3rd layer.
Matrix
Fiber
Interface of
thickness Li
Figure 5.1: Interface of thickness Li added as a 3rd layer between matrix and
fiber.
can also be expressed in terms of a convective heat transfer coefficient ‘ ’
or a wall of thermal conductivity ( ) and thickness ( ). These are defined by
53
4.2
Therefore,
5.1
5.2
Where,
= Interface thermal resistance
= Interface thickness
= Interfacial coefficient of heat transfer
= Interface thermal conductivity
In this way, the interfacial resistance can be expressed interchangeably in
terms of
or
.
The effect of change in Ri on the bulk thermal conductivity as well as the
difference in results using the two types of model was studied using a range of Ri
values. The selected range of those values was between
= 10-7 to 10-10 m2K/W.
For the purpose of comparing results obtained by both these methods, all other
parameters and boundary conditions that the model is subjected to are maintained
the same. The model used to study different ways of imparting interface thermal
resistance between matrix and reinforcements is a ‘unit cell’ that represents the
composite as discussed in the previous chapter. This square shaped unit cell model
represents a typical composite with a single fiber surrounded by matrix having a
54
fiber volume fraction of 60%, the fiber conductivity being 100 W/mK and matrix
conductivity being 5 W/mK. The dimensions of the square unit cell can be calculated
based on the fiber volume fraction (60%) and fiber diameter of 7 µm.
Heat transfer is modeled in the y-direction. The top side of the matrix is kept
at 305 K and the bottom at 300 K; inducing a heat flux in downward y direction. The
results are independent of the selected temperatures. Therefore, in the model
described in this section as well as in later chapters, any two temperatures are
selected satisfying the condition: Temperature at the top > Temperature at bottom,
for a downward heat flux. The exact value of heat flux is obtained as an output after
running the simulation. The only parameter to be varied is the interface thermal
resistance between matrix and fiber which is modeled in two different ways
mentioned above making two cases for comparison. However, in each case there are
subcases due to varying thickness of the interface.
The two ways and subcases of modeling interface to be studied are:
1. Insert physical interface wall between matrix and fiber
The thickness of the wall being varied to two different values

0.035 µm

0.07 µm
2. Virtual wall between matrix and fiber
The thickness of the virtual being varied to three different values

2 µm
55

1 µm

0.05 µm
The interface in each of these subcases is assigned over a range of thermal
conductivity values to impart a range of thermal resistance values. The chosen range
of interface resistances;
= 10-7 to 10-10 m2K/W. gives thermal conductivities in the
models according to interface thicknesses and variable thermal resistances. They
are shown in Table 5.1.
Table 5.1: Variable interface thermal resistances and corresponding thermal
conductivities for various thicknesses.
Thermal resistance
for different
h (W/m2K)
(m2K/W) 0.035 µm
0.07 µm
1 µm
2 µm
0.5 µm
107
10-7
0.35
0.70
10.00
20.00
5.00
107.5
10-7.5
1.11
2.21
31.62
63.25
15.81
108
10-8
3.50
7.00
100.00
200.00
50.00
108.5
10-8.5
11.07
22.14
316.23
632.46
158.11
109
10-9
35.00
70.00
1000.00
2000.00
500.00
109.5
10-9.5
110.68
221.36
3162.28
6324.56
1581.14
1010
10-10
350
700
10000
20000
5000
An appropriate thermal conductivity
to produce the required value of
can be selected for the wall in order
using a wall of a known thickness.
56
The output of FLUENT simulations is in terms of Heat flux across the
laminate in y direction. This heat flux is then used to compute thermal conductivity
of the sample.
(4.3)
Where,
(4.4)
is the thermal conductivity of the laminate of polymer matrix
composite,
is the cross-sectional area of an element face, and
is the temperature difference across thickness of laminate
5.1 Variable Thicknesses of the Physical Wall
In the models simulated, the thicknesses selected for such a layer are 0.07µm
and 0.035 µm. 0.07 µm will be referred to as the thick interface and 0.035µm as the
thin one. These values are 1/100th and 1/200th of the fiber diameter, respectively.
The results for both the thicknesses are presented in Table 5.2.
57
Table 5.2: Thermal conductivities obtained after simulating for a thin wall interface
Interface
Thermal conductivity
Serial #
Heat flux (W)
resistance
(W/mK)
Thin
Thick
Thin
Thick
hi (W/m2K)
interface
interface
interface
interface
0.035 µm
0.07 µm
1
107
60.75
62.00
12.15
12.40
2
107.5
78.21
80.71
15.64
16.14
3
108
86.71
90.01
17.34
18.00
4
108.5
89.90
93.58
17.98
18.71
5
109
91.01
94.96
18.20
18.99
6
109.5
91.49
95.92
18.29
19.18
7
1010
92.00
97.68
18.40
19.53
It can be seen from Table 5.2, the thicker interface gives greater overall
thermal conductivity of the composite as compared to the thinner one even though
the equivalent hi for both is the same. This difference occurs due to the difference in
the fraction of volume that the two interface layers occupy. In the case of thicker
interface, it replaces a greater amount of low conductivity matrix adjacent to the
fiber.
