Thermal and Mechanical Analysis of Carbon Foam
A dissertation presented to
the faculty of
the Russ College of Engineering and Technology of Ohio University
In partial fulfillment
of the requirements for the degree
Doctor of Philosophy
Mihnea S. Anghelescu
March 2009
© 2009 Mihnea S. Anghelescu. All Rights Reserved.
2
This dissertation titled
Thermal and Mechanical Analysis of Carbon Foam
by
MIHNEA S. ANGHELESCU
has been approved for
the Department of Mechanical Engineering
and the Russ College of Engineering and Technology by
M. Khairul Alam
Moss Professor of Mechanical Engineering
Dennis Irwin
Dean, Russ College of Engineering and Technology
3
ABSTRACT
ANGHELESCU, MIHNEA S., Ph.D., March 2009, Integrated Engineering
Thermal and Mechanical Analysis of Carbon Foam (122 pp.)
Director of Dissertation: M. Khairul Alam
Carbon foams are porous materials which are attractive for many
engineering applications because their thermal and mechanical properties can be
customized by varying manufacturing process parameters. However, a highly random
geometry at pore level makes it very difficult to analyze the properties and the behavior
of this material in an application. Published research work on the analysis of foams has
employed various ideal geometries to approximate the pore microstructure. However,
these models are unable to predict accurately the foam properties and behavior in
engineering applications.
The objective of this research work is to determine thermal and
mechanical properties of carbon foam on the basis of its true microstructure. A new
approach is proposed by creating a three dimensional (3D) solid model based on an
accurate representation of the real geometry of carbon foam. Finite element models are
then developed to investigate the bulk thermal and mechanical properties of carbon foam
using the three dimensional solid model.
On the basis of the true 3D model of carbon foam, a study is undertaken to
examine the effect of the unique microstructure on the flow field within the foam pores
and the resultant convective heat transfer. A finite volume model is developed using the
accurate representation of carbon foam microstructure inside a flow channel. The fluid
4
flow and heat transfer is simulated to evaluate pressure drop and heat transfer
capabilities. The carbon foam permeability, inertial coefficient and friction coefficient are
determined and found to be in good agreement with experimental and semi-empirical
models. The results also show a large enhancement in the heat transfer due to the
presence of carbon foam in the channel. These results are comparable to the experimental
results available in published literature.
Another application that has been analyzed in this study is the use of carbon foam
as tooling material for manufacturing advanced composite materials. Finite element
simulations are carried out to predict the process induced residual stresses and
deformations when a composite part is manufactured on conventional tooling versus
carbon foam tooling. The results show that both the lower coefficient of thermal
expansion and the elastic modulus of carbon foam contribute to the reduction of residual
stress and deformation of the composite part.
Approved: _____________________________________________________________
M. Khairul Alam
Moss Professor of Mechanical Engineering
5
ACKNOWLEDGMENTS
I would like to express my deepest gratitude to my academic advisor, Professor
Khairul Alam for his support and outstanding guidance during my PhD program. I am
extremely indebted to him for giving me the opportunity to work on such an interesting
research project.
I would like to thank Drs. Dusan Sormaz, Daniel Gulino, David Ingram and Liwei
Chen, for serving on my dissertation committee.
I would like to thank the Air Force Research Laboratory (AFRL, Dayton, OH) for
support and for providing the 3D rendering of the carbon foam microstructure. I am also
thankful to Drs. Adriana and Calin Druma for their helpful suggestions during the first
stage of this research project.
I would like to acknowledge the support provided by GrafTech International Ltd.,
and the allocation of computing time and the software products Hyperworks, Abaqus and
Fluent from the Ohio Supercomputer Center. I would also like to acknowledge the use of
the software product Geomagic Studio from Geomagic, Inc..
I am extremely grateful to my family for their continuous support and
encouragement in pursuing my career. I am especially thankful to my parents for
supporting and helping me during various stages of my life.
6
TABLE OF CONTENTS
Page
Abstract ............................................................................................................................... 3
Acknowledgments............................................................................................................... 5
List of Tables ...................................................................................................................... 9
List of Figures ................................................................................................................... 10
Chapter 1: INTRODUCTION TO CARBON FOAM ...................................................... 13
Chapter 2: SOLID MODELING OF CARBON FOAM MICROSTRUCTURE ............. 18
Chapter 3: THERMAL AND MECHANICAL CHARACTERIZATION OF CARBON
FOAM MICROSTRUCTURE.......................................................................................... 26
3.1 Thermal analysis ..................................................................................................... 26
3.1.1 Introduction and objective ............................................................................... 26
3.1.2 Methodology and simulations .......................................................................... 29
3.1.3 Results and conclusions ................................................................................... 32
3.2 Mechanical analysis ................................................................................................ 35
3.2.1 Introduction and objective ............................................................................... 35
3.2.2 Methodology and simulations .......................................................................... 36
3.2.3 Results and conclusions ................................................................................... 39
Chapter 4: FLUID FLOW AND CONVECTION HEAT TRANSFER IN CARBON
FOAM ............................................................................................................................... 43
4.1 Introduction ............................................................................................................. 43
4.2 Volume averaged model ......................................................................................... 45
7
4.2.1 Fluid flow model .............................................................................................. 46
4.2.2 Heat transfer model ......................................................................................... 49
4.3 Direct simulation of fluid flow and heat transfer .................................................... 52
4.3.1 Ideal pore geometry ......................................................................................... 52
4.3.2 Real pore geometry .......................................................................................... 53
4.3.3 Fluid flow computational model ...................................................................... 54
4.3.4 Convection heat transfer computational model ............................................... 56
4.4 Results and conclusions .......................................................................................... 57
4.4.1 Fluid flow analysis ........................................................................................... 57
4.4.2 Convection heat transfer analysis .................................................................... 65
Chapter 5: CARBON FOAM TOOLING FOR ADVANCED COMPOSITE
MANUFACTURING ....................................................................................................... 73
5.1 Introduction ............................................................................................................. 73
5.2 Thermo-chemical model ......................................................................................... 80
5.3 Stress-displacement model ..................................................................................... 85
5.4 Simulations and results ........................................................................................... 90
5.4.1 Results for curved tooling geometry (convex and concave) ............................ 94
5.4.2 Results for flat tooling geometry ...................................................................... 98
5.4.3 Effects of thermo-mechanical properties of tooling....................................... 100
5.5 Summary of analyses of carbon foam tooling ...................................................... 103
Chapter 6: CONCLUSIONS AND RECOMMENDATIONS FOR FUTURE WORK . 105
References ....................................................................................................................... 108
8
Appendix A: RELATIONS FOR THERMAL AND MECHANICAL PROPERTIES OF
COMPOSITE MATERIALS .......................................................................................... 116
Appendix B: MATHEMATICAL MODEL FOR TURBULENT FLUID FLOW AND
HEAT TRANSFER ........................................................................................................ 120
9
LIST OF TABLES
Page
Table 3.1: Non-dimensional bulk thermal conductivity of
90% porosity carbon foam .............................................................................34
Table 3.2: Non-dimensional bulk Young’s modulus of
90% porosity carbon foam .............................................................................41
Table 4.1: Permeability and inertial coefficient of various porous materials .................62
Table 4.2: Increase in effective heat transfer coefficient of porous channel
relative to clear channel ..................................................................................70
Table 5.1: Thermal properties of AS4/3501-6 prepreg and its components
(Lee et al., 1982; Loos & Springer 1983).......................................................84
Table 5.2: Thermal properties of tooling materials
(“GrafoamTM Carbon Foam Solutions”, n.d.; Wiersma et al., 1998) .............84
Table 5.3: Weight factors and relaxation times for 3501-6 epoxy resin
(Kim & White, 1996) .....................................................................................88
Table 5.4: Mechanical properties of AS4/3501-6 prepreg components
(Bogetti & Gillespie, 1992; Kim & White, 1996; White & Kim, 1998) ........88
Table 5.5: Mechanical properties of various tooling materials
(“GrafoamTM Carbon Foam Solutions”, n.d.; Wiersma et al., 1998) .............89
Table 5.6: Bent composite part spring-in angle ..............................................................93
Table 5.7: Flat composite part curvature ........................................................................94
10
LIST OF FIGURES
Page
Figure 1.1: SEM picture of graphitic carbon foam (Source: AFRL) ..............................14
Figure 2.1 SEM picture of (a) graphitic carbon foam (Source: AFRL) and
(b) Duocel® aluminum foam (Source: “Metal foam” n.d.) ..........................19
Figure 2.2: Various ideal geometry models used to approximate carbon foam
microstructure: (a) tetrahedron (Source: Sihn & Roy, 2004),
(b) tetrakaidecahedron (Source: Li et al., 2005), (c) centered cube (Source:
Yu et al., 2006), (d) BCC type cube (Source: Druma et al., 2004),
(e) BCC type ellipse, vertical and horizontal
(Source: Druma et al., 2004) .........................................................................20
Figure 2.3: 3D rendering of graphitic carbon foam by serial sectioning
technique (Source: AFRL) ............................................................................21
Figure 2.4: 3D solid model of (a) carbon foam and (b) carbon foam
saturated with fluid .......................................................................................23
Figure 3.1: Finite element discretization of 3D carbon foam microstructure .................31
Figure 3.2: (a) Temperature and (b) heat flux distributions in carbon foam
microstructure when applying the heat flux in the x-direction .....................33
Figure 3.3: (a) Displacement and (b) stress distributions in carbon foam
microstructure when applying the compressive load in the x-direction .......40
Figure 4.1: 3D solid model of porous channel................................................................45
Figure 4.2: Boundary conditions applied on the porous channel....................................54
Figure 4.3: Computational mesh for porous channel ......................................................58
Figure 4.4: Fluid flow pathlines in the porous channel colored by velocity
magnitude for a free stream velocity of 0.5 m/s ...........................................60
Figure 4.5: Pressure drop per unit length as a function of free stream velocity .............62
Figure 4.6: Friction coefficient as a function of modified Reynolds number .................65
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Figure 4.7: Fluid temperature distribution in (a) clear channel (no foam) and in
(b) porous channel for a free stream velocity of 0.5 m/s ..............................67
Figure 4.8: Effective heat transfer coefficient as a function of free stream
velocity and foam solid phase thermal conductivity.....................................69
Figure 4.9: Pumping power per unit volume of fluid as a function of
thermal resistance of the channel ..................................................................72
Figure 5.1: CTE of different tooling materials and carbon fiber (Burke, 2003;
Burden, 1989; “GrafoamTM Carbon Foam Solutions”, n.d.).........................75
Figure 5.2: Carbon foam tooling for a simple part
(Source: GrafTech International Ltd.) ..........................................................77
Figure 5.3: Cure cycle for AS4/3501-6 (Kim & Hahn, 1989) ........................................83
Figure 5.4: (a) 3D solid model and (b) 2D finite element discretization of the
composite part on convex and concave tooling ............................................91
Figure 5.5: The deflections and angles due to deformations from processing:
(a) spring-in for a 90 degree angle and (b) warpage of a flat
composite part. The values are determined after removing
the part from the tooling ...............................................................................92
Figure 5.6: Stress distribution along fiber direction at the end of cure cycle before
removing the composite part from convex tooling made of
(a) carbon foam and (b) steel ........................................................................95
Figure 5.7: Stress distribution along fiber direction at the end of cure cycle after
removing the composite part from convex tooling made of
(a) carbon foam and (b) steel. A deformation scale factor of 10 is
used for displacement ...................................................................................96
Figure 5.8: Stress distribution along fiber direction at the end of cure cycle before
removing the composite part from concave tooling made of
(a) carbon foam and (b) steel. .......................................................................97
Figure 5.9: Stress distribution along fiber direction at the end of cure cycle after
removing the composite part from concave tooling made of
(a) carbon foam and (b) steel. A deformation scale factor of 10 is
used for displacement ...................................................................................97
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Figure 5.10: Stress distribution along fiber direction at the end of cure cycle before
removing the composite part from flat tooling made of
(a) carbon foam and (b) steel ........................................................................99
Figure 5.11: Stress distribution along fiber direction at the end of cure cycle after
removing the composite part from flat tooling made of
(a) carbon foam and (b) steel. A deformation scale factor of 10 is
used for displacement ...................................................................................99
Figure 5.12: Curing process of the composite part on convex tooling made of
different materials. The stress distribution along fiber at point (A) is
shown for different tooling materials. The curves for the degree of
cure at points (A) and (B) overlap, even though the point (B) is
insulated by the carbon foam tooling ..........................................................101
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CHAPTER 1: INTRODUCTION TO CARBON FOAM
Carbon foams are rigid, porous materials consisting of an interconnected network
of ligaments. They can have open cell structure where pores are interconnected to one
another or closed cell structure where pores are isolated from one another. There are
several types of carbon foams, depending on the raw material (precursor) and the
manufacturing process employed to produce the foam. The major categories are:
reticulated vitreous carbon foam (RVC), graphitic carbon foam and non-graphitic carbon
foam.
In general, porous materials can be considered a subclass of cellular materials. By
definition, a cellular material is a mechanical structure made of interconnected struts or
plates which form the cells (Gibson & Ashby, 1997). All cellular materials have a certain
degree of porosity.
The first carbon foams were developed by W. Ford in the 1960s as reticulated
vitreous carbon foams by carbonizing thermosetting polymer foams (Gallego & Klett,
2003). In the 1990’s, scientists at Air Force Research Laboratory (AFRL, Dayton, OH)
developed graphitic carbon foams by blowing a melted mesophase pitch precursor. After
the foaming process, the carbon foam is usually stabilized at 170°C and a heat treatment
consisting of carbonization at 1000°C and graphitization at 2700°C is applied (Brow et
al., 2003). Thermal conductivity of graphitic carbon foams generally ranges between 1
W/m°C and 250 W/m°C depending on the microstructure, porosity and process
parameters (Druma, 2005). An SEM picture of a graphitic carbon foam produced by
14
AFRL is shown in Figure 1.1. The open cell structure due to the windows between
adjacent pores is seen in this figure. In this particular case the pores are spherical or
elliptical with diameter ranging between 100 and 350 μm.
Scientists at Oak Ridge National Laboratory developed an alternative process to
manufacture graphitic carbon foams and obtained bulk thermal conductivities as high as
180 W/m°C (Gallego & Klett, 2003). Potential utilizations of highly thermal conductive
graphitic carbon foams with open cell structure include thermal management applications
such as heat sink and heat exchanger cores.
Figure 1.1. SEM picture of graphitic carbon foam (Source: AFRL).
A method to manufacture non-graphitic carbon foams was developed by
researchers at West Virginia University (Chen et al., 2006). They used inexpensive
precursors such as coal, petroleum pitch, coal tar pitch and the result is a carbon foam
15
that can be either very strong mechanically or thermally conductive, depending on the
manufacturing process parameters. In addition, the foam is nearly isotropic and it can
have open cell structure or closed cell structure (Chen et al., 2006). Potential utilizations
of non-graphitic carbon foams include structural applications such as tooling for
composite materials manufacturing, stiffener inserts and core materials for composite
sandwich structures. Non-graphitic carbon foams are also attractive as thermal protection
materials, as they can be produced with very low thermal conductivity (Spradling &
Guth, 2003).