5.2 Simulation with Virtual Interface Layer
Another way to impart a thermal resistance at the interface between the
matrix and fiber is to model a coupled-wall between the surface of fiber and matrix.
The thickness and thermal conductivity of the material of this virtual wall can be
input in FLUENT to give the required value of interface resistance. Since this wall is
virtual or fictitious, it does not compromise the geometry of the model but serves
58
the purpose of imparting thermal resistance in the path of heat flux between matrix
and the fiber.
It is seen from Table 5.3 that as long as the equivalent heat-transfer
coefficient ‘hi’ is the same, changing the thickness and conductivity of the wall to
various values does not make any difference to the overall heat flux and thermal
conductivity. This is because it imparts thermal resistance when heat flux crosses
over the interface without any physical change in the geometry, as opposed to the
thin wall resistance that changes the geometry due to its material thickness and
occupies appreciable volume fraction in composite material domain.
Table 5.3: Thermal conductivity with variable thickness of interface modeled as a
virtual wall
Thermal
Serial Interface
Heat flux (W)
conductivity
#
resistance
(W/mK)
hi
Virtual interface Virtual interface
(W/m2K)
thickness
thickness
2 µm 0.5 µm
2 µm
0.5 µm
1
107
59.84
59.84
11.96
11.96
2
107.5
76.69
76.69
15.33
15.33
3
108
84.83
84.83
16.96
16.96
4
108.5
87.86
87.86
17.57
17.57
5
109
88.88
88.88
17.77
17.77
6
109.5
89.21
89.21
17.84
17.84
7
1010
89.31
89.31
17.86
17.86
59
5.3 Physical Wall Vs. Virtual Interface Thickness
Table 5.4 compares results obtained by physical wall resistances against
virtual resistances. Ideally, as long as Ri remains constant, regardless of the
thickness of the interface, bulk conductivity should remain constant. It is seen from
the table as well as Figure 5.2 that for any constant value of Ri, the physical wall
interface results vary with the thickness of the wall. On the other hand, even with
varying thickness, the coupled-wall interface model gives consistent results as long
as the equivalent Ri remains constant. Therefore, the virtual coupled-wall interfacial
resistance is used in this project.
Table 5.4: Comparison of thermal conductivities with thin wall and virtual wall
models with variable interfacial resistance
Interface
Heat flux (W)
Thermal conductivity (W/mK)
resistance
Thin
Thick
Virtual
Thin
Thick
Virtual
hi
Interface Interface Interface Interface
Interface Interface
(W/m2)
0.035µm
0.07µm
1µm
0.035 µm
0.07 µm
1 µm
1
107
60.75
62.00
59.84
12.15
12.40
11.97
2
107.5
78.22
80.71
76.70
15.64
16.14
15.34
3
108
86.71
90.01
84.83
17.34
18.00
16.97
4
108.5
89.91
93.59
87.87
17.98
18.72
17.57
5
109
91.02
94.96
88.89
18.20
18.99
17.78
6
109.5
91.49
95.92
89.21
18.30
19.18
17.84
7
1010
92.01
97.68
89.32
18.40
19.54
17.86
60
Thermal conductivity (W/mK)
22
Thermal Conductivities by two different models and variable thermal
resistances
20
18
16
Thin
Interface
14
Thick
Interface
Virtual
Interface
12
10
7.0
7.5
8.0
8.5
log(hi)
9.0
9.5
10.0
Figure 5.2: Comparison of thermal conductivities obtained by different models with
variable interfacial resistance.
61
CHAPTER 6:
RESULTS AND DISCUSSION
The results obtained for each sample in the form of heat flux, for various
estimated interface thermal conductivities, were used to calculate thermal
conductivities of the sample as discussed in the previous chapter. This chapter
details the results and further calculations to obtain actual interface thermal
conductivities.
6.1 Temperature Distribution
The temperature distribution in the model after simulation with previously
discussed boundary conditions is shown in Figure 6.1. The temperature is seen
gradually decreasing from top to bottom due to the boundary conditions:
Temperature at the top at 310 K and bottom at 300 K. The contours inside the fibers
are radially fanning out. This pattern is a result of the fiber’s strong radial
conductivity and weak tangential conductivity.
Figure 6.1: Temperature-distribution in the 5-layer model of a composite.
62
6.2 Thermal Conductivity of Samples
Table 6.1 gives thermal conductivities of composites (kcomposite) obtained by
simulations using a range of interfacial heat transfer coefficient (hi) values, as well
as thermal conductivities obtained from experimental measurement. The
experimental values were obtained from P2SI. In simulations, the value of
is
raised from 103 to 108 which covers the range between a nearly infinite thermal
resistance to a near zero thermal resistance. Therefore, the real value of
to be found out, must lie within those limits. The
, which is
input in the simulations is
increased step by step. It was expected that, when it reaches the real unknown
value, the simulation conductivity would equal the experimental conductivity. That
particular
can be then acknowledged as the real
.
It can be seen from Table 6.1 that the conductivity of each sample increases
with increasing interfacial thermal conductivity. However, the simulation
conductivity reaches a maximum limit which is lower than the experimental
conductivity in case of each sample. For example, in the case of sample 1, after
varying
from 103 to 108 W/m2K, the maximum sample conductivity (0.752
W/mK) reached was considerably lower than the experimental conductivity (1.16
W/mK). It can be concluded that the simulations under-estimate the composite
conductivities. Therefore, they cannot be used directly to determine the unknown
real
.
63
Table 6.1: Thermal conductivities of samples obtained by simulations at varying
interface heat transfer coefficients and by experiments
hi (W/m2K)
SAMPLE DETAILS
Sample
#
Fiber
Type
Fiber
Conducti-vity
Fiber
Dia
Fiber
Volume
fraction
Experimental
Conductivity
kf W/mk
D
Vf
kcomposite
10
3
10
4
10
5
10
6
10
7
10
8
Simulation Composite Conductivity
W/mK
W/mK
1
CN-50
(N)
140
10
57.49
1.16
0.055
0.096
0.328
0.661
0.752
0.763
2
CN-60
(N)
180
10
68.79
2.57
0.035
0.079
0.363
0.977
1.248
1.286
3
CN-80
(N)
320
10
57.54
1.52
0.055
0.095
0.331
0.674
0.770
0.782
4
CN-90
(N)
500
10
68.83
3.37
0.035
0.079
0.364
0.980
1.254
1.293
5
YSH60A60S
(N)
180
7
68.5
2.09
0.034
0.065
0.290
0.883
1.215
1.266
6
YS-80A
(N)
320
7
65.8
2.82
0.039
0.07
0.286
0.805
1.055
1.091
7
K1352U
(M)
140
10
62.15
1.59
0.047
0.089
0.342
0.764
0.899
0.916
8
K6356U
(M)
130
11
59.21
1.53
0.053
0.097
0.350
0.705
0.802
0.814
9
K1392U
(M)
210
10
60.48
1.58
0.050
0.091
0.337
0.725
0.843
0.857
10
K13C2U
(M)
620
10
58.32
2.15
0.054
0.094
0.331
0.681
0.780
0.792
11
K13D2
U (M)
800
11
56.09
2.02
0.102
0.102
0.654
0.733
0.743
0.744
12
CN-80
(2) (N)
320
10
70.86
2.94
0.031
0.096
0.372
1.094
1.477
1.536
13
CN-50
(2) (N)
140
10
72.31
2.81
0.028
0.074
0.377
1.169
1.645
1.723
14
K13C2U
(2) (M)
620
10
63.4
4.24
0.045
0.087
0.348
0.811
0.970
0.991
15
K13D2
U (2)
(M)
800
11
52.37
2.33
0.065
0.107
0.329
0.590
0.651
0.658
16
YS-95A
(N)
600
7
58.87
1.98
0.051
0.081
0.275
0.655
0.793
0.811
64
A typical variation in the thermal conductivity of the sample (sample # 1)
against varying interface heat transfer coefficient (hi) is shown in Figure 6.2.
Sample# 1
0.9
kcomposite W/mK
0.8
0.7
0.6
0.5
Sample# 1
0.4
0.3
0.2
0.1
0
1
2
3
4
5
6
7
8
9
10
11
log(hi)
Figure 6.2: Variation of sample conductivity with increasing interfacial thermal
conductivity.
The matrix conductivity was shown be extremely important for the
conductivity of the composite. However, for the composite conductivity to match
the experimental value, the matrix conductivity must be increased in the simulation
by a factor of about 2 to 4.5, which would be an increase of 100% to 350%. This is
as shown in the Table 6.2 where the matrix conductivity are listed needed for the
simulation results to match experimental data. These three composites were
selected to represent low, medium and high values of experimental thermal
conductivity.
65
It can be seen from Table 6.2 that the matrix conductivity must be raised to
the range of 0.33 W/mK to 0.9 W/mK for the numerical results to match the
experimental data. Since the thermal conductivity of epoxy tends to be in the range
of 0.17 to 0.20W/mK; it seems unlikely that the uncertainty in the matrix
conductivity would account for the discrepancy between simulation and
experimentation.
Table 6.2: Matrix conductivities needed to match composite conductivity with
experimental conductivity
SAMPLE
#
1
2
4
10
12
14
Experimental
Matrix
Conductivity Conductivity
W/mK
W/mK
1.16
2.57
3.37
2.15
2.94
4.24
0.33
0.415
0.51
0.55
0.85
0.9
The experimental data for the samples were provided by Performance
Polymers, Inc. (P2SI); and the measurement was done in a fully calibrated
instrument. Therefore, it was concluded that the reason for the discrepancy must be
in the simplified model. An obvious source of error in this model is the ideal
arrangement of the fibers in the matrix. In this two-dimensional numerical model,
the assumption is that the fibers are perfectly aligned in the axial direction,
maintaining the same distance from neighboring fibers. In practice, this is not
66
correct, since fibers are dispersed from their ideal geometrical locations. In the next
section, the error due to fiber dispersion effects is discussed.