Overall, carbon foam is a very versatile material whose properties can be tailored
by controlling three major factors: the precursor, the foaming process and the heat
treatment conditions (Rowe et al., 2005). Moreover, carbon foams can have very low
density and coefficient of thermal expansion (CTE), and can be produced with a wide
range of porosities. These properties make carbon foams suitable for utilization in many
engineering applications. However, designing with this material requires a good
understanding of its thermal and mechanical properties, and the characteristics of fluid
flow and convection heat transfer through the foam. Because this foam was developed
within the last 10 – 15 years, the complex tridimensional (3D) geometry of its
microstructure and associated fluid flow phenomena have not been studied in detail.
Bulk thermal and mechanical properties of carbon foam, as well as its fluid flow
and convection heat transfer characteristics are strong functions of the pore shape,
dimensions and distribution in the solid matrix. They also depend upon the thermal and
mechanical properties of the solid phase.
16
The objective of this research work is to develop an understanding of the effect of
the microstructure and properties of carbon foam so that the feasibility of applications
such as thermal management, composite tooling, etc. can be evaluated. This will be done
by first modeling the properties of the carbon foam and then applying the models to the
specific applications. Therefore, the objectives in this research are to investigate thermal
and mechanical behavior of carbon foam by the finite element method (FEM). Based on
the model of the foam, the fluid flow phenomenon through the foam is then investigated
by finite volume method. Finally, the application of the foam as a tooling material for
manufacturing composite materials is studied.
Chapter 2 presents the development of the 3D solid model of carbon foam
microstructure and calculation of its geometry parameters (solid volume, porosity and
surface area). Chapter 3 presents the thermal and mechanical characterization of carbon
foam microstructure. Finite element simulations to evaluate bulk thermal conductivity
and bulk Young’s (elastic) modulus of carbon foam microstructure are carried out using
the 3D solid model developed in Chapter 2. The finite element results are compared with
experimental results from literature in order to assess their accuracy. Fluid flow and
convection heat transfer in carbon foam are investigated in Chapter 4. In this chapter,
finite volume simulations are used to evaluate pressure drop and effective heat transfer
capabilities when fluid flows through the carbon foam. Fluid flow results are compared
with experimental results based on various porous materials available in literature in
order to assess their accuracy. The convection heat transfer capabilities are evaluated by
comparing the effective heat transfer coefficient obtained in a porous channel with that
17
obtained in a clear channel (no carbon foam). The utilization of carbon foam as tooling
material for manufacturing carbon fiber reinforced epoxy composites is investigated in
Chapter 5. In this chapter, the process induced residual stresses and geometric
deformations of composite parts manufactured on traditional tooling materials and carbon
foam are compared in terms of warpage and spring-in.
A unique aspect of this research is the development of an accurate solid model
representation of 3D carbon foam microstructure, which is used to investigate the bulk
(effective) thermal and mechanical properties of carbon foam as well as the fluid flow
and heat transfer phenomena through the open pores of the foam.
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CHAPTER 2: SOLID MODELING OF CARBON FOAM MICROSTRUCTURE
One of the potential applications of carbon foam is its use in thermal
management, including convection heat transfer through its pores. However, because of
the manufacturing process, the microstructure of carbon foam is quite complex.
Therefore, the study of the relationship between its microstructure and bulk properties is
a difficult problem. Carbon foam obtains its microstructure through the processing
method in which a pitch precursor is heated to the melting point while being pressurized
in an inert atmosphere and then is blown by releasing the gas pressure (Druma, 2005;
Brow et al., 2003). The blowing process produces pores that are bubbles which have
grown either to form closed cell or open cell structure; in the latter case the walls between
the cells have openings that can allow a flow through the structure (see Figure 2.1(a)).
Metal foams used for heat exchanger, such as Duocel® produced by ERG Aerospace (see
Figure 2.1(b)) have a more open reticulated structure and the pore size and the pore
window are virtually identical. Reticulated structures are often modeled as a set of
ligaments with a geometric relationship (Calmidi, 1998).
19
Figure 2.1. SEM picture of (a) graphitic carbon foam (Source: AFRL) and (b)
Duocel® aluminum foam (Source: “Metal foam” n.d.).
Due to its complexity and randomness in the pore shape, dimensions and
distribution in the solid matrix, it is difficult to obtain an accurate representation of 3D
microstructure of carbon foam in solid modeling software. This is the reason why the
carbon foam microstructure is often approximated by idealized geometry. Sihn and Roy
(2004) approximated the carbon foam with a unit cell obtained by subtracting four
identical spheres from a regular tetrahedron (see Figure 2.2(a)). The spheres are located
at the corners of the tetrahedron. By varying the diameter of the spheres, the porosity of
the unit cell is varied. Li et al. (2005) extended this concept and proposed a
tetrakaidecahedral unit cell for carbon foam (see Figure 2.2(b)). Yu et al. (2006) proposed
the unit cell obtained by subtracting a sphere from a cube for the representative
elementary volume of carbon foam (see Figure 2.2(c)). The sphere is located at the center
of the cube. Druma et al. (2004) proposed a body centered cubic (BCC) type structure for
the unit cell of carbon foam where nine spheres of equal volumes are subtracted from a
cube (see Figure 2.2(d)). The spheres are located at the corners and the center of the cube.
They also used ellipses (horizontal and vertical) to create the pores (see Figure 2.2(e)).
20
(a)
(b)
(c)
(d)
(e)
Figure 2.2. Various ideal geometry models used to approximate carbon foam
microstructure: (a) tetrahedron (Source: Sihn & Roy, 2004), (b) tetrakaidecahedron
(Source: Li et al., 2005), (c) centered cube (Source: Yu et al., 2006), (d) BCC type
cube (Source: Druma et al., 2004), (e) BCC type ellipse, vertical and horizontal
(Source: Druma et al., 2004).
21
In this research work, an accurate representation of the 3D microstructure of
carbon foam will be used. The foam was manufactured at AFRL and it had a bulk density
of 0.24 g/cc (Alam & Maruyama, 2004) which correspond to approximately 89%
porosity. The digital image of the foam was obtained by automated serial sectioning
technique with Robo-Met.3D, a process presented in detail by Maruyama et al. (2006)
and briefly described as follows: (i) the pore space in the carbon foam sample is filled
with epoxy resin to give structural support and serial layers of approximately 3.5μm
thickness are removed; (ii) a digital image of the cross-section of the carbon foam is
taken after each cut; (iii) the collection of digital images obtained is assembled using a
custom software. The result is a file containing a point cloud that resembles the geometry
of the carbon foam, as shown in Figure 2.3. The AFRL supplied the 3D rendering of the
carbon foam structure for this research work.
Figure 2.3. 3D rendering of graphitic carbon foam by serial sectioning technique
(Source: AFRL).
22
The computational effort required to set up and run a model that reflects accurate
details at the pore level of the microstructure requires large computing resources. The
ideal model should be exactly representative of the bulk material, i.e. it should be large
enough to have the same properties (density, microstructure, thermal and mechanical
properties, etc.) as the bulk material. Since the thermal and mechanical properties of the
bulk foam material itself tend to be non-homogeneous and anisotropic, it is quite difficult
to select a reasonably small volume of the foam that can reflect all of the bulk properties.
To keep the computational effort feasible and within the constraints of the computational
resources, only a quarter of the 3D rendering of carbon foam shown in Figure 2.3 was
converted to a solid model.
Surface reconstruction using reverse engineering software (Geomagic Studio)
starts from the point cloud representation of the carbon foam (see Figure 2.3). The final
result is a file containing a closed surface that resembles the geometry of the carbon foam
microstructure. The file is imported into a solid modeling software and the 3D solid
model of carbon foam microstructure is obtained, as shown in Figure 2.4(a). It is
estimated that the model contains approximately 20 to 40 pores. The faces of the carbon
foam solid model are trimmed to obtain flat surfaces. For the convection model with fluid
flow, a boolean operation is carried out to subtract the shape of the foam from a
parallelepiped representing the fluid. The foam is subsequently fitted into this empty
space and aligned to the sides. The assembly consisting of carbon foam saturated with the
fluid is shown in Figure 2.4(b).
23
(a)
(b)
Figure 2.4. 3D solid model of (a) carbon foam and (b) carbon foam saturated with
fluid.
24
The geometric parameters of the carbon foam can be calculated using the 3D solid
model representation of the microstructure. The dimensions of the carbon foam
microstructure bounding parallelepiped in Figure 2.4(a) are 1.502 x 1.482 x 1.540 mm 3 .
The accuracy of this measurement is determined by the scanning process in which 3.5 μm
slices were used to build the digital point cloud representation. Therefore, the accuracy of
the length scale in the solid model is of the order of 1 μm. The calculations in the model
will have this scale of accuracy; however experimental results are expected to have much
lower accuracy in the length scale.
The porosity (void content) of a porous material is defined as
φ (% ) =
Vf
V
⋅ 100 =
Vf
V f + Vs
⋅ 100
(2.1)
where V f is the volume of the pores and Vs is the volume of the solid phase. The
porosity of the carbon foam calculated using the solid model in Figure 2.4(a) is 90.31%.
This is very close to the experimental porosity, which was measured to be approximately
89%. Therefore this sample was considered to be representative of the foam.
Experimental determination of porosity can be done by using the following relation,
which can be derived from Equation (2.1):
⎛
φ (% ) = ⎜⎜1 −
⎝
ρb ⎞
⎟ ⋅ 100
ρ s ⎟⎠
(2.2)
where ρ b is the bulk (effective) density and ρ s is the solid phase density (Druma, 2005).
The surface area per unit volume of a porous material is defined as
25
a sf =
Asf
V
=
Asf
V f + Vs
(2.3)
where Asf is the total pore surface area. The surface area per unit volume of the carbon
foam calculated using the solid model in Figure 2.4(a) is 4339.5 m 2 m 3 . Since the
length scale accuracy is of the order of 1 μm the surface area calculation has high
accuracy for a given solid model that is used in the calculation. However it should be
recognized that there are inaccuracies that are inherent in the transformation of the digital
cloud representation to the surfaces in the solid model. It should be mentioned that
experimental determination of surface area per unit volume of a porous material is a
complex measurement with significant errors.
26
CHAPTER 3: THERMAL AND MECHANICAL CHARACTERIZATION OF
CARBON FOAM MICROSTRUCTURE
3.1 Thermal analysis
3.1.1 Introduction and objective
A significant amount of research on thermal analysis of porous materials has been
done using highly porous aluminum foams. Calmidi and Mahajan (1999) developed a
correlation for bulk thermal conductivity of aluminum foam saturated with air and water
using an analytical model validated with experimental results. The foam was
approximated with a 2D periodic hexagonal structure. Boomsma and Poulikakos (2001)
extended this procedure and developed a bulk thermal conductivity model for aluminum
foam saturated with fluid by approximating the aluminum foam with a 3D structure, the
tetrakaidecahedron.
Thermal conductivity analysis of idealized structure of carbon foam was carried
out by Druma et al. (2004). They developed finite element models and calculated bulk
thermal conductivity of carbon foam as a function of porosity, however they did not
account for the effect of convection and radiation within the pores. The pore shapes were
taken to be spherical and elliptical (horizontal and vertical orientation).
Yu et al. (2006) developed a representative elementary volume model for heat
transfer and fluid flow in carbon foam. An analytical correlation for the bulk (stagnant)
thermal conductivity of carbon foam saturated with fluid is developed by taking the
27
porous structure to be a cube with a spherical centered pore and equating it to a square
bar structure with the same porosity.
Klett et al. (2004) proposed the following correlation for bulk thermal
conductivity of graphitic carbon foam by fitting a curve through a set of experimental
data points:
⎛ρ
k e = 0.734⎜⎜ b
⎝ ρs
⎞
⎟⎟
⎠
1.427
ks
(3.1)
where ρ b is the bulk density of carbon foam, ρ s is the density of the solid phase of the
carbon foam and k s is the thermal conductivity of the solid phase of the carbon foam.
The thermal conductivity of the solid phase depends on the degree of graphitization and
ranges from ~1 to 2000 W/m°C (Alam & Maruyama, 2004). As mentioned earlier, the
bulk thermal conductivity of graphitic carbon foams generally ranges between 1 W/m°C
and 250 W/m°C depending on the microstructure, porosity and process parameters
(Druma, 2005).
Bauer (1993) developed an analytical model for thermal conductivity of porous
media by applying a perturbation to a uniform temperature gradient. The perturbation is
given by pores distributed in a homogeneous and isotropic material. For the particular
case of spherical pores and thermal conductivity of gas in the pores much smaller than
that of solid phase, the bulk thermal conductivity of the porous material is:
φ ⎞
⎛
k e = ⎜1 −
⎟
⎝ 100 ⎠
1.295
ks
(3.2)
28
Alam and Maruyama (2004) tested the bulk thermal conductivity of carbon foam
samples made by various manufacturers using laser flash and hot plate techniques. Their
results showed a wide range of thermal conductivity of foams made by different
manufacturers. The differences were due to differences in starting materials and foaming
processes. It is known that graphitic foams made by a foaming process are thermally
anisotropic and this was confirmed by bulk thermal conductivity measurements (Alam &
Maruyama, 2004).
The objective of the thermal analysis in the present work is to determine the bulk
thermal conductivity of the carbon foam microstructure. The 3D solid model of carbon
foam microstructure developed in Chapter 2 (see Figure 2.4(a)) is used to carry out the
analysis. A study by Druma et al. (2005) investigated the bulk thermal conductivity of
carbon foam microstructure using a smaller solid model. In the current analysis, the solid
phase of the carbon foam is assumed homogeneous and isotropic. It is important to note
that this is a simplification of the micro-level properties of graphitic carbon foam. In
graphitic low density foams the cell walls can be highly anisotropic along the ligaments
because they are composed of aligned graphene planes (Druma, 2005). These ligaments
meet at thicker nodes where the structure is much less graphitic (Druma, 2005). In the
current analysis, the ligaments and nodes will be assumed to be isotropic and
homogeneous. Most of the studies on the modeling of carbon foam have used this
assumption because of the computational complexity of a model that incorporates the
variation of properties along the ligaments and nodes.
29
3.1.2 Methodology and simulations
The carbon foam microstructure subjected to a thermal load is analyzed by finite
element method. The results of the analysis allow for calculation of bulk thermal
conductivity of carbon foam microstructure. The 3D solid model of carbon foam is
sandwiched between two solid plates which will be used for applying the boundary
conditions. A perfect thermal contact (zero thermal contact resistance) is assumed
between plates and carbon foam. The pores are assumed empty and convection and
radiation heat transfer are not considered in this analysis. Constant thermal conductivity
of solid phase of the carbon foam and solid plates is assumed throughout the analysis.
The temperature field in the system (carbon foam microstructure and plates) is governed
by the steady state conduction heat transfer equation without internal heat generation
∂
∂xi
⎛ ∂T
⎜⎜ k
⎝ ∂xi
⎞
⎟⎟ = 0
⎠
i = 1, 2, 3
(3.3)
where T is the temperature and k is the thermal conductivity of the domain analyzed.