6.3 Dispersion Factor
As discussed in the last section, the values from the simulation were
significantly less than the experimental values for the thermal conductivity. Since
this discrepancy cannot be explained by variations in the thermal conductivity of the
matrix or experimental measurements, it is assumed that the values of thermal
conductivity in a composite are significantly influenced by the dispersion of fibers
from the ideal locations in the model.
The difference between experiments and simulation can be explained by the
fact that the idealized model uses perfectly parallel and straight fibers in the matrix.
The true arrangement of fibers in the transverse plane is not a perfect array of
parallel straight fibers; instead, it is observed that the fibers are dispersed from
their ideal positions. Consequently, fibers can come quite close to each other, and
may even touch other fibers at several points along their lengths. The model cannot
account for these factors and therefore, resulted in a different (lower) thermal
conductivity than the actual/experimental conductivity value at all values of
interfacial resistance. In other words, the fibers in the actual composite are
dispersed from their ideal positions. Since the samples are 3-dimensional, there will
be fibers that are very close to each other (and in physical contact) at many points
over the length of the fiber, which can greatly change the thermal conductivity of the
composite.
67
It should be noted that all the composite samples have the same matrix and
were manufactured using the exact same process of fiber winding on the same
mandrel, and all the samples are made of unidirectional plies. Most fiber volume
fractions are within the range of approximately 60% to 70%. Therefore, it is
reasonable to assume that the level of dispersion is approximately similar in all the
samples.
To compensate for the fiber dispersion, the sample conductivities obtained
from experimental measurement need to be lowered by a certain factor to represent
the thermal conductivities for the ideal unidirectional laminate as represented in
the numerical model. This will be termed as a “dispersion factor” and it will
calculated based on the experimental value of sample thermal conductivity and the
simulation conductivity obtained at a perfect interface. It is given by the following
equation.
6.1
Since the resulting composite thermal conductivities from the simulations
reach a maximum limit near
, this value can be considered to be
the maximum value of interface heat transfer coefficient. This is consistent with a
previous study in which the interface heat transfer coefficient values in composites
were found to be between 105 to 106W/m2K (Macedo & Ferreira, 2003).
68
The dispersion factor, as given by Eqn. 6.1, cannot be calculated by dividing
the experimental conductivity with the simulation conductivity since experimental
values are dependent on two characteristics:
(a) The dispersion of the fibers from the ideal positions in the composite
(b) The interfacial coefficient of heat transfer
All samples in this study were processed in identical manner with similar fiber
volume fractions.
In order to evaluate and compare the samples in terms of
interface heat transfer, it was assumed that the dispersion factor for all samples is
approximately the same, and the “best sample” has an experimental conductivity
with the highest
. In other words, the “best sample” is one that
shows the biggest percentage difference between the experimental and the
simulation value. This sample is then assumed to have
(the
maximum value), and it is used to calculate the dispersion factor.
This analysis may not actually produce very accurate values of
, but at the
very least will produce a good comparison of all the samples on the scale of the
interface heat transfer coefficient
. It will be shown later that the results are not
strongly dependent on the value of
. Sample #14, which had the highest
thermal conductivity in the samples, was identified as the “best sample” and it gave
a dispersion factor of 5.22. All the thermal conductivities obtained by experiment
were reduced by this dispersion factor. After this step, the “corrected” experimental
values of all samples fell between the minimum and maximum limits of the
simulation thermal conductivities as can be seen in Table 6.2. Now, at some value of
69
hi, the corrected conductivities will match the experimental value. That value of hi
will be denoted by h*. Table 6.3 shows the simulation conductivities for sample#1.
Table 6.3: Simulation conductivities of sample#1
hi
W/m2K
Simulation
conductivities
102
103
104
105
106
0.0510
0.0554
0.0960
0.3279
0.6607
The corrected experimental conductivity of sample 1 is 0.22 W/mK. It is
obvious from Table 6.3 that the h* value will lie in between 104 and 105 W/m2K.
After further simulations, h* value was found to be 0.48x105 W/m2K. The same
procedure was followed for rest of the samples. The results for all samples are given
in Table 6.4 below. It is important to note that the effect of the dispersion factor is
quite large (~ factor of 5), and is stronger than the effect of the interfacial heat
transfer coefficient.
These results will be compared with another set of results obtained by
theoretical calculations (Nan et al., 1997). To assess the sensitivity of h* on the
assumed value of
values.
, two other basis values for
were used to calculate h*
70
Table 6.4: Experimental conductivities (corrected with dispersion factor of 5.22)
and corresponding values of interfacial heat transfer coefficient (h*)
Sample
#
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
Experimental
Conductivity
Conductivity by simulation as a function of hi (W/m2K)
W/mK
(without
dispersion)
h* x 10-5
W/mK
hi=102
hi = 103
hi = 104
hi = 105
hi = 106
hi = 107
W/m2K
0.222
0.492
0.291
0.646
0.400
0.540
0.305
0.293
0.303
0.412
0.387
0.563
0.538
0.812
0.446
0.379
0.051
0.029
0.051
0.029
0.03
0.035
0.042
0.048
0.045
0.049
0.053
0.0512
0.022
0.04
0.061
0.048
0.055
0.035
0.055
0.034
0.033
0.038
0.047
0.052
0.05
0.054
0.058
0.056
0.027
0.044
0.065
0.051
0.096
0.079
0.095
0.079
0.065
0.07
0.089
0.097
0.091
0.094
0.102
0.096
0.074
0.087
0.107
0.081
0.327
0.363
0.329
0.365
0.279
0.28
0.341
0.33
0.337
0.332
0.338
0.326
0.374
0.347
0.326
0.274
0.660
0.9800
0.6700
1.0000
0.7700
0.7600
0.7620
0.6180
0.7290
0.6900
0.6410
0.6570
1.1400
0.8130
0.5800
0.6500
0.752
1.257
0.766
1.29
1
0.978
0.897
0.689
0.848
0.793
0.718
0.748
1.58
0.974
0.638
0.786
0.48
1.70
0.78
2.90
2.40
5.30
0.81
0.71
0.82
1.60
0.13
2.00
1.80
10.00
2.20
1.90
As mentioned earlier, the interface heat transfer coefficient values in
composites were found to be between 105 to 106W/m2K (Macedo & Ferreira, 2003).
Therefore, following values within that range were selected as a basis for dispersion
factors and the same procedure was followed as described for the previous case.
o
(Dispersion factor = 5.45)
o
(Dispersion factor = 5.8)
The results for h* obtained using all the 3 dispersion factors are shown in Table 6.5.
71
Table 6.5: The h* values for each sample obtained by simulations using three
different dispersion factors
Sample
#
Dispersion
factor
Experimental
5.22
5.45
5.8
5.22 5.45
5.8
Conductivity w/o dispersion
h* x10-5 W/m2K
W/mK
0.222
0.213
0.200 0.48 0.45 0.4
1
1.16
2
2.57
0.492
0.472
0.443
1.7
3
1.52
0.291
0.279
0.262
0.78 0.75 0.65
4
3.37
0.646
0.618
0.581
2.9
2.6
2.3
5
2.09
0.400
0.383
0.360
2
1.8
1.6
6
2.82
0.540
0.517
0.486
5.3
3.2
2.8
7
1.59
0.305
0.292
0.274
0.81
0.8
0.7
8
1.53
0.293
0.281
0.264
0.71
0.7
0.65
9
1.58
0.303
0.290
0.272
0.82
0.8
0.7
10
2.15
0.412
0.394
0.371
1.6
1.4
1.3
11
2.02
0.387
0.371
0.348
1.5
1.25 1.11
12
2.94
0.563
0.539
0.507
4.1
3.5
3
13
2.81
0.538
0.516
0.484
1.8
1.7
1.5
14
4.24
0.812
0.778
0.731
10
8
6
15
2.33
0.446
0.428
0.402
2.2
2
1.8
16
1.98
0.379
0.363
0.341
1.9
1.7
1.5
1.55
1.4
It can be observed that the h* values generally appear to converge as
approaches
W/m2K. The exceptions are sample#6, sample#12, and sample#14,
which have high conductivity compared to the rest. Figure 6.3 compares the h*
values obtained for each sample using different dispersion factors.
72
h* x 10-5 W/m2K
12
10
Using disperson factor of
5.22
8
Using dispersion factor of
5.45
6
Using dispersion factor of
5.8
4
2
0
0
1
2
3
4
5
6
7
8
9
10 11 12 13 14 15 16
Sample #
Figure 6.3: Comparison of h* values obtained using three different dispersion
factors.
It can be seen from Figure 6.3 that the results with different dispersion
factors are mostly consistent except for the sample #14, which had the highest
thermal conductivity in the experimental measurements. Since there was only a
single sample available for each type of composite, it is difficult to conclude whether
this sample is an outlier, or was simply a very superior sample with very high
dispersion and interface conductivity coefficient due to its processing method.
Therefore, a comparative study was undertaken to evaluate the results using the
theoretical results of (Nan et al., 1997).
73
6.4 Analysis of Results Based On Fiber Characteristics
The samples that were analyzed can be classified according to their fiber
volume fraction, the fiber manufacturer and the fiber diameter. The samples were
grouped according to this classification to examine the correlation that may exist
between these characteristics and the interface heat transfer coefficient. Table 6.6
and Figure 6.4 show the results of grouping by fiber volume fraction.