The following boundary conditions are applied for thermal analysis:
- a uniform heat flux on the top surface of the upper plate
- a uniform temperature on the bottom surface of lower plate
- temperature and heat flux continuity at the plate – carbon foam
microstructure interface
- all of the other surfaces are thermally insulated
The thermal conductivity of the two solid plates is chosen to be three orders of
magnitude higher than that of the solid phase of carbon foam in order to ensure constant
temperature boundary conditions on the carbon foam top and bottom surfaces.
30
The assembly consisting of the 3D solid model of carbon foam microstructure and
the two solid plates was meshed using 10-node quadratic tetrahedral elements. The
general purpose finite element code ABAQUS is used to solve Equation (3.3) using
Galerkin weighted residual method (Abaqus Inc., 2006). The solution of finite element
analysis yields the temperature and heat flux distributions in the carbon foam
microstructure and solid plates. Mesh independence of the finite element solution was
established by solving the problem using two different mesh densities: 519,397 elements
(see Figure 3.1) and 1,357,568 elements. The difference between the two meshes in terms
of the bulk thermal conductivity is less than 0.1%. The results presented in the next
section are obtained using the coarser mesh.
31
Figure 3.1. Finite element discretization of 3D carbon foam microstructure.
The bulk thermal conductivity of carbon foam can be calculated by applying
Fourier’s law for 1D conduction heat transfer:
q x = −k e
dT
ΔT
= ke
dx
Δx
(3.4)
where qx is the heat flux applied on the top surface of the upper plate, ΔT is the
temperature difference between the two solid plates and Δx is the thickness of the carbon
foam in the heat flux direction.
32
The non-dimensional bulk thermal conductivity of carbon foam is calculated as
follows
k eff =
ke
ks
(3.5)
to facilitate comparison with results reported in literature.
Independent analyses are carried out in the x- y- and z- directions of the 3D solid
model of carbon foam to account for randomness in pore shape, dimensions and
distribution in the solid matrix. The thermal conductivity of the solid phase has been
varied to study its influence on the bulk thermal conductivity of the microstructure.
3.1.3 Results and conclusions
Figure 3.2(a) and (b) show the temperature and heat flux distributions in carbon
foam microstructure when the heat flux is applied in the x- direction. It can be noticed
that the temperature gradient is predominantly one dimensional in the heat flux direction.
The heat flux distribution shows higher values in the thin ligaments of the carbon foam
microstructure. Similar temperature and heat flux distributions are obtained when
applying the heat flux in the y- and z- directions.
33
(a)
(b)
Figure 3.2. (a) Temperature and (b) heat flux distributions in carbon foam
microstructure when applying the heat flux in the x-direction.
34
The thermal analysis results are summarized in Table 3.1 for comparison with
similar results reported in literature. The bulk thermal conductivity values obtained by
FEM analysis on real geometry of carbon foam microstructure are in very good
agreement with the experimental results in Klett et al. (2004). It should be noted that
these experimental values were obtained for a graphitic foam which is similar but not
identical to the foam modeled in this study. The analytical solution proposed by Bauer
(1993) and FEM analysis on ideal geometry developed by Druma et al. (2004)
overpredict the experimental value in Klett et al. (2004) except for the case of ellipsoidal
pores perpendicular to the heat flux.
Table 3.1
Non-dimensional bulk thermal conductivity of 90% porosity carbon foam
Type of
k eff (% of solid phase)
Source
analysis
Klett et al. (2004)
Experimental
2.6
(Equation 3.1)
6.25 (spherical pores)
FEM on
2.2 (ellipsoidal pores – perpendicular to heat flux)
Druma et al. (2004)
ideal
9.2 (ellipsoidal pores – aligned with heat flux)
geometry
Bauer (1993)
Analytical
4.9 (spherical pores)
(Equation 3.2)
2.78 (x-direction)
FEM on real
2.93 (y-direction)
2.72 (average)
Present study
geometry
2.44 (z-direction)
It can be concluded that, for the same porosity, the bulk thermal conductivity is a
strong function of the particular geometry of the microstructure. Ideal geometries used by
researchers do not seem to have the capability of predicting the bulk thermal conductivity
35
accurately. The lower value of the non-dimensional thermal conductivity in the true
geometry is probably due to the higher degree of tortuosity caused by the randomness in
pore shape, dimensions and distribution in the solid matrix.
The differences between the bulk thermal conductivity values obtained when
analyzing the foam in the three directions (see Table 3.1) indicate that the particular solid
model of carbon foam microstructure analyzed is not thermally isotropic. This raises the
question whether the model volume is representative of the bulk foam. It is well known
that most carbon foams do not exhibit bulk isotropy in thermal conductivity (Alam &
Maruyama, 2004); however the origin of the anisotropy could be a combination of the
anisotropy of the geometry and the anisotropy of the thermal properties in the ligaments
and nodes. Therefore, it is quite difficult to select a reasonably small volume of the foam
that can reflect all of the bulk properties. The FEM analysis also exhibited a linear
relationship between the bulk thermal conductivity and the thermal conductivity of the
solid phase of the carbon foam.
3.2 Mechanical analysis
3.2.1 Introduction and objective
Stress-strain analysis of idealized structure of carbon foam was carried out by
Sihn and Roy (2004). They developed a finite element model and calculated bulk
Young’s modulus and Poisson’s ratio of carbon foam as a function of porosity. The
carbon foam unit cell was approximated with tetrahedral type geometry. Li et al. (2005)
approximated the carbon foam unit cell with a tetrakaidecahedron and calculated bulk
36
Young’s modulus, Poisson’s ratio and shear modulus as a function of porosity for various
cross-sectional shapes of the cell struts.
Sihn and Rice (2003) tested the bulk Young’s modulus of carbon foam samples
made by various manufacturers under compressive loading. For a 90% porosity foam
they reported a bulk value that ranged from 0.4% to 1% of the intrinsic solid phase
modulus.
The objective of the mechanical analysis in the present work is to determine the
bulk Young’s modulus of carbon foam microstructure. The analysis is carried out by
using the solid model of carbon foam microstructure developed in Chapter 2 (see Figure
2.4(a)). A study by Druma et al. (2005) investigated the bulk Young’s modulus of carbon
foam microstructure using a smaller solid model. The solid phase is assumed to have
homogeneous isotropic modulus value. This is a simplification of the non-homogeneous
properties of the solid phase. This assumption has been made by Sihn and Roy (2004), Li
et al. (2005) and several other researchers.
3.2.2 Methodology and simulations
The carbon foam microstructure subjected to a mechanical load is analyzed by
finite element method. A linear elastic model with constant material properties and
accounting only for small deformations (no large deformations and geometric
nonlinearities) is employed, which is consistent with the brittle nature of graphitic carbon
foams. The results of the analysis allow for calculation of bulk Young’s (elastic) modulus
of carbon foam microstructure. Two solid plates are used to apply the boundary
37
conditions on the carbon foam microstructure sandwiched between them. A rigid
mechanical contact (no penetration) is assumed between the solid plates and carbon foam
in the normal direction. In the tangential direction, the mechanical interaction between
the carbon foam and plates is frictionless in order not to introduce restrictions in
deformation. The displacement field in the system (carbon foam microstructure and
plates) is governed by the force equilibrium equation
∂
(σ ij ) = 0
∂x j
i = 1, 2, 3
(3.6)
where σ ij is the stress tensor (symmetric). The body forces are neglected.
The constitutive behavior of the solid phase of the carbon foam relates stress and
strain and is modeled using linear elasticity:
σ ij = Cijkl ε kl
i, j , k , l = 1,2,3
(3.7)
where C ijkl are the stiffness matrix components and ε kl are the displacement vector
components. The C ijkl values are calculated using the relations in Appendix A with
isotropic material properties.
The following boundary conditions are applied for stress-strain analysis:
- the bottom surface of lower plate is fully constrained
- two adjacent sides of the carbon foam are constrained in their normal
directions in order to avoid translation
- two adjacent sides of the upper plate are constrained in their normal
directions in order to avoid translation
- a normal uniform compressive load on the top surface of the upper plate
38
In order to reduce the influence of the plates on the results, the Young’s modulus
of the plates material was chosen to be three orders of magnitude higher than that of solid
phase of the foam. Therefore, the strain of the plates is negligible in comparison with the
strain of the carbon foam.
The mesh created for thermal analysis (see Figure 3.1) is also used for mechanical
analysis. The general purpose finite element software ABAQUS is used to solve Equation
(3.6) using the virtual work principle (Abaqus Inc., 2006). The solution of the finite
element analysis yields the displacement and stress distributions in the carbon foam
microstructure and solid plates. Mesh independence of the finite element solution was
established in a manner similar to the one used for thermal analysis. The difference
between the two meshes (519,397 elements vs. 1,357,568 elements) in terms of the bulk
Young’s modulus is less than 0.4%. The results in the next section were obtained by
performing the analysis with the coarser mesh.
The Young’s modulus of bulk foam can be obtained by applying Hook’s law for
1D stress situation:
σ x = Eeε x = Ee
ΔL
L0
(3.8)
where σ x is the uniform normal compressive load on the top surface of the upper plate,
ε x is the strain in the x- direction, L0 is the initial (undeformed) thickness of the carbon
foam and ΔL is the displacement of the top plate, both in the x- direction.
The non-dimensional bulk value for Young’s modulus of carbon foam is obtained
as
39
E eff =
Ee
Es
(3.9)
where E s is the Young’s modulus of the solid phase of carbon foam. The nondimensional value is used to facilitate comparison with the results reported in literature.
By following the same approach as in the thermal analysis, independent
simulations are carried out in the x- y- and z- directions of the 3D solid model of carbon
foam. The Young’s modulus and Poisson’s ratio of the solid phase have been varied to
study their influence on the bulk Young’s modulus of the microstructure. The two values
used for solid phase Poisson’s ratio are ν s = 0.2 and 0.33 (Sihn & Roy, 2004). The
values used for Young’s modulus of the solid phase are chosen between 10 and 20 GPa,
which is consistent with the values in Sihn and Roy (2004) and Li et al. (2005). The
magnitude of the compressive load applied to the model was selected such that the
maximum resulted strain is smaller than 0.2%.
3.2.3 Results and conclusions
Figure 3.3(a) and (b) show the x- displacement and stress distributions in carbon
foam microstructure when the compressive load is applied in the x- direction. It can be
noticed that the x-displacement gradient is mostly uniform in the compressive stress
direction. The stress distribution shows higher values in the thin ligaments of carbon
foam microstructure, which act as stress concentrators. Similar displacement and stress
distributions are obtained when applying the compressive stress in the y- and zdirections.
40
(a)
(b)
Figure 3.3. (a) Displacement and (b) stress distributions in carbon foam microstructure
when applying the compressive load in the x-direction.
41
The stress-strain analysis results are summarized in Table 3.2 for comparison with
similar results reported in literature. It needs to be mentioned that a range is specified in
Table 3.2 for the analysis by Li et al. (2005) because different values are calculated for
various cross-sectional shapes of the ligament. It can be seen that the simple regular
tetrahedral type geometry used by Sihn and Roy (2004) produces a very different result
for Young’s modulus when compared with the experimental values for a variety of foams
(Sihn & Rice, 2003). The results are different by a factor of about 6. It is probably due to
the complex deformation process that takes place at the pore level when a mechanical
load is applied. However, the tetrakaidecahedron, a much more complex geometrical
model used by Li et al. (2005), produces much better agreement with experimental
values; therefore this particular idealized geometry appears to reflect the stresses and
strains in the foam microstructure.
Table 3.2
Non-dimensional bulk Young’s modulus of 90% porosity carbon foam
E eff (% of solid phase)
Source
Type of analysis
Sihn & Rice (2003)
Experimental
0.4 – 1.0
Sihn & Roy (2004) FEM on ideal geometry
3.4
Li et al. (2005)
Analytical
0.45 – 0.77
0.596 (x-direction)
Present study
FEM on real geometry 0.76 (y-direction)
0.623 (average)
0.513 (z-direction)
The bulk Young’s modulus values for the true geometry in the present study are
in good agreement with both the experimental results in Sihn and Rice (2003) and
42
solution of analytical model by Li et al. (2005). The results of this analysis on the true
geometry produced different values of bulk Young’s modulus in the three directions (see
Table 3.2). The differences for the three directions could be due to the inherent geometric
anisotropy of the foam or due to the small sample size of the foam used in analysis. As
discussed earlier, the foam sample has the same porosity that is obtained by experimental
measurement of bulk foam samples, so that the foam sample was a taken to be a
reasonable representation of the bulk material.
As expected, the bulk Young’s modulus for the true geometry showed a linear
variation with Young’s modulus of the solid phase; so the non-dimensional values in
Table 3.2 did not change. The change of Poisson’s ratio of solid phase from ν = 0.2 to
0.33 showed negligible variation in Young’s modulus of the bulk foam. This has also
been reported by Sihn and Roy (2004).
43
CHAPTER 4: FLUID FLOW AND CONVECTION HEAT TRANSFER IN CARBON
FOAM
4.1 Introduction
In this chapter, the application of carbon foam in thermal management is studied
by considering a coolant flow through the pores of the foam. This study is motivated by
the continuous increase in thermal power dissipated by electronic devices, which requires
more efficient cooling solutions. Traditionally, the high density electronic circuits use
heat sinks made of a metallic finned structure (copper or aluminum) cooled by natural or
forced convection. However, the requirement for increasingly powerful cooling devices
has boosted the research and development of advanced heat sinks such as micro-channels
and heat pipes.
The interconnected pore structure of porous materials allows for fluid flow and
also offers a significant increase in surface area available for convective heat transfer.
Highly porous aluminum foams have been investigated as possible solutions for thermal
management of electronics (Bhattacharya & Mahajan, 2002; Boomsma et al., 2003).
Because of its reduced weight and high thermal conductivity, carbon foam is considered
as potential candidate for heat sinks and heat exchangers core. Typical porous media heat
exchange system consists of a channel filled with an open cell porous material and
saturated with a coolant that flows through the pores. In order to be able to design cooling
systems based on a porous material, its characteristics must be known in terms of
pressure drop and convective heat transfer coefficient when a fluid flows through it.
44
This chapter is focused on the analysis of fluid flow and convection heat transfer
in 3D porous channel using the finite volume method as implemented in the FLUENT
software (Fluent Inc., 2006). Carbon foam permeability, inertial coefficient and friction
coefficient will be calculated from the simulation results of pressure drop and free stream
fluid velocity. Comparison with experimental results available in literature will be used to
validate the results of our simulations. The effective heat transfer coefficient will also be
calculated from the results of heat transfer analysis in the porous channel. It will be
compared with the convection heat transfer coefficient obtained for the case of clear
channel (no foam) in order to quantify the enhancement in heat transfer due to the
presence of the carbon foam in the channel.
The assembly consisting of carbon foam saturated with fluid discussed in Chapter
2 (see Figure 4.1) is used to carry out this analysis. The fluid part of the model is
extended beyond the carbon foam on two opposite sides in order to create clear inlet and
outlet for the porous channel.
45
Carbon foam
Interstitial fluid
Flow direction
Figure 4.1. 3D solid model of porous channel.
4.2 Volume averaged model
Analytical and numerical modeling of transport phenomena (heat transfer and
fluid flow) in porous media is difficult due to its totally irregular structure. Not only is the
typical porous media non-homogeneous, but the pores are irregular in geometry and
dimension, and randomly distributed in the solid matrix.
The traditional approach in modeling transport phenomena in porous media
employs local volume averaging in formulation of mass, momentum and energy
conservation equations (Vafai & Tien, 1981; Calmidi, 1998). Under this approach, the
46
quantities associated with the fluid flow are averaged over a representative elementary
volume consisting of both interstitial fluid and porous material (Vafai & Tien, 1981).
When modeling the heat transfer between porous material and interstitial fluid, the two
energy equation approach must be used (one for each phase) and the temperatures are
averaged separately for solid phase and fluid phase (Calmidi, 1998). This averaging
approach reduces the complexity of the general problem, but information about the
transport phenomena at pore level and the influence on the overall transport phenomena
are lost (Vafai & Tien, 1981; Calmidi, 1998).
4.2.1 Fluid flow model
The steady state mass and momentum conservation equations for incompressible
flow through porous media are (Hunt & Tien, 1988)
∇⋅ u = 0
ρ
u ⋅ ∇u = −∇ p
φ2
where
(4.1)
f
+
ρ cf
μ 2
μ
∇ u−
u −
u u
φ
K
K
(4.2)
is the volume average symbol, u is the fluid velocity vector, p is the fluid
pressure, ρ is the fluid density, μ is the fluid viscosity, φ is the porous media porosity,
K is the porous media permeability and c f is the porous media inertial coefficient. The
mass and momentum conservation equations can be solved numerically for pressure and
velocity fields provided that φ , K and c f are known.
It needs to be mentioned that in comparison to Navier-Stokes equations, the
momentum conservation equation for flow through porous media contains two extra
47
terms, which are the last two terms on the right hand side of Equation (4.2). They account
for additional pressure loss due to the presence of the porous media. The second term on
the right hand side of Equation (4.2) accounts for pressure loss due to the presence of the
solid wall boundary.
For steady state fully developed flow through porous media without a solid wall
boundary, the momentum conservation equation can be expressed in a simpler 1D form
(Darcy-Forchheimer equation) which gives the pressure drop when fluid flows through
porous media
ρ cf 2
Δp μ
= uD +
uD
Δx K
K
(4.3)
where the free stream velocity in the clear channel before (or after) the porous region is
given by
uD =
m&
ρAch
and Δx is the length of the porous region in the direction of the flow, m& is the mass flow
rate and Ach is the cross sectional area of the channel filled with porous material.
The fluid flow regime in clear pipes and channels with constant flow area is
established as either laminar or turbulent using the Reynolds number based on the
equivalent hydraulic diameter which is the length scale of the flow. In porous channels it
is difficult to define an equivalent hydraulic diameter because the flow area is
continuously changing. The square root of the permeability
( K ) represents the length
scale of flow through porous media and has been used in defining the “modified”
48
Reynolds number (Beavers & Sparrow, 1969; Paek et al., 2000; Boomsma & Poulikakos,
2002)
Re K =
ρu D K
μ
(4.4)
The fluid flow in porous media is in the “Darcy regime” when the velocities are
small enough so that the inertial effects are insignificant and the quadratic term in DarcyForchheimer equation is negligible. At high velocities, the quadratic term in DarcyForchheimer equation cannot be neglected and the flow regime is “non-Darcy”.
Equation (4.3) has been used to determine K and c f from experimental
measurements of pressure drop and free stream velocity for a large variety of porous
materials: aluminum, nickel and carbon foams (Hunt & Tien, 1988), compressed
aluminum foam (Antohe et al., 1997) and carbon foam (Straatman et al., 2007).
Experimental testing on aluminum foams showed that permeability of a porous
material is a strong function of porosity and pore size, and inertial coefficient is
influenced by the solid phase shape and pore structure (Paek et al., 2000).
In an attempt to unify the pressure loss characteristics for various porous media, a
dimensionless friction coefficient has been used (Beavers & Sparrow, 1969; Paek et al.,
2000)
Δp
K
Δ
x
f =
ρu D2
(4.5)
49
Equation (4.5) was used to determine f from experimental measurements of
pressure drop and free stream velocity for aluminum foams (Paek et al., 2000) and
Foametal (Vafai & Tien, 1982).
Beavers and Sparrow (1969) combined together Equations (4.3), (4.4) and (4.5)
and obtained the following relation for friction coefficient:
f = cf +
1
Re K
(4.6)
It must be noticed that in the Darcy flow regime, the quadratic term in Equation
(4.3) can be neglected, so that the friction coefficient becomes equal to the inverse of
Reynolds number.
Based on Equation (4.6), semi-empirical correlations for friction coefficient were
developed for aluminum foams by Paek et al. (2000)
f = 0.105 +
1
Re K
(4.7)
and for Foametal by Vafai and Tien (1982)
f = 0.057 +
1
Re K
(4.8)
4.2.2 Heat transfer model
Two distinct approaches have been used in studying convection heat transfer in
porous media: (i) thermal equilibrium of the solid and fluid phases when only one
equation can be used to model the thermal transport (Vafai & Tien, 1981) and (ii) thermal
non-equilibrium, when the two phases have different temperatures and one equation is
50
necessary for each phase (Calmidi, 1998; Calmidi & Mahajan, 2000). In the first
approach, there is only one temperature in the system, which is averaged over a
representative volume consisting of both solid and fluid, so there is no thermal transport
between phases. This approach is not accurate when the thermal conductivities of the two
phases are very different, such as aluminum foam and air (Calmidi, 1998).
The steady state energy equations for porous material and fluid flowing through it
in the thermal non-equilibrium approach are (Calmidi, 1998)
0 = k se∇ 2Ts − hsf Asf (Ts − T f )
(4.9)
ρc p u ⋅ ∇T f = (k fe + kd )∇ 2T f + hsf Asf (Ts − T f )
(4.10)
where Ts and T f are the solid and fluid temperatures, c p is the fluid specific heat, k se
and k fe are the solid and fluid bulk (stagnant) thermal conductivities, kd is the fluid
dispersion thermal conductivity, hsf is the interfacial heat transfer coefficient and Asf is
the interfacial surface area available for heat transfer in the porous channel. The energy
conservations equations can be solved for solid and fluid temperature fields provided that
k se , k fe , kd , hsf and Asf are known.
Calmidi and Mahajan (2000) developed semi-empirical correlations for dispersion
thermal conductivity and interfacial heat transfer coefficient. The empirical parameters
were determined by curve fitting through experimental results for aluminum foam.
Straatman et al. (2007) used the same type of correlation for interfacial heat transfer
coefficient as Calmidi and Mahajan (2000) but they determined the empirical parameter
by curve fitting through experimental results for carbon foam.
51
An overall energy balance is generally employed in order to calculate the
effective heat transfer coefficient from the heated surface area in a porous channel:
Q = m& c p (Tout − Tin ) =
∫ h (T
sf
s
− T f )dA = heff Ab ΔTLM
(4.11)
Atot
where the log mean temperature difference between solid and fluid is
ΔTLM =
Tout − Tin
T − Tin
ln 0
T0 − Tout
and Q is the heat flux transferred to the fluid in the porous channel, Tin and Tout are the
channel inlet and outlet fluid temperatures and T0 is the heat source constant temperature
applied to the porous channel. The local interfacial heat transfer coefficient hsf is highly
dependent on fluid velocity and porous material geometry and varies from point to point
on the porous material surface. It is calculated taking into account the entire surface area
available for heat transfer in the porous channel, which is:
Atot = Asf + φAb
where Asf is the porous material surface area and Ab is the upper (base) surface area of
the channel.
The effective heat transfer coefficient heff is calculated by taking into account the
base surface area of the porous channel Ab rather than the entire surface area available
for heat transfer. This reflects the enhancement in heat transfer relative to the base area
due to the presence of the porous material in the channel.
52
In this research we will use the direct simulation model because the 3D
microstructure of the porous material can have significant effect on the thermal and
mechanical properties and also affects the heat transfer and fluid flow through the foam.
4.3 Direct simulation of fluid flow and heat transfer
4.3.1 Ideal pore geometry
The pore level simulation of fluid flow and heat transfer in porous media accounts
for the microscopic phenomena by modeling the geometry of the porous material
microstructure. The geometry of the porous media is often approximated with an ideal
geometry. For example, Krishnan et al. (2006) calculated permeability and friction factor
from direct simulation of fluid flow through metal foam with periodic boundary
conditions. Their representative elementary volume is created by subtracting nine spheres
of equal volumes from a cube. The spheres are located at the corners and the center of the
cube. This geometry is the same as one of the several geometries studied by Druma et al.
(2004). Krishnan et al. (2006) also calculated the heat transfer in terms of Nusselt number
by applying a constant heat flux boundary condition on the foam surface, neglecting the
conduction heat transfer through the solid phase. Karimian and Straatman (2008) studied
the hydraulic and thermal characteristics of porous material assuming spherical pores and
developed correlations for pressure drop and heat transfer based on simulations results.
The most important simplification of these two models is that they use ideal periodic
geometries to represent the porous material microstructure.
53
4.3.2 Real pore geometry
The porous material – fluid system used in the present analysis (see Figure 4.1)
consists of a rectangular channel filled with a real 3D representation of carbon foam
microstructure and saturated with coolant (air) flowing through the foam. The channel
walls can be used as virtual walls in the simulations by using symmetry conditions at
these boundaries. The effect of such a boundary condition is to extend the flow regime
beyond the channel walls and to eliminate the effect of the channel walls on the flow.
Symmetry conditions will be used on all surfaces of the channel (see Figure 4.2). For the
heat transfer, a constant temperature heat source is applied on the upper surface of the
channel to both carbon foam and fluid. The surfaces belonging to channel inlet and outlet
are not heated. Heat is transferred by convection from the channel upper wall to the fluid
and also by conduction through the solid phase of the foam and then convection from the
foam to the fluid. The system is analyzed considering steady state fluid flow and heat
transfer by direct simulation on the real scale model. Fluid velocity and pressure
distributions, as well as temperature distribution in carbon foam and fluid at pore level
are obtained from numerical simulation by commercial software FLUENT (Fluent Inc.,
2006). The mathematical model used for fluid flow and heat transfer calculations, as
implemented in FLUENT is summarized in the next section.
The above approach was adopted in a preliminary study by Anghelescu and Alam
(2006) to investigate the convection heat transfer in carbon foam microstructure by using
a smaller solid model.
54
Surface not heated
Surfaces heated
(solid and fluid)
Surface not heated
Flow
direction
Symmetry
(also on the bottom and on the back surfaces)
Figure 4.2. Boundary conditions applied on the porous channel.
4.3.3 Fluid flow computational model
The velocity and pressure fields of an incompressible, steady state, Newtonian
fluid flow in laminar regime are governed by the instantaneous mass conservation
equation
∂ui
=0
∂xi
i = 1, 2, 3
and the Navier-Stokes (momentum conservation) equations
(4.12)
55
ρu j
∂ui
∂p
∂ ⎛⎜ ∂ui ⎞⎟
=−
+μ
∂x j
∂xi
∂x j ⎜⎝ ∂x j ⎟⎠
i, j = 1, 2, 3
(4.13)
where ui are the fluid velocities, p is the fluid pressure, μ is the fluid viscosity and ρ
is the fluid density. The fluid flow is assumed viscous and the body forces on the fluid are
neglected. Due to the complexity of the geometry, a large number of finite volume
elements have to be used. Therefore, to reduce computational effort, the thermophysical
properties of the fluid are assumed constant with temperature.
The following boundary conditions are applied for fluid flow analysis:
- constant fluid velocity at the inlet of the porous channel
- constant fluid pressure at the outlet of the porous channel
- zero fluid velocity (no-slip) on carbon foam surfaces
- symmetry on top, bottom and side surfaces of the porous channel
Symmetric boundary conditions are applied on all outer surfaces of the porous
channel except for inlet and outlet. As explained earlier, this approach extends the
dimensions of the porous media and avoids solid wall boundaries which would increase
the pressure drop in the porous channel (due to additional friction) and affect the
accuracy of permeability, inertial coefficient and friction coefficient calculations for the
porous microstructure.
The fluid flow model yields the velocity and pressure distributions in the fluid as
a function of space coordinates x, y and z . Simulations are carried out by varying the
velocity at the inlet of the channel. The carbon foam permeability, inertial coefficient and
the relation between the friction coefficient and the modified Reynolds number
56
(f
− Re K ) will be determined from simulation results using Equations 4.3 and 4.5. The
f − Re K relation will be compared with Equations (4.7) and (4.8).
4.3.4 Convection heat transfer computational model
The steady state fluid temperature distribution in laminar flow regime is governed
by the energy conservation equation
ρc p
∂
(uiT f ) = ∂
∂xi
∂x j
⎛ ∂T f ⎞
⎜k f
⎟
⎜ ∂x ⎟
j
⎝
⎠
(4.14)
where k f is the fluid thermal conductivity.
The steady state temperature distribution in the carbon foam is governed by the
energy conservation equation
∂ ⎛⎜ ∂Ts ⎞⎟
=0
ks
∂x j ⎜⎝ ∂x j ⎟⎠
(4.15)
where k s is the thermal conductivity of the solid phase.
The following boundary conditions are applied for heat transfer analysis:
- constant temperature on the upper surface of the porous channel (carbon
foam and fluid)
- constant fluid temperature at the inlet of the channel
- symmetry on the sides and lower surface of the channel
- temperature and heat flux are continuous at the solid – fluid interface
57
Different from fluid flow computational model, a wall boundary condition is used
on the top surface of the porous channel in order to apply the constant temperature heat
source (see Figure 4.2).
The two energy equations solved together yield the temperature distributions in
carbon foam and fluid as a function of space coordinates x, y and z . The fluid flow
computational model provides the velocity distributions necessary for solving fluid
energy conservation equation. The thermophysical properties of the fluid are assumed
constant with temperature during the convection heat transfer analysis so that the fluid
flow governing equations and the energy conservation equations can be decoupled and
solved independently.
Simulations are carried out by varying the intrinsic thermal conductivity of the
solid phase in the carbon foam to study its influence on the effective heat transfer
coefficient in the porous channel. The effective heat transfer coefficient at the base area
of the porous channel will be calculated from simulation results using Equation (4.11).
4.4 Results and conclusions
4.4.1 Fluid flow analysis
The solid model assembly of porous channel (carbon foam microstructure and
interstitial fluid) is discretized using unstructured tetrahedral mesh in commercial
software HYPERMESH (Altair Engineering Inc., 2007) as showed in Figure 4.3. The
mesh created is imported into computational fluid dynamics (CFD) software FLUENT
for fluid flow and heat transfer calculations. The unstructured tetrahedral mesh is first
58
converted to polyhedral mesh in order to improve the mesh quality (eliminate bad
elements) and reduce the computational time. In the polyhedral mesh 367,462 elements
are generated for the solid phase and 1,148,766 elements for the fluid. The second order
upwind scheme is used for discretizing the momentum and fluid energy conservation
equations. The SIMPLE algorithm is used for pressure-velocity coupling.