Table 6.6: Results grouped on the basis of volume fraction
Sample
#
Volume
Fraction
%
52.37
55-60
60-65
>65
Dispersion
factor
Experimental
Conductivity
5.22
5.45
5.8
5.22
Conductivity w/o dispersion
5.45
5.8
h* x10-5 W/m2K
W/mK
15
2.33
0.446
0.428
0.402
2.2
2
1.8
1
1.16
0.222
0.213
0.2
0.48
0.45
0.4
3
1.52
0.291
0.279
0.262
0.78
0.75
0.65
8
1.53
0.293
0.281
0.264
0.71
0.7
0.65
10
2.15
0.412
0.394
0.371
1.6
1.4
1.3
11
2.02
0.387
0.371
0.348
1.5
1.25
1.11
16
1.98
0.379
0.363
0.341
1.9
1.7
1.5
7
1.59
0.305
0.292
0.274
0.81
0.8
0.7
9
1.58
0.303
0.29
0.272
0.82
0.8
0.7
14
4.24
0.812
0.778
0.731
10
8
6
2
2.57
0.492
0.472
0.443
1.7
1.55
1.4
4
3.37
0.646
0.618
0.581
2.9
2.6
2.3
5
2.09
0.4
0.383
0.36
2
1.8
1.6
6
2.82
0.54
0.517
0.486
5.3
3.2
2.8
12
2.94
0.563
0.539
0.507
4.1
3.5
3
13
2.81
0.538
0.516
0.484
1.8
1.7
1.5
74
Results grouped by Volume fraction
10
8
6
-
h* x 10 5 W/m2K
12
4
2
0
15 1
3
8 10 11 16 7
9 14 2
4
5
6 12 13
Sample #
Figure 6.4: h* results obtained by dispersion factor of 5.22, grouped according to
volume fraction
It can be seen that the lower fiber volume fraction samples tend to have
lower values of the interface heat transfer coefficient. This may be due to the fact
that the fibers are not packed as tightly and the interface contact is consequently not
as strong as the high fiber volume fraction samples.
The results grouped by fiber manufacturers are shown in Tables 6.7, and
Figure 6.5. It can be observed that the Mitsubishi fibers tended to have lower heat
transfer coefficients at the interface compared to fibers made by Nippon. The
exception is Sample #14.
75
Table 6.7: Results grouped on the basis of fiber manufacturer
Sample
#
Fiber
make
CN
(Type)
Manufacturer
Nippon
YS/YSH
Manufacturer
Nippon
K
Manufacturer
Mitsubishi
Dispersion
factor
5.22
Experimental
1
2
3
4
12
13
5
6
16
7
8
9
10
11
14
15
1.16
2.57
1.52
3.37
2.94
2.81
2.09
2.82
1.98
1.59
1.53
1.58
2.15
2.02
4.24
2.33
5.45
5.8
Conductivity w/o dispersion
W/mK
0.222
0.492
0.291
0.646
0.563
0.538
0.4
0.54
0.379
0.305
0.293
0.303
0.412
0.387
0.812
0.446
0.213
0.472
0.279
0.618
0.539
0.516
0.383
0.517
0.363
0.292
0.281
0.29
0.394
0.371
0.778
0.428
0.2
0.443
0.262
0.581
0.507
0.484
0.36
0.486
0.341
0.274
0.264
0.272
0.371
0.348
0.731
0.402
5.22 5.45
5.8
h* x10-5 W/m2K
0.48
1.7
0.78
2.9
4.1
1.8
2
5.3
1.9
0.81
0.71
0.82
1.6
1.5
10
2.2
0.45 0.4
1.55 1.4
0.75 0.65
2.6
2.3
3.5
3
1.7
1.5
1.8
1.6
3.2
2.8
1.7
1.5
0.8
0.7
0.7 0.65
0.8
0.7
1.4
1.3
1.25 1.11
8
6
2
1.8
Results grouped by Fiber manufacturer
-
h* x 10 5 W/m2K
12
10
8
6
4
2
0
1
2
3
4 12 13 5
6 16 7
8
9 10 11 14 15
Sample #
Figure 6.5: h* results obtained by dispersion factor of 5.22, grouped according to
fiber manufacturer
76
The results grouped by fiber diameter are shown in Table 6.8, and Figure 6.6.
From these results, it appears that the fiber diameter does not seem to be a factor in
the interface thermal property.