Mesh independence of the finite volume solution was established by
solving the problem using a different mesh density: 355,088 elements for solid and
756,387 for fluid. The difference between the two discretizations in terms of permeability
and inertial coefficient of carbon foam microstructure is 1.6%, and 1.1%, respectively.
The results presented in this chapter are obtained using the finer mesh.
Figure 4.3. Computational mesh for porous channel.
59
It must be mentioned that the fluid flow and heat transfer simulation results
presented in this chapter were obtained using the mathematical models for laminar flow
regime as described in sections 4.3.3 and 4.3.4. The reason behind this is that the
Reynolds number calculated based on the equivalent hydraulic diameter of the flow
channel is smaller than Re cr = 2320 for the entire range of velocities used in simulations.
However, it can be argued that, even at low velocities, the flow in the porous channel
might have a certain degree of turbulence at pore level due to the presence of ligaments
and pore edges which obstruct part of the flow area and creates tortuous pathlines. In
order to account for this possibility the simulations were also run by including turbulence
in the flow model. The fluid flow turbulence was modeled using the Reynolds-Averaged
Navier-Stokes (RANS) method. The turbulent viscosity and turbulence kinetic energy
were modeled by employing the k − ε method as implemented in the commercial code
FLUENT (Fluent Inc., 2006). The mathematical model for turbulence is presented in
Appendix B. The difference between the laminar and turbulent flow simulation results in
terms of permeability and inertial coefficient of carbon foam microstructure is 0.7% and
5.3%, respectively. This is an indication that for higher velocities, the turbulent model
needs to be used for accurate simulation results.
Fluid flow simulations for inlet (free stream) velocities ranging between 0.01 m/s
and 1.5 m/s are run in order to determine the carbon foam microstructure permeability,
inertial coefficient and friction coefficient. Fluid thermophysical properties at 20°C are
used in the simulations.
60
Figure 4.4 shows the fluid flow pathlines colored by velocity magnitude for a free
stream fluid velocity of 0.5 m/s. The fluid flows along the x- axis in the positive
direction. The presence of the carbon foam in the channel reduces the cross-sectional area
available for fluid flow and increases the fluid velocity. It can be noticed that the
maximum fluid velocity in the porous channel is about 3.6 times higher than free stream
velocity. This, in turn increases the pressure drop per unit length. The tortuosity of the
fluid flow pathlines around the foam ligaments is observed in the simulations as shown in
Figure 4.4
Flow direction
Figure 4.4. Fluid flow pathlines in the porous channel colored by velocity magnitude
for a free stream velocity of 0.5 m/s.
61
The simulation results of pressure drop per unit length across the carbon foam as a
function of free stream fluid velocity are shown in Figure 4.5 as discrete points. Pressure
drop per unit length for the clear channel is plotted for comparison. The pressure drop
obtained in the porous channel filled with ideal model of porous material with the same
porosity has also been plotted in this figure. It consists of a matrix of 20 by 5 BCC type
spherical pores (see Figure 2.2(d)) with the same boundary conditions as porous channel
filled with the real model. Quadratic curves are fitted through the data points using the
least square method and their equations are also shown in Figure 4.5. It can be noticed
that the simulation results of pressure drop per unit length can be represented by a
quadratic dependence on free stream velocity, which is in agreement with the DarcyForchheimer equation. The quadratic behavior also shows that, for this particular
geometry of carbon foam microstructure, the fluid flow makes the transition from Darcy
to non-Darcy flow regime in the velocity range of the simulations. By comparing the
curve equations in Figure 4.5 with the Darcy-Forchheimer equation (Equation (4.3)) the
values of permeability and inertial coefficient for carbon foam can be calculated. The
values of these two quantities are listed in Table 4.1 along with similar values reported in
literature for comparison.
62
30000
Porous channel - real geometry
Clear channel
Pressure drop per unit length (Pa/m)
25000
Porous channel - ideal geometry
20000
y = 6160.7x2 + 7527.1x
15000
10000
y = 2749.4x2 + 4010.4x
5000
0
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
Free stream fluid velocity (m/s)
Figure 4.5. Pressure drop per unit length as a function of free stream velocity.
Table 4.1
Permeability and inertial coefficient of various porous materials
K ( x 10 8 m 2 )
Source
Porous material Type of analysis
Calmidi &
Aluminum
Mahajan
90.05%
Experimental
9.0
(2000)
20 PPI
CFD on real
0.4462
geometry
Carbon
Present study
90.31%
CFD on ideal
0.2377
geometry
cf
0.088
0.15
0.245
63
It can be noticed from Table 4.1 that there are significant differences in
permeability and inertial coefficient between carbon foam and aluminum foam. The
permeability of aluminum foam is generally much higher than that of carbon foam for the
same porosity because of the significant differences in pore structure. In aluminum foams
the pore opening is of the same dimension as the pore (cell) size. On the other hand, the
carbon foam pores tend to have the shape of bubbles and the pore openings are smaller
than the bubble (cell) size. The aluminum foams tend to have larger pores than carbon
foams for the same porosity. For instance, the 90.05% porous aluminum foam studied by
Calmidi and Mahajan (2000) has an average pore diameter of 1.27 mm (20 pores per
inch) in comparison with carbon foam which usually ranges between 0.1 and 0.5 mm.
This makes the aluminum foam much more permeable to fluid than carbon foam at
similar porosity. The inertial coefficient of carbon foam is higher than that of aluminum
foam because of the differences in pore structure: the windows between pores are much
smaller in carbon foams and the ligaments of carbon foam have much more complex
geometry. It can also be noticed from Table 4.1 that the ideal geometry gives about half
the permeability and a 63% higher inertial coefficient in comparison with the real
geometry of carbon foam. It has already been shown that the thermal conductivity of the
true foam is lower than that of the idealized geometry, which can be explained by higher
tortuosity of the true foam. Therefore, the permeability of the true foam could be
expected to be lower. However, the idealized geometry has much sharper edges than the
real geometry, and this is reflected in the lower permeability and higher inertial
coefficient.
64
Figure 4.6 shows the dependence between the friction coefficient calculated from
Equation (4.5) and the modified Reynolds number, as discrete points. For the purpose of
comparison, the experimental correlations from Paek et al. (2000) (Equation (4.7)) and
Vafai and Tien (1982) (Equation (4.8)) are also plotted. It can be seen from Figure 4.6
that for Re K < 1 (when the fluid flow is in the Darcy regime) the friction coefficient
calculated from simulation results follows very accurately as the inverse of the modified
Reynolds number. Our results are also in very good agreement with experimental results
reported by Paek et al. (2000) and Vafai and Tien (1982). It should be noticed that for
Re K > 1 the friction coefficient starts to deviate from 1 Re K because as the higher
velocity changes the flow to non-Darcy regime, the inertial effects can no longer be
neglected.
65
100
Present study - real geometry
Paek et al., 2000
Vafai & Tien, 1982
Friction coefficient
10
1
0.1
0.01
0.1
1
10
Modified Reynolds number
Figure 4.6. Friction coefficient as a function of modified Reynolds number.
4.4.2 Convection heat transfer analysis
In order to reduce the computational effort for the convection heat transfer
analysis, the fluid flow and heat transfer problems are decoupled and solved
independently. The fluid (air) and solid (carbon) thermophysical properties are assumed
constant with temperature during both fluid flow and heat transfer simulations. The fluid
flow analysis is first carried out with fluid properties calculated at 20°C and the fluid
pressure and velocity distributions are obtained for inlet fluid velocities ranging between
0.1 m/s and 1.5 m/s. The heat transfer analysis is then carried out using the fluid velocity
field obtained from the flow simulations and the temperature distributions in fluid and
66
solid are then obtained. The fluid temperature at the channel inlet is 20°C. The constant
temperature heat source applied on the upper surface of the porous channel is at 50°C.
Figure 4.7 shows the temperature distribution in the fluid in a clear channel (no
carbon foam) and porous channel (solid phase thermal conductivity of 50 W/m°C) for an
inlet fluid velocity of 0.5 m/s. It can be noticed that the fluid average temperature at the
channel outlet is higher for the porous channel in comparison with the clear channel. This
is due to the fact that a higher heat flux can be dissipated into the fluid when the porous
material is present in the channel. The porous material acts like highly efficient fins,
therefore the heat is conducted through the solid phase and then dispersed deep into the
fluid. Overall, the effective heat transfer in the porous channel is increased by two means:
higher velocity due to restrictions in flow area and extended surface area available for
convection. At higher flow velocity, the effect of turbulence also enhances the heat
transfer.
67
Flow direction
Flow direction
(a)
(b)
Figure 4.7. Fluid temperature distribution in (a) clear channel (no foam) and in (b)
porous channel for a free stream velocity of 0.5 m/s.
68
The plots in Figure 4.8 are drawn to show the effective heat transfer coefficient
calculated at the base area of the porous channel as a function of flow velocity and the
intrinsic solid phase thermal conductivity of the foam. Three different values for thermal
conductivity of the solid phase in carbon foam are used in simulations, which are 10, 50
and 100W/m°C. The results, as shown in Figure 4.8, demonstrate the influence on the
effective heat transfer coefficient due to the presence of the foam. A simulation of
convection heat transfer in clear channel (no foam) was also performed in order to
provide a baseline that would show the enhancement in the effective heat transfer
coefficient due to the presence of carbon foam in the channel.
There is a significant increase in the effective heat transfer coefficient obtained in
the porous channel in comparison with the clear channel (no foam). As expected, the
effective heat transfer coefficient increases with the solid phase thermal conductivity.
However, there is a saturation effect in heat transfer enhancement when the solid phase
thermal conductivity rises from 50 W/m°C to 100 W/m°C. The saturation limit is due to
the convection heat transfer limitation and is therefore dependent on the flow velocity. At
higher flow rates the convection heat transfer increases and the effect of higher solid
phase thermal conductivity shows up in the enhancement of the thermal transport.
69
1200.0
no foam
Effective heat transfer coefficient (W/m2°C)
1000.0
k=10 W/m°C
k=50 W/m°C
800.0
k=100 W/m°C
600.0
400.0
200.0
0.0
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
Free stream fluid velocity (m/s)
Figure 4.8. Effective heat transfer coefficient as a function of free stream velocity and
foam solid phase thermal conductivity.
Table 4.2 lists the ratios between the effective heat transfer coefficients obtained
with carbon foam and without carbon foam for various free stream velocity and solid
phase thermal conductivity. It can be noticed that for high values of thermal conductivity
of the solid phase and velocity, the heat transfer due to the presence of carbon foam in the
channel is about 4.5 times higher than the value in the clear channel. This result compares
well with experimental results reported by Ultramet (Pacoima, CA) on their website.
They tested a cooling system made of a tungsten tube filled with tungsten foam which
was cooled by helium and noticed that “the heat flux performance of the tungsten
70
foam/tungsten tube structure was approximately five times better than that of a tungsten
tube with no foam” (“Refractory open-cell,” n.d.).
Table 4.2
Increase in effective heat transfer coefficient for porous channel relative to clear channel
Solid phase thermal conductivity (W/m°C)
10
50
100
0.1
2.81
2.92
2.96
Free stream
0.5
3.65
4.79
5.01
fluid velocity
1
3.03
4.51
4.65
(m/s)
1.5
2.96
4.46
4.65
As it can be noticed from the fluid flow and heat transfer simulation results
presented above, the presence of carbon foam in the channel increases the heat transfer,
but also causes a much higher pressure drop. When designing a heat sink, the energy
required to flow the coolant needs to be compared against the thermal performance in
order to ensure a certain level of efficiency.
The energy required to flow the coolant through a heat sink can be quantified by
pumping power per unit volume:
Δp
W& =
uD
Δx
(4.16)
The thermal performance of a heat sink can be quantified by its thermal resistance
Rθ =
ΔTLM
Q
(4.17)
and a lower thermal resistance is desirable for a cooling system as it indicates good heat
transfer capability.
71
Figure 4.9 shows the dependence between the thermal resistance and pumping
power per unit volume of fluid for clear channel and porous channel with various solid
phase thermal conductivities. It can be noticed that for a pumping power in the range of
40 to 600 W/m3 the porous channel gives a lower thermal resistance, which means better
heat transfer performance. As expected, a higher thermal conductivity of the solid phase
of carbon foam reduces the thermal resistance of the system. For instance, in order to
obtain a thermal resistance of 2000 °C /W, the pumping power needed for the clear
channel is about 8 times higher than that needed for the porous channel with a solid phase
thermal conductivity of 100 W/m°C. It can be concluded that by adding carbon foam into
channel, the heat transfer performance of the system can be increased in an efficient
manner.
72
8000
7000
k=10 W/m°C
k=50 W/m°C
Thermal resistance (°C/W)
6000
k=100 W/m°C
no foam
5000
4000
3000
2000
1000
0
1
10
100
1000
10000
3
Pumping power per unit volume (W/m )
Figure 4.9. Pumping power per unit volume of fluid as a function of thermal
resistance of the channel.
100000
73
CHAPTER 5: CARBON FOAM TOOLING FOR ADVANCED COMPOSITE
MANUFACTURING
5.1 Introduction
A typical advanced composite material for aerospace application is made of
carbon fiber reinforcement embedded in epoxy matrix. For example, clothes of woven
carbon fiber impregnated with partially cured resin (prepreg) are commercially available
for making composite parts. The composite part can be manufactured by manually laying
up the prepreg on a tooling material and curing it under a pressure-temperature cycle in
an autoclave. Alternatively, the automatic lay up of the carbon fiber yarn can be used
when large composite parts are to be made. A critical problem for manufacture of
composite parts is the geometrical and dimensional stability associated with process
induced residual stresses. The severity of residual stresses and geometrical deformation is
strongly influenced by the mismatch of thermal expansion of the composite part and the
tool.
A recent development in composites processing is the tendency towards
manufacture of increasingly larger components, including fuselage sections, as a onepiece composite part. This requires larger tools and the handling of the massive tool can
become an issue. The heating and cooling of the large thermal mass of a dense tool can
also increase production costs. Consequently, lightweight materials would be preferred in
composite tooling.
Metals have been traditionally used as tooling materials for composite processing;
in particular, steel and aluminum are quite popular. Steel is a reasonably cheap material,
74
with very good durability. Unfortunately, it has a coefficient of thermal expansion (CTE)
higher than composite materials and the high density results in a high thermal mass.
Aluminum is lighter than steel but has higher thermal conductivity and a higher CTE.
Therefore, it exhibits a good heat up rate but the high CTE is detrimental to the
dimensional stability (see Figure 5.1). In general, materials with high CTE are not used as
tooling material for composite structures when geometrical and dimensional stability is a
critical issue. To overcome this drawback, nickel base alloys, such as Invar, have been
introduced as tooling for advanced composite materials. For example, Invar 36 closely
matches the CTE of carbon fiber reinforced composites (see Figure 5.1). However, it is
very expensive and dense. Due to its higher cost, Invar 36 is used primarily in special
applications of advanced composites to aerospace structures such as wing skin
(Campbell, 2004).
Because of good CTE, carbon fiber reinforced epoxy laminates have also been
used as composite tooling. A master tooling is first prepared, and then composite
laminates are laid on the master tool and cured in an autoclave. By laying up many
laminae at different orientation angles, the composite laminate tooling can be tailored to
match closely the CTE of carbon fiber reinforced composite parts. Such composite
toolings have the advantage of being light, but their use is limited by the fact that the
matrix can crack after repeated thermal cycles in autoclave (Campbell, 2004).
A tooling made of a layer of syntactic epoxy resin molded over an aluminum
honeycomb core has been described by Cloud and Norton (2001) as a low cost tooling
alternative. A similar tooling system, based on aluminum honeycomb core and Ren
75
patties is described by Burke (2003). Both of these types of tooling system have the
disadvantage of having a CTE almost as high as aluminum (Figure 5.1).
25
20
CTE (10-6/°C)
15
10
5
0
-5
Carbon Fiber Carbon Fiber
(longitudinal) (transverse)
Invar 36
Carbon
Foam
Monolithic
Graphite
Steel
Ren Patties
Aluminum
Figure 5.1. CTE of different tooling materials and carbon fiber (Burke, 2003;
Burden, 1989; GrafoamTM Carbon Foam Solutions).
Carbon as a tooling material is attractive because it can match the CTE of carbon
based composites. Burden (1989) has discussed the advantages of monolithic graphite in
composite tooling due to its CTE match with carbon fiber composites, high thermal
conductivity, low thermal mass and a good dimensional stability at high temperatures.
However, monolithic graphite is dense enough to create weight problems for large
tooling.
76
Non-graphitic carbon foam is a good candidate for a tooling material by virtue of
its wide range of properties. Unlike most traditional tooling materials, it is quite easy to
tailor its thermal and mechanical properties to specific application requirements (Rowe et
al., 2005). By decreasing the porosity of the foam the elastic modulus and mechanical
strength will increase. On the other hand, by increasing the porosity, the bulk density
decreases, which in turn reduces thermal mass and allows for better heat up rate. A major
consideration in the application of carbon foam to tooling is its CTE which can match the
CTE of carbon fiber reinforced composites.
Rowe et al. (2005) provides a discussion of the advantages of carbon foam
tooling. Experimental work on carbon foam tooling has been carried out by some carbon
foam manufacturers and efforts are underway to commercialize the tooling application. A
carbon foam tooling example is shown in Figure 5.2. This tool has two types of features
(i) flat surfaces where curvature (warpage) of the composite part can occur due to stresses
induced by mismatch between CTE of the tooling and the composite part, and (ii) curved
surfaces (convex and concave) where spring-in of the composite part can result from
anisotropic nature of the composite as well CTE mismatch with the tooling (Twigg et al.,
2004a; Twigg et al., 2004b). This chapter focuses on the comparative analysis of the
carbon foam vs. traditional tooling on dimensional stability of the composite; caused by
the interaction of tooling and composite part, and strongly influenced by the thermomechanical properties of the tooling (Twigg et al., 2004a; Twigg et al., 2004b).
77
Figure 5.2. Carbon foam tooling for a simple part (Source: GrafTech
International Ltd.).
Even though the CTE of carbon foam is well matched to aerospace composites,
and the thermal and mechanical properties of carbon foam can span a wide range, it is
important to note that these properties are dramatically different from traditional tooling
materials. In general, the density, modulus, and strength of typical carbon foam are orders
of magnitude lower than steel. For example, carbon foam can be made in a wide range of
bulk densities, such as 0.03 g/cc to 0.56 g/cc (“GrafoamTM Carbon Foam Solutions”,
n.d.), which is much lighter than steel (~7.8 g/cc). The tensile strength of carbon foam
can vary from 0.37 MPa to 6.3 MPa (“GrafoamTM Carbon Foam Solutions”, n.d.), which
is much lower than steel (~710 MPa). The typical modulus of carbon foam is two orders
of magnitude lower than steel. The bulk thermal conductivity of carbon foam can range
from 1 W/m°C to 250 W/m°C, and is strongly dependent on degree of graphitization
(Druma, 2005). The effect of all these properties on the performance or feasibility of
78
carbon foam tooling is not well documented since there are very few published studies of
carbon foam tooling.
The lack of process simulation on carbon foam tooling is in contrast to several
analytical studies of composite processing using traditional tooling materials that have
been published. For the carbon foam tooling design, it is important that process analysis
must be available to identify the foam properties that would ensure the best geometrical
and dimensional stability of the composite part at the lowest cost. Such an analysis
should include the effect of tooling material properties on process induced stress
development and deformation in fiber reinforced composites during autoclave curing. In
particular, the effect of low values of CTE, thermal conductivity, mechanical strength,
modulus and density of the carbon foam tooling on the product dimensional changes and
process feasibility is of great interest. A preliminary study by Anghelescu and Alam
(2008) studied non-graphitic carbon foam tooling for a simple shape. This chapter
extends these results to predict the deformation associated with the different features
shown in the tooling in Figure 5.2. The stresses in the carbon foam tooling are examined
to evaluate the limitations due to lower mechanical strength of the foam. The effect of the
distinct carbon foam properties on the stresses and deformations of the composite part are
determined and compared with process simulations that use Invar and steel tooling.
The numerical simulation in this study is based on the studies of thermal and
mechanical behavior of composite materials that have been developed over several
decades (Springer & Tsai, 1967; Bogetti & Gillespie, 1992). Over the last several years,
experimental and numerical work has been carried out in order to understand the causes
79
of process induced residual stresses and their effect on the final geometry and dimensions
of the composite part. Mechanical behavior of these materials is complex, therefore
constitutive models have been developed with different simplifying assumptions and
degree of detail. An important aspect of the model is to predict the behavior of
thermosetting resins that changes from a highly viscous liquid to an elastic solid.
Johnston et al. (2001) proposed a “cure-hardening instantaneously linear elastic” model
for the development of the elastic modulus of the resin. Kim and White (1996) developed
a stress relaxation model for an epoxy resin (3501-6) during cure, assuming a
thermorheologically simple behavior and considering both time and cure-dependent
effects. White and Kim (1998) developed a combined model in which thermal history is
predicted by finite difference method and the stresses and deformations of composite part
are simulated by finite element method. This approach was extended to a 3D finite
element model by Zhu et al. (2001) in which process induced residual stresses
accumulated during matrix curing are determined, and the deformations that result when
the cured composite part is removed from tooling are then calculated. A transversely
isotropic linear viscoelastic constitutive equation is used to model the composite material.
It is shown that the mismatch between coefficients of thermal expansion (CTE) of
composite part and tooling is an important contributor to process induced residual
stresses developed during autoclave curing (Zhu et al., 2001).
In the present research work, the approach developed by Zhu et al. (2001) is
followed and the focus is on the effect of carbon foam tooling properties on the cured
composite part. The numerical model is developed and validated by comparing with prior
80
results on traditional tooling. Then the model is used to examine the residual stresses and
deformations of different features of composite parts. The stresses and deformations for
the different thermo-mechanical properties of carbon foam, steel, and invar tooling are
then evaluated.
The composite material system used in this study is an AS4/3501-6 prepreg which
has been widely used for making aerospace structures. Typically, this prepreg has AS4
carbon fiber with volume fraction ν = 0.6 . AS4 is a continuous, high strength, PAN
based carbon fiber (“HexTowTM AS4 Carbon Fiber”, n.d.) and 3501-6 is a typical
aerospace grade epoxy resin. This prepreg is selected because it has been used in prior
studies, so the current numerical model could be validated by comparison with prior
results. In the curing process, the composite material is assumed to be homogeneous and
transversely isotropic. The carbon foam tooling material GRAFOAMTM FPA-35
(“GrafoamTM Carbon Foam Solutions”, n.d.) is taken to be homogeneous and isotropic.
The thermal, chemical and mechanical processes for the processing of the
composite-tooling system in this study have been described by Anghelescu and Alam
(2008), which follows the approach of Zhu et al. (2001). The models are summarized
below.
5.2 Thermo-chemical model
The temperature distribution in the composite part during autoclave curing can be
found by solving transient conduction heat transfer equation with internal heat generation
dc
∂ 2T
∂ 2T
∂T
∂ 2T
ρc p
= k x 2 + k y 2 + k z 2 + ρ (1 − ν ) H R
dt
∂t
∂z
∂y
∂x
(5.1)
81
where ρ and c p are the composite material density and specific heat, T is the composite
material temperature, k x , k y and k z are the composite material thermal conductivities in
x, y and z directions, v is the fiber volume fraction in the composite material, c is the
degree of cure of epoxy resin and H R is the ultimate heat of reaction of epoxy resin.
The degree of cure of a thermosetting polymer at time t is defined as the ratio of
the heat of reaction released up to time t to ultimate heat of reaction
c=
H (t )
HR
and lies between 0 for a completely uncured polymer and 1.0 for a fully cured polymer.
The heat released by a thermosetting polymer during curing is described by the
cure kinetics as a function of time, temperature and degree of cure:
dc
= f (T , c )
dt
The rate of degree of cure for 3501-6 epoxy resin used in this research work is
(Lee et al., 1982)
dc
= (K 1 + K 2 c )(1 − c )(0.47 − c )
dt
dc
= K 3 (1 − c )
dt
c ≤ 0.3
(5.2)
c > 0.3
where
K i = Ai e
− ΔEi
RT
i = 1, 2, 3
and R = 8.31 J/mol K is the gas constant and Ai and ΔEi are pre-exponential factors and
activation energies, respectively:
82
A1 = 2.101 ⋅ 10 9 min −1
A2 = −2.014 ⋅ 10 9 min −1
A3 = 1.96 ⋅ 10 5 min −1
ΔE1 = 8.07 ⋅ 10 4 J / mol
ΔE 2 = 7.78 ⋅ 10 4 J / mol
ΔE3 = 5.66 ⋅ 10 4 J / mol
The initial conditions at each point in the composite
T = T0 and c = 0 at t = 0
and the convective boundary condition on the surface of the composite exposed to
autoclave environment
− ki
∂T
= h(T − T∞ )
∂xi
i = 1, 2, 3
must be imposed when solving Equations (5.1) and (5.2) for temperature and degree of
cure distributions, where T∞ is the autoclave environment temperature and h is
convection heat transfer coefficient from autoclave environment to composite surface.
The autoclave environment is assumed to follow the typical pressure-temperature cure
cycle described by Kim and Hahn (1989) which is shown in Figure 5.3. The value of the
heat transfer coefficient associated with the convective boundary condition is 140
W/m2°C (“Autoclave and tooling effects”, 2003). A perfect thermal contact between
composite part and tooling material was assumed during autoclave curing.
The temperature distribution in the tooling during autoclave curing can be found
by solving Equation (5.1) with two simplifications: (a) there is no degree of cure and
83
internal heat generation in tooling and, (b) tooling material is assumed to be isotropic so
k x = k y = k z . The boundary conditions are changed accordingly.
200
1200
180
1100
160
1000
Temperature (°C)
140
900
120
800
100
700
80
600
60
500
40
400
20
300
0
0
50
100
150
200
250
Pressure (KPa)
Temperature (°C)
Pressure (KPa)
200
300
Time (min)
Figure 5.3. Cure cycle for AS4/3501-6 (Kim & Hahn, 1989).
Thermal properties of the composite material are shown in Table 5.1. These are
calculated using rule of mixture and models analogous to electrical circuits as introduced
by Springer and Tsai (1967) (see Appendix A). Thermal properties of tooling materials
are shown in Table 5.2. Thermal properties of carbon fiber, epoxy resin and tooling
material have been assumed to be constant during the curing cycle.
84
Table 5.1
Thermal properties of AS4/3501-6 prepreg and its components (Lee et al., 1982; Loos &
Springer, 1983)
AS4 Carbon
3501-6 Epoxy
AS4/3501-6
Property
Fiber
Resin
Composite
Density,
1790
1260
1578
ρ (kg/m3)
Specific heat,
712
1260
887
c p (J/kg°C)
Thermal conductivity,
k (W/m°C)
Heat of reaction,
H R (J/kg)
26
0.167
15.7 (longitudinal)
0.687 (transverse)
-
473600
-
Table 5.2
Thermal properties of tooling materials (“GrafoamTM Carbon Foam Solutions”, n.d.;
Wiersma et al., 1998)
GRAFOAMTM FPA-35
Property
Steel
Invar 36
carbon foam
Density,
7833
8055
560
ρ (kg/m3)
Specific heat,
434
515
710
c p (J/kg°C)
Thermal conductivity,
k (W/m°C)
Thermal diffusivity,
α t (m2/s)
60.5
10.5
0.3
1.8·10-5
2.5·10-6
7.5·10-7
A 2D version of the thermo-chemical model was used to study the stresses and
deformations. A 2D version is preferred because the causes and effects can be readily
identified in the results. For the same reason, the composite part is taken to be made of a
simple unidirectional laminate. The model was implemented in the finite element
85
computer software ABAQUS (Abaqus Inc., 2006). A user subroutine was written to
calculate the internal heat generation and the degree of cure of composite material for the
ABAQUS model. The thermo-chemical model yields the temperature and degree of cure
distributions in composite as well as the temperature distribution in tooling as a function
of time t and space coordinates x and y . These results are used as input data for the
stress-displacement model.