Table 6.8: Results grouped based on fiber diameter
Sample
#
Fiber
diameter
µm
10
11
7
Dispersion
factor
Experimental
conductivity
5.22
5.45
5.8
Conductivity w/o dispersion
W/mK
5.22
5.45
5.8
h* x10-5 W/m2K
1
1.16
0.222
0.213
0.2
0.48
0.45
0.4
2
2.57
0.492
0.472
0.443
1.7
1.55
1.4
3
1.52
0.291
0.279
0.262
0.78
0.75
0.65
4
3.37
0.646
0.618
0.581
2.9
2.6
2.3
7
1.59
0.305
0.292
0.274
0.81
0.8
0.7
9
1.58
0.303
0.29
0.272
0.82
0.8
0.7
10
2.15
0.412
0.394
0.371
1.6
1.4
1.3
12
2.94
0.563
0.539
0.507
4.1
3.5
3
13
2.81
0.538
0.516
0.484
1.8
1.7
1.5
14
4.24
0.812
0.778
0.731
10
8
6
8
1.53
0.293
0.281
0.264
0.71
0.7
0.65
11
2.02
0.387
0.371
0.348
1.5
1.25
1.11
15
2.33
0.446
0.428
0.402
2.2
2
1.8
5
2.09
0.4
0.383
0.36
2
1.8
1.6
6
2.82
0.54
0.517
0.486
5.3
3.2
2.8
16
1.98
0.379
0.363
0.341
1.9
1.7
1.5
77
Results grouped by Fiber diameter
12
h* x 105 W/m2K
10
8
6
4
2
0
1
2
3
4
7
9 10 12 13 14 8 11 15 5
6 16
Sample #
Figure 6.6: h* results obtained by dispersion factor of 5.22, grouped according to
fiber diameter
6.5 Theoretical Calculations
An equation for calculating the overall thermal conductivity of a composite
was derived that accounts for interfacial thermal resistance (Nan et al., 1997) as
discussed in chapter 3. This equation, repeated below, was used to calculate the
theoretical interfacial heat transfer coefficient ( ) from experimental thermal
conductivities (corrected for dispersion). The values of
obtained from these
calculations were compared with values obtained from simulations. The composite
thermal conductivity formula was discussed in chapter 3, and is repeated below:
(
)
(
)
(
)
(
)
3.3
78
Where the parameters are simplified due to the high aspect ratio of the fiber
as follows:
is the fiber conductivity
is the matrix conductivity
is the fiber volume fraction
is the theoretical interfacial heat transfer coefficient
is the fiber radius
Equation 3.3 was re-written to calculate the interface heat transfer coefficient as
follows:
(
) (
)
(
) (
)
The above expression can be used to find the
3.4
value from the experimental
value after correcting for dispersion. However, the theoretical expression was
derived under the assumption that the fibers have an isotropic conductivity in the
transverse direction, unlike the anisotropic thermal conductivity of the pitch carbon
fibers used in the composite samples in this project. The results from the numerical
model are based on a model that includes the anisotropic thermal conductivity of
the fibers and the radial arrangement of the graphene planes.
Therefore, the direct results of
obtained from the theoretical calculations
cannot be compared with simulation results unless an equivalence is established
between fibers with radial arrangement and isotropic fibers. This makes it
necessary to find an equivalent isotropic conductivity value that will result in the
79
same transverse conductivity when the radial arrangement of graphene is used in
the model. This was done by carrying out model simulations as follows.
The transverse conductivity was calculated using a 5 layer model of a typical
composite, as described in Chapter 4. As an example, a fiber with radial graphene
arrangement was modeled with a diameter of 10 µm and fiber volume fraction of
65%. The thermal conductivity is then changed to isotropic and the simulation is
carried out again. By multiple trials, the equivalent isotropic conductivity, i.e. the
isotropic conductivity which gives same results as the radial graphene arrangement,
is obtained. This isotropic conductivity is then used in the theoretical calculations.
By repeated simulations, it was determined that the equivalent transverse
conductivity is equal to a radial graphene arrangement when the isotropic
conductivity is approximately 12.5% of the radial conductivity in the anisotropic
model. It was observed that, the ratio of the equivalent isotropic conductivity to the
radial conductivity remains constant for any fiber. Therefore, this ratio was used as
to find the transverse conductivity which is then used to determine the interface
coefficient
in the theoretical calculation. These values are then used in equation
3.3 to calculate theoretical interface heat transfer coefficient ( ) values. The results
are compared with simulation results, shown in Table 6.9.
80
Table 6.9: Values of h* and ht* obtained using the three dispersion factors
106
hi,max 
Dispersion
factor 
Interfacial heat transfer coefficient
8x105
6x105
5.22
5.45
5.8
Sample #
Simulation
h*x10-5
W/m2K
Theory
ht*x10-5
W/m2K
Simulation
h*x10-5
W/m2K
Theory
ht*x10-5
W/m2K
Simulation
h*x10-5
W/m2K
Theory
ht*x10-5
W/m2K
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
0.48
1.7
0.78
2.9
2
5.3
0.81
0.71
0.82
1.6
1.5
4.1
1.8
10
2.2
1.9
0.49
1.73
0.79
3.07
1.70
3.30
0.82
0.72
0.82
1.58
1.32
2.08
1.87
18.69
2.32
1.85
0.45
1.55
0.75
2.6
1.8
3.2
0.8
0.7
0.8
1.4
1.25
3.5
1.7
8
2
1.7
0.45
1.60
0.73
2.77
1.58
2.99
0.76
0.67
0.76
1.44
1.20
1.92
1.73
12.30
2.02
1.69
0.4
1.4
0.65
2.3
1.6
2.8
0.7
0.65
0.7
1.3
1.11
3
1.5
6
1.8
1.5
0.40
1.42
0.65
2.39
1.42
2.61
0.68
0.60
0.68
1.25
1.05
1.70
1.55
7.96
1.67
0.40
It can be seen from Table 6.9 that the interface heat transfer coefficient does
not change very much for most of the samples for the three cases considered. It is
also observed that most of the samples have the interface heat transfer coefficient in
the neighborhood of approximately 106 W/m2K.
Figure 6.7 (a)-(c) show plots comparing h* values obtained from theoretical
calculations and simulations using a particular dispersion factor. The match is poor
for Samples #6, 12 and 14, but the rest of the samples show consistent results.