5.3 Stress-displacement model
The time-dependent mechanical behavior of epoxy resin is modeled using linear
viscoelasticity while the mechanical behavior of carbon fiber is modeled using linear
elasticity. The constitutive equation in hereditary integral form is therefore based on the
model of a linear anisotropic viscoelastic material undergoing changes in temperature and
degree of cure (White & Kim, 1998; Zhu et al., 2001):
t
σ ij (t ) = ∫ C ijkl (c, T , t − t ')
−∞
d (ε kl (t ') − ε kl (t '))
dt '
dt '
i, j , k , l = 1, 2, 3
(5.3)
where σ ij are the stress components, Cijkl are the composite material stiffness matrix
components, ε kl are the total strain components, t is the current time and t ' is the past
time. The material stiffness Cijkl has a strong variation with time, temperature and degree
of cure. Assuming that the linear anisotropic viscoelastic material exhibits
thermorheologically simple behavior at constant degree of cure (White & Kim, 1998;
Zhu et al., 2001), reduced time ‘ ξ ’, can be introduced:
86
t
ξ =∫
0
dt"
aT (c, T )
(5.4)
t'
dt"
ξ '= ∫
a (c, T )
0 T
where the shift factor aT (c, T ) allows for time-temperature superposition. For the 3501-6
epoxy resin the shift factor is (Kim & White, 1996)
1
⎞
⎛
aT (c, T ) = ⎜⎜ − 1.4e c −1 − 0.0712 ⎟⎟ ⋅ (T − 30)
⎠
⎝
Assuming that the linear anisotropic viscoelastic material has been free of strain
before t = 0 , the constitutive equation ca be written in a simpler form (Zhu et al., 2001):
t
σ ij (t ) = ∫ C ijkl (ξ − ξ ')
0
d (ε kl (t ') − ε kl (t '))
dt '
dt '
i, j , k , l = 1, 2, 3
(5.5)
The thermo-chemical strain is the sum of the effects of thermal and chemical
expansion/shrinkage of linear viscoelastic material, and is expressed as:
ε kl = α kl ΔT + β kl Δc
where α kl and β kl are thermal and respectively, chemical expansion/shrinkage
coefficients. The relaxation modulus of a thermoreologically simple viscoelastic material
is modeled using an assembly of n Maxwell elements in parallel, which gives a Prony
series with n terms (Kim & White, 1996)
[
]
n
E (c, ξ ) = E (c ) + E (c ) − E (c ) ∑ wi (c )e
∞
0
∞
−ξ (c ,T )
τ i (c )
(5.6)
i =1
where E 0 and E ∞ are the unrelaxed and the relaxed elastic modulus of viscoelastic
material, respectively; and wi and τ i are the weight factor and the relaxation time of the
87
i th Maxwell element. According to experimental results reported by Kim and White
(1996), E 0 , E ∞ and wi do not depend significantly on degree of cure so the relaxation
modulus can be written in the simpler form
(
E (ξ ) = E + E − E
∞
0
∞
)∑ w e
n
i =1
− ξ ( c ,T )
τ i (c )
(5.7)
i
where the wi and τ i values for 3501-6 epoxy resin are given in Table 5.3. Following the
approach of White and Kim (1998) it has been assumed that the relaxation behavior of
the composite material has the same thermorheologically simple behavior as its polymer
matrix material in pure state. Under this assumption, the relaxation behavior of the
0
composite material can be written in terms of its unrelaxed and relaxed stiffnesses Cijkl
∞
and Cijkl
C ijkl (ξ ) = C
∞
ijkl
(
+ C
0
ijkl
−C
∞
ijkl
)∑ w e
n
i =1
−ξ ( c ,T )
τ i (c )
i
(5.8)
The unrelaxed and relaxed composite material stiffnesses are calculated using the
values of mechanical properties shown in Table 5.4 and the micromechanics model
introduced by Bogetti and Gillespie (1992) (see Appendix A).
The mechanical properties of carbon fiber are assumed to be constant during the
autoclave temperature-pressure cure cycle. Poisson’s ratio and thermal and chemical
expansion/shrinkage coefficients of 3501-6 epoxy resin are also assumed constant during
cure (Kim & White, 1996).
88
Table 5.3
Weight factors and relaxation times for 3501-6 epoxy resin (Kim & White, 1996)
τi (min)
wi
i
1
1
2.92·10
0.059
3
2
2.92·10
0.066
3
1.82·105
0.083
7
4
1.10·10
0.112
8
5
2.83·10
0.154
6
7.94·109
0.262
11
7
1.95·10
0.184
8
3.32·1012
0.049
14
9
4.92·10
0.025
Table 5.4
Mechanical properties of AS4/3501-6 prepreg components (Bogetti & Gillespie, 1992;
Kim & White, 1996; White & Kim, 1998)
AS4
3501-6 Epoxy Resin 3501-6 Epoxy Resin
Property
Carbon
Unrelaxed
Relaxed
Fiber
Longitudinal elastic
206.8
modulus, E1 (GPa)
3.2
0.032
Transverse elastic
20.68
modulus, E 2 = E3 (GPa)
In-plane Poisson’s ratio,
0.2
ν 12 = ν 13
0.35
Transverse Poisson’s
0.3
ratio, ν 23
In-plane shear modulus,
27.58
G12 = G13 (GPa)
1.185
0.01185
Transverse shear modulus,
6.894
G23 (GPa)
Longitudinal CTE,
-9·10-7
α 1 (1/°C)
5.76·10-5
Transverse CTE,
7.2·10-6
α 2 = α 3 (1/°C)
Chemical shrinkage
0
-0.01695
coefficient, β1 = β 2 = β 3
89
The mechanical behavior of tooling material can be described using linear
elasticity. The constitutive equation of a linear isotropic elastic material undergoing
changes in temperature is:
σ ij = Cijkl (ε kl − αΔT )
i, j , k , l = 1, 2, 3
(5.9)
where α is thermal expansion coefficient of tooling material. The tooling material
stiffnesses are calculated using mechanical properties in Table 5.5 and relations in
Appendix A. Mechanical properties of tooling material are assumed to be constant during
the autoclave temperature-pressure curing cycle.
Table 5.5
Mechanical properties of various tooling materials (“GrafoamTM Carbon Foam
Solutions”, n.d.; Wiersma et al., 1998)
GRAFOAMTM FPA-35
Property
Steel
Invar 36
carbon foam
Elastic modulus,
210
145
3.5
E (GPa)
Poisson’s ratio
0.3
0.3
0.3
ν
Coefficient of thermal expansion
12·10-6
1.6·10-6
2.3·10-6
α (1/°C)
60 (compressive)
Strength
710
450
6.3 (tensile)
(MPa)
A 2D generalized plane strain version of the stress-displacement model was
implemented in the general purpose finite element computer code ABAQUS. As the z
dimension of the composite part is much longer than the other two dimensions and there
are no mechanical constrains in this direction a generalized plane strain behavior can be
assumed. A user subroutine was written to implement the constitutive mechanical
90
behavior of transversely isotropic linear viscoelastic material. The stress-displacement
model yields the stress distribution in the composite part throughout the curing process
and the deformation of the composite part after removing from the tooling. This model
was first validated by simulating several test cases from prior studies (Zhu et al., 2001;
Wiersma et al., 1998). The simulations for the carbon foam tooling were then carried out
with the model, as described below.
5.4 Simulations and results
The feasibility and carbon foam tooling-composite part interaction properties
during composite processing is investigated by analyzing the deformation and residual
stress generated in three cases of different geometric features. The composite part is a 2
mm thick [016]T laminate of the following geometry: (i) laminate bent at 90° on a convex
tooling, (ii) laminate bent at 90° on a concave tooling and, (iii) flat laminate on a flat
tooling. For the first two cases of 90° bend, the inner radius is 4 mm and the length of
straight section is 15 mm (Figure 5.4) and carbon fibers are oriented along the tooling
surface parallel to the length direction. The flat composite part has a length of 30 mm,
with fibers oriented along the tooling surface parallel to the length direction. The tooling
is 8 mm thick in all cases. Because of symmetry, just half of the geometry is analyzed by
applying appropriate boundary conditions on the symmetry line. A schematic
representation of geometries, finite element meshes, spring-in and warpage for all three
cases (i) - (iii) are shown in Figures 5.4 and 5.5.
91
Symmetry line
Convex tooling
Convex tooling
Composite part
Composite part
Symmetry line
Composite part
Composite part
Concave tooling
Concave tooling
(a)
(b)
Figure 5.4. (a) 3D solid model and (b) 2D finite element discretization of the composite
part on convex and concave tooling.
The spring-in angle was calculated from the displacements of the composite part
side facing the tooling, which means the angle Δθ1 for convex tooling and Δθ 2 for
concave tooling. These angles are defined by the lines through the corners of the straight
section of the composite part, as shown in Figure 5.5(a), so they account for the spring-in
of the curved section along with the warpage of the flat section. The warpage of the flat
composite part was calculated by taking the displacement δ at the bottom of the
composite. After removing from the tooling, the flat composite part is assumed to be an
arc of a circle, where the curvature is calculated as Kim and Hahn (1989)
92
κ=
1
2δ
=
R ⎛ L ⎞2
2
⎜ ⎟ +δ
2
⎝ ⎠
(5.10)
where R is the radius of the curved composite part and δ and L are the dimensions
given in Figure 5.5(b).
Symmetry line
Δθ1
Δθ2
Symmetry line
(a)
δ
L
2
(b)
Figure 5.5. The deflections and angles due to deformations from processing: (a)
spring-in for a 90 degree angle and (b) warpage of a flat composite part. The values
are determined after removing the part from the tooling.
Two limiting cases of tooling-composite interactions were investigated: (a) in the
first case the composite part is assumed perfectly bonded to the tooling throughout the
cure cycle, and (b) the composite and the tooling are assumed to have frictionless sliding
contact with no separation. For the first case, the composite part is separated from the
tooling at the end of the cure cycle to determine the residual stresses and the deformation
93
of the free part. In the model this is simulated by deactivating the mechanical interaction
between the two.
The spring-in angles and curvatures predicted by the model for both convex and
concave cases are listed in Tables 5.6 and 5.7, respectively. As expected, the tooling
material does not affect the final spring-in of the composite when there is no bonding
between the tool and the composite. However, even for this “ideal” case, there is stress
accumulation in the composite part that produces a spring-in for the case of curved
geometry due to anisotropic nature of the composite material.
The remainder of the results discussed below is based on perfect bonding between
the tooling and the composite, so that the effect of tooling material properties can be
evaluated. At the end of the cure cycle, after removing from the tooling, the curved
section of the composite part will spring-in and the flat section will become curved
(warpage) as shown in Figure 5.5(a) and (b).
Table 5.6
Bent composite part spring-in angle
Tooling
material/geometry
Carbon Foam
Convex
0.27
Bonding
Concave
0.24
Convex
0.27
No
Bonding
Concave
0.27
Spring-in angle (deg.)
Invar 36
0.26
0.2
0.27
0.27
Steel
0.37
0.05
0.26
0.27
94
Table 5.7
Flat composite part curvature
Tooling
material/geometry
Carbon Foam
Bonding
2.4 ⋅ 10−5
No Bonding
0
Curvature ( mm −1 )
Invar 36
5.6 ⋅ 10−5
0
Steel
29.2 ⋅ 10−5
0
5.4.1 Results for curved tooling geometry (convex and concave)
As it can be seen from Figure 5.6(a), a good CTE match between composite part
and tooling, which is the case for carbon foam, will generate almost no stress
accumulation in the straight section of the composite laminate; while the stress
distribution in the curved section of the composite laminate is mostly due to the
composite material anisotropy. It can be noticed from Figure 5.6(b) that, curing on steel
tooling generates compressive stresses up to 167 MPa in the mid-section of the composite
part. There are also tensile stresses up to 9 MPa generated in the composite part at the
interface with tooling. This high stress variation across the thickness increases the risk of
composite part delamination. After removing from the tooling, part of this complex state
of stress will relax generating a permanent deformation, namely the curved section of the
composite part will spring-in and the straight section will warp as shown in Figure 5.7(b).
The total spring-in angle, as given in Table 5.6 represents the superposition of these two
effects. The two effects add-up, as both of them tend to deform the composite part
towards the tooling. The angle is smaller for carbon foam because there is almost no
warpage when curing the composite part on carbon foam tooling as shown in Figure
95
5.7(a). Simulations have also been carried out which show that an increased warpage will
be obtained when the composite part is thinner or the straight section is longer.
1
1
2
2
(a)
(b)
Figure 5.6. Stress distribution along fiber direction at the end of cure cycle before
removing the composite part from convex tooling made of (a) carbon foam and (b) steel.
96
1
1
2
(a)
2
(b)
Figure 5.7. Stress distribution along fiber direction at the end of cure cycle after
removing the composite part from convex tooling made of (a) carbon foam and (b) steel.
A deformation scale factor of 10 is used for displacement.
The stress accumulation process in the case of concave tooling is similar to the
case presented for convex tooling. The stress distribution is shown in Figure 5.8(a) and
(b) for curing the composite part on carbon foam tooling and steel tooling, respectively.
The only relevant difference from the convex tooling case is the following: during the
stress relaxation process after removing the composite part from tooling, the curved
section spring-in will deform the composite part away from the tooling and the straight
section will warp towards the tooling. The two effects tend to deform the composite part
in opposite directions. For this reason, a higher CTE tooling material (steel) generates a
smaller spring-in angle after removing from tooling (see Table 5.6), but also a
compressive residual stress up to 42.3 MPa, which is 5.5 times higher than when using
carbon foam (see Figure 5.9(a) and (b)).
97
1
1
2
2
(a)
(b)
Figure 5.8. Stress distribution along fiber direction at the end of cure cycle before
removing the composite part from concave tooling made of (a) carbon foam and (b)
steel.
1
1
2
(a)
2
(b)
Figure 5.9. Stress distribution along fiber direction at the end of cure cycle after
removing the composite part from concave tooling made of (a) carbon foam and (b)
steel. A deformation scale factor of 10 is used for displacement.
98
The history of stresses in the carbon foam tooling was also studied for these cases
of perfect bonding, and the stresses were observed to be much lower (<2.5 MPa) as seen
in Figure 5.6(a) because of the lower CTE and lower elastic modulus values of carbon
foam. Since the stresses in the carbon foam tooling are low, the tooling stresses have
been omitted in the rest of the figures. This helps to clearly delineate the partition line
between the tooling and the part. The following discussion focuses first on the stresses
and deformation in the flat composite part; the issues of the lower modulus and tensile
strength of the carbon foam tooling will be discussed later.
5.4.2 Results for flat tooling geometry
When curing flat composite laminates, the tooling material has a very strong
influence on the amount of stress accumulated at the end of the autoclave curing cycle as
it can be seen in Figure 5.10(a) and (b). The stress patterns are similar but the maximum
compressive stress in case of steel tooling (147 MPa) is much higher than in case of
carbon foam tooling (4.6 MPa). This results in a much higher deformation after removing
the composite part from the steel tooling as shown in Figure 5.11(a) and (b). The results
obtained for curvature are presented in Table 5.7. It can be noticed that both Invar 36 and
carbon foam tooling give a smaller curvature of composite part in comparison with steel
tooling because their CTEs are much smaller than steel.
99
1
2
(a)
(b)
Figure 5.10. Stress distribution along fiber direction at the end of cure cycle before
removing the composite part from flat tooling made of (a) carbon foam and (b) steel.
1
2
(a)
(b)
Figure 5.11. Stress distribution along fiber direction at the end of cure cycle after
removing the composite part from flat tooling made of (a) carbon foam and (b) steel. A
deformation scale factor of 10 is used for displacement.
100
5.4.3 Effects of thermo-mechanical properties of tooling
The history of the curing process, as shown by the degree of cure at points (A)
and (B) for carbon foam tooling in Figure 5.12, demonstrates that the lower thermal
conductivity of the foam does not affect the composite part significantly. This is due to
the fact that the thermal transport can occur through the composite part, which has
conductive carbon fiber reinforcement. It has been shown that thermal transport issues
can become significant for thicker laminates (Bogetti & Gillespie, 1992), even when the
tooling material is a conductive metal. The stress variation along the fiber direction
throughout the cure cycle for the three different tooling is also quite similar at the specific
point (A) with the three different tooling materials until the cool down process begins,
when stresses start to develop due to CTE mismatch and material anisotropy.
101
20
1.00
0
0.89
-20
0.78
-40
0.67
-60
0.56
Degree of cure at
points (A) and (B)
-80
0.44
Degree of cure
Stress (MPa)
Carbon
Foam
Invar
-100
0.33
-120
0.22
B
-140
A
0.11
Steel
-160
0
50
100
150
200
250
300
0.00
350
Time (min)
Figure 5.12. Curing process of the composite part on convex tooling made of different
materials. The stress distribution along fiber at point (A) is shown for different tooling
materials. The curves for the degree of cure at points (A) and (B) overlap, even though the
point (B) is insulated by the carbon foam tooling.