81
h* results with dispersion factor at hi,max=106 W/m2K
h* x 10-5 W/m2K
20
18
Simulation result
16
Theoretical result
14
12
10
8
6
4
2
0
0
1
2
3
4
5
6
7
8
9 10 11 12 13 14 15 16
Sample #
Figure 6.7(a): Comparison of h* obtained from theoretical calculations and
simulations at dispersion factor based on hi,max=106 W/m2K.
h* results with dispersion factor at hi,max= 8x105 W/m2K
14
Simulation result
Theoretical result
10
6
h* x
8
10-5
W/m2K
12
4
2
0
0
1
2
3
4
5
6
7
8
9 10 11 12 13 14 15 16
Sample #
Figure 6.7(b): Comparison of h* obtained from theoretical calculations and
simulations at dispersion factor based on hi,max=8x105 W/m2K.
82
h* results with dispersion factor at hi,max=6x105 W/m2K
9
Simulation result
8
Theoretical result
h* x 10-5 W/m2K
7
6
5
4
3
2
1
0
0
1
2
3
4
5
6
7
8
9
10 11 12 13 14 15 16
Sample #
Figure 6.7(c): Comparison of h* obtained from theoretical calculations and
simulations at dispersion factor based on hi,max=6x105 W/m2K.
6.6 Discussion of h* Results
From the results in Table 6.5 and Figures 6.7(a)-(c), it can be concluded that
the majority of the numerical model results are consistent with the values from the
theoretical analysis. The two approaches did not agree in some of the samples
which had higher thermal conductivity. It is difficult to pinpoint the causes of this
discrepancy, since results from multiple samples of the same composition were not
tested to determine if some of the data points were outliers.
Since the results are based on an assumed maximum (hi,max=1x106 W/m2K),
the results of this study may not provide accurate absolute results for the interface
heat transfer coefficient; but these results can be useful in comparing the interface
83
conductivities of the samples. However, it does appear that the analysis provides
reasonable estimates of the interface heat transfer coefficients for such composite
samples. All samples showed the value of this coefficient to be in the range of 105 to
106W/m2K, which is consistent with the results of (Macedo & Ferreira, 2003). The
interface heat transfer coefficient for most of the samples is seen to be quite similar
by numerical simulation and by the theoretical analysis.
84
CHAPTER 7:
SUMMARY AND CONCLUSIONS
A numerical model for pitch fiber composites having strong radial fiber
conductivities was created in the FLUENT solver. The model was based on a unit cell
composed of a single fiber surrounded by the matrix. This model was then used to
analyze 16 samples of composite laminates. The approximate values of matrix-fiber
interfacial resistance were predicted based on experimental data and simulation
results.
The initial simulation results showed that the laminates were strongly
affected by the dispersion of the fibers from the ideal location in the unit cell. As a
result of this, the experimental results were much higher than the simulations.
Therefore, a dispersion factor was introduced to account for the dispersion of fibers.
The adjusted values were then used to find the interface heat transfer coefficients,
which were then compared with theoretical calculations from a prior study. The
comparison showed good agreement between the two sets of results. The maximum
interface heat transfer coefficient h* was determined to be in the range between 105
& 106 W/m2K. The value tended to be lower at lower fiber volume fractions and for
Mitsubishi fibers when compared with Nippon fibers. The variation in fiber
diameter did not show a consistent trend in the interface heat transfer coefficient.
The dispersion factor introduced in the numerical model was based on the
maximum interface heat transfer coefficient (hi,max) results of a previous study.
However, three different dispersion factors were used and the values of interfacial
heat transfer coefficient showed convergence as the hi,max limit was raised.
85
The numerical model also demonstrated that the interface heat transfer
coefficient is not the dominant factor in the heat transfer process within the
composite. Much of the resistance to heat transfer comes from the polymer matrix;
therefore, increasing the heat transfer coefficient beyond the hi,max=1x106 W/m2K
does not improve the conductivity significantly. A polymer matrix can reduce the
thermal conductivity by about a factor of ten.
Therefore, to improve the
conductivity of the composite, the improvement in the interface heat transfer
coefficient should be accompanied by improvement in the conductivity of the
matrix.
The third factor that controls the composite conductivity is the dispersion of
fibers from the ideal geometric layout, where each fiber location varies along the
direction of the fiber axis and fibers touch each other randomly along their length.
The effect of the deviation of the fiber location from the ideal geometry has a very
strong effect on the thermal conductivity of the composite. In this study, this effect
resulted in the conductivity changing by a factor of five, which appears to be higher
than the effect of the interface thermal conductivity. Therefore, it is recommended
that, in future work, additional experiments be carried out on samples with known
interfacial properties, and the conductivities be compared with simulation results to
determine the exact value of the dispersion factor.
If the exact value of the dispersion factor is known, this model can be used to
predict either the composite sample conductivity when the interface heat transfer
coefficient hi is known or the value of hi when the composite conductivity is known.
86
An analytical or numerical approach to the study of the dispersion factor is to add
fiber dispersion model in a two or three dimensional geometry.
87
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