It can be seen in Figure 5.12 that only a small part of the total process induced
stress is accumulated in the composite part before the final cool down stage because of
viscoelastic stress relaxation. When the composite part is fully cured and the system cools
down, it will cause an increase in compressive stress along fiber. At this point the CTE
mismatch between the fully cured composite part and tooling becomes important. As the
system is cooling down, the tooling shrinks due its positive CTE, while the composite
part tends to expand since it has negative CTE. This generates a high compressive stress
in composite part at the end of cure cycle before the composite part is removed from
102
tooling, as shown in Figure 5.12. The stress generated for the case of steel tooling is
much higher in comparison with the case of carbon foam or Invar 36 tooling because the
CTE mismatch is higher with steel.
The stress distribution in carbon foam tooling for all three tooling configurations
was also analyzed. The maximum value, as discussed earlier, is about 2.5 MPa and it was
obtained for convex configuration. This value is low because of (i) better CTE match
with the composite and, (ii) lower modulus of the carbon foam. In other words, the low
CTE produces a better match with the CTE of the composite part resulting in smaller
strains in the carbon foam tooling, and this, in combination with the lower modulus of the
foam gives very low stress in the foam tooling. This is an important issue because the
stresses of the same order as in the composite part (~90 MPa) can not be sustained in the
foam tooling.
The effect of the lower modulus of carbon foam tooling is evident in Table 5.6,
where the curvature produced by carbon foam tooling is less than what is produced by
Invar tooling, even though the carbon foam has a higher CTE than Invar 36. This is due
to the lower bulk modulus of carbon foam, which reduces the stress and the strain on the
composite part. While the lower modulus is seen to be helpful in reducing residual
stresses, it can be an issue in terms of deformation of the tooling-composite part system.
For example, the composite part in Figure 5.6 exerts a moment on the tooling; resulting
in a curvature that is a “spring-in” of the total tooling-composite part system. This
curvature was determined to be 0.017 degree, which is more than an order of magnitude
less than the final spring-in of 0.27 degree of the composite part. Therefore it can be
103
expected that the deformation of the tooling will not have a significant effect as long as
the tooling is properly supported. In practice, the carbon foam tooling may have an
adhesive seal and/or composite face sheet on the surface which can also reduce
deformation of the tooling (Roy, 2008).
The tensile strength of this carbon foam is about 6.3 MPa (“GrafoamTM Carbon
Foam Solutions”, n.d.). This is higher than the highest stress (~2.5 MPa) in the
simulation; but they are of similar order of magnitude. Therefore, even though the
mechanical integrity of the tooling in this particular case should not be affected during
autoclave curing of the composite, the strength of the foam is a possible limitation for
designing complex, large tooling with carbon foams of low strength. It should be noted
that the compressive strength of the same foam is 60 MPa (“GrafoamTM Carbon Foam
Solutions”, n.d.), which would provide adequate strength in compression mode for most
tooling applications.
5.5 Summary of analyses of carbon foam tooling
A detailed analysis has been carried out for a small tooling-composite part system
to compare the residual stresses and deformations when processing composite laminates
using carbon foam versus traditional tooling materials. The results have shown that the
residual stresses can be reduced significantly by using carbon foam tooling. The stresses
are reduced due to lower CTE and lower modulus of the carbon foam. Because of lower
residual stresses, carbon foam tooling material generated smaller deformation of
composite parts. An exception was seen when large spring-in with steel tooling was
104
compensated by high warpage in a concave system. As regards the spring-in angles of
curved composites, the values obtained for carbon foam tooling are similar to those
obtained for Invar tooling. These results are obtained for the simple geometries and small
parts that were analyzed. Additional studies need to be done to determine the thermomechanical effects in curing thicker and/or larger parts with complex geometry. Care
must be taken to evaluate the mechanical integrity of the tooling and in selection of a
carbon foam with adequate strength. Experimental research also has to be carried out to
validate these results and develop a reliable carbon foam tooling design for composite
processing.
Even though the dimensional stability of the composite part is a major factor in
the selection of the tooling material, it is just one of the many requirements for composite
processing. Additional requirements include a long-term resistance to high temperature
and heat-up/cool-down cycles in autoclave, ability to maintain vacuum integrity and
suitable heat-up rate (Campbell, 2004; Rowe et al., 2005). All these factors play a key
role in production of geometrical and dimensionally accurate composite structures;
therefore, these issues must also be considered in making a tooling material selection.
105
CHAPTER 6: CONCLUSIONS AND RECOMMENDATIONS FOR FUTURE WORK
The objective of this research work was two fold: (i) to use an accurate three
dimensional solid model of carbon foam microstructure in developing finite element
models to calculate bulk thermal and mechanical properties; and (ii) to study the
application of carbon foam in convection heat transfer and composite tooling. The results
from the model compare favorably with experimental and analytical results available in
literature.
One of the limiting factors of the modeling effort is the computational effort
required to set up and run a model that reflects accurate details at the pore level of the
microstructure. In order to have a model that is representative of the bulk material, the
model should to be large enough to have the same properties (density, thermal and
mechanical properties, etc.) as the bulk material. Since the thermal and mechanical
properties of the bulk material tend to be non-homogeneous, it is quite difficult (if not
impossible) to select a reasonably small volume of the foam that can reflect all of the
bulk properties. To keep the computational effort reasonable, the small volume of the
model in this study was selected only on the basis of similarity to the bulk density. The
validity of the model is checked by comparing with experimental values from the
literature.
A high performance CAD workstation was used for solid modeling of carbon
foam. The number of mesh elements in the model is typically 1 million or more,
especially when the fluid flow was included. Because of the large number of mesh
106
elements, a 64-bit Linux machine was used to run the FEA and CFD simulations With the
continuous improvement in the capabilities of computational systems, it is expected that
larger volume models would be handled more easily in future work and this would result
in greater confidence in the results of the numerical model.
The results of the solid model and the fluid flow simulations compared well with
the experimental values. This is particularly significant considering that the idealized
geometry models used in prior research produced results that are typically very different
from experimental values. For example, the thermal conductivity of carbon foam is
higher in the idealized model by a factor of 2 to 3 when compared to experimental values.
The elastic modulus has a higher discrepancy, a factor of about 6. But the model
developed in this current study predicts values that are in the range of experimental
values. Similar differences in the flow model of idealized vs. true three dimensional
carbon foam geometry have significant implications for the design of heat exchangers
and heat sinks using carbon foam.
The solid model developed in Chapter 2 was then used in Chapter 4 to develop a
fluid flow and convection heat transfer model to study the behavior of thermal transport
of carbon foams in presence of a fluid flow. For future work, the method and the results
of this research should be used to model a heat sink or heat exchanger core which is
typically much larger than the solid model in this study. In general, a heat sink or heat
exchanger core is a periodic structure that can be modeled by identifying a repeating cell
and applying periodic boundary conditions. In this approach, the solid model of carbon
107
foam used in this study would be mirrored in different directions, so that it represents a
periodic cell and periodic boundary conditions would be applied on each of its faces.
For a solid model which is at least several millimeters in dimension, it will be
more accurate to include a bulk variation in conductivity that is often the case in graphitic
foams. The complex task of incorporating the anisotropic and inhomogeneous material
properties of the ligaments and nodes can be modeled when greater computational
capability becomes available.
108
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116
APPENDIX A: RELATIONS FOR THERMAL AND MECHANICAL PROPERTIES
OF COMPOSITE MATERIALS
In the following relations, subscript “f” denotes a carbon fiber property and
subscript “m” denotes an epoxy matrix property.
Density and specific heat of the composite material can be calculated from
constituents’ density and specific heat using the rule of mixture:
ρ = vρ f + (1 − v )ρ m
cp =
vρ f c pf + (1 − v )ρ m c pm
ρ
Longitudinal and transverse thermal conductivities of the composite material can
be calculated from constituents’ thermal conductivity using the rule of mixture and the
relations introduced by Springer and Tsai (1967):
k l = vk f + (1 − v )k m
⎛
D 2 v ⎞⎟
⎜
1−
⎛
v ⎞ km ⎜
4
π ⎟
⎟+
k t = k m ⎜⎜1 − 2
tan −1
π−
⎜
⎟
⎟
π⎠ D
v ⎟
D 2v
⎝
⎜
1+ D
1−
⎜
π ⎟⎠
π
⎝
⎛k
⎞
D = 2⎜ m − 1⎟
⎜k
⎟
⎝ f
⎠
Lamina composite mechanical properties can be calculated from constituents’
mechanical properties using the following micromechanics model introduced by Bogetti
and Gillespie (1992). The relations are written for a composite material that is
transversely isotropic in the 2-3 plane (the fibers oriented in the 1-direction).
117
The longitudinal elastic modulus:
E1 = E1 f v + E m (1 − v ) +
(
)
4 ν m − ν 122 f K f K m Gm (1 − v )v
(K
f
+ Gm )K r + (K f − K m )Gm v
The in-plane Poisson’s ratio:
ν 12 = ν 13 = ν 12 f v + ν m (1 − v ) +
(ν
(K
− ν 12 f )(K m − K f )Gm (1 − v )v
m
f
+ Gm )K m + (K f − K m )Gm v
The in-plane shear modulus:
G12 = G13 = Gm
G12 f + Gm + (G12 f − Gm )v
G12 f + Gm − (G12 f − Gm )v
The transverse elastic modulus:
E 2 = E3 =
1
ν 122
1
1
+
+
4 K T 4G23 E1
The transverse Poisson’s ratio:
ν 23 =
2 E1 K T − E1 E 2 − 4ν 122 K T E 2
2 E1 K T
The transverse shear modulus:
G23 = Gm
K m (G23 f + Gm ) + 2G23 f Gm + K m (G23 f − Gm )v
K m (G23 f + Gm ) + 2G23 f Gm − (K m + 2Gm )(G23 f − Gm )v
The longitudinal coefficient of thermal expansion
α1 =
vα 1 f E1 f + (1 − v )α m E m (ξ )
α 1 f E1 f + α m E m (ξ )
118
The transverse coefficient of thermal expansion
α 2 = α 3 = (α 2 f + ν 12 f α 1 f )v + α m (1 + ν m )(1 − v ) − [ν 12 f v + ν m (1 − v )]α 1
The longitudinal and transverse coefficients of chemical expansion are obtained
by replacing fiber and matrix coefficients of thermal expansion by their respective
coefficients of chemical expansion.
In the above relations, the isotropic plane strain bulk modulus and the effective
plane strain bulk modulus of composite are (Bogetti & Gillespie, 1992):
K=
E
2 1 − ν − 2ν 2
(
KT =
(K
f
)
+ Gm )K m + (K f − K m )Gm v
(K
f
+ Gm ) − (K f − K m )v
The stiffness matrix of the transversely isotropic composite material is:
C ijkl
⎡C1111
⎢C
⎢ 1122
⎢C
= ⎢ 1122
⎢ 0
⎢ 0
⎢
⎣⎢ 0
C1122
C1122
0
0
C 2222
C 2233
0
0
C 2233
C 2222
0
0
0
0
0
0
C1212
0
0
C1212
0
0
0
0
where the matrix elements are:
C1111 = E1
1 − ν 23ν 32
Λ
C 2222 = E 2
1 − ν 13ν 31
Λ
C1122 = E1
ν 21 + ν 31ν 23
Λ
0 ⎤
0 ⎥⎥
0 ⎥
⎥
0 ⎥
0 ⎥
⎥
C 2323 ⎦⎥
119
C 2233 = E 2
ν 32 + ν 12ν 31
Λ
C1212 = G12
C 2323 = G23
Λ = 1 − ν 12ν 21 − ν 23ν 32 − ν 31ν 13 − 2ν 21ν 32ν 13
ν ij =
Ei
ν ji
Ej
i, j = 1, 2, 3
120
APPENDIX B: MATHEMATICAL MODEL FOR TURBULENT FLUID FLOW AND
HEAT TRANSFER
The mathematical model for turbulent fluid flow and heat transfer as implemented
in commercial CFD software FLUENT (Fluent Inc., 2006) is used for performing the
simulations in this research. This model is summarized below.
In the Reynolds-Averaged Navier-Stokes (RANS) method for modeling fluid
flow turbulence, the instantaneous fluid velocities and pressure are decomposed into the
time-averaged components (overbar) and the fluctuating components (prime)
ui = ui + ui '
p = p + p'
and then substituted into Equations (4.12) and (4.13). After taking the time-average and
omitting the overbar on ui and p , the fluid flow governing equations are
∂u i
=0
∂xi
ρu j
∂u i
∂p
∂
=−
+μ
∂x j
∂xi
∂x j
⎛ ∂u i
⎜
⎜ ∂x
⎝ j
⎞ ∂
⎟+
− ρu i ' u j '
⎟ ∂x
j
⎠
(
)
The Boussinesq hypothesis gives the Reynolds stresses as a function of the
average velocity gradients (Hinze, 1975)
⎛ ∂u ∂u j
− ρ ui ' u j ' = μ t ⎜ i +
⎜ ∂x
⎝ j ∂xi
⎞ 2
⎟ − ρkδ ij
⎟ 3
⎠
121
where μ t is the turbulent viscosity, k is the turbulence kinetic energy and δ ij is the
Kroneker delta.
The turbulent viscosity and turbulence kinetic energy are modeled by employing
the k − ε method. This method introduces two additional equations:
⎡
∂
(ρku j ) = ∂ ⎢⎛⎜⎜ μ + μ t
∂x j
∂x j ⎢⎣⎝
σk
⎞ ∂k ⎤
⎟⎟
⎥ + Gk − ρε
⎠ ∂x j ⎥⎦
⎡
∂
(ρεu j ) = ∂ ⎢⎛⎜⎜ μ + μ t
∂x j
∂x j ⎣⎢⎝
σε
⎞ ∂ε ⎤
ε2
⎟⎟
⎥ + ρC1 Sε − ρC 2
k + νε
⎠ ∂x j ⎦⎥
where the constants are given by
⎡
η ⎤
C1 = max ⎢0.43,
η + 5 ⎥⎦
⎣
σ k = 1.0
C 2 = 1.9
η=S
k
ε
σ e = 1.2
and ε is the turbulence dissipation rate.
The generation of turbulence kinetic energy is given by
Gk = μ t S 2
S = 2S ij S ij
where S ij =
1 ⎛⎜ ∂u j ∂u i
+
2 ⎜⎝ ∂xi ∂x j
⎞
⎟ is the rate of strain tensor of the fluid.
⎟
⎠
The turbulent viscosity is given by
μ t = ρC μ
k2
ε
122
Cμ =
1
A0 + As
~ ~
U = S ij S ij + Ω ij Ω ij
kU
ε
~
Ω ij = Ω ij − 2ε ijk ω k
Ω ij = Ω ij − ε ijk ω k
where Ω ij is the is the mean rate of rotation tensor viewed in a rotating reference frame
with the angular velocity ω k . The model constants are computed from
As = 6 cos ϕ
A0 = 4.04
1
3
(
ϕ = cos −1 6W
)
W =
S ij S jk S ki
~
S3
~
S = S ij S ij
The fluid temperature distribution in turbulent fluid flow is governed by the
energy conservation equation (4.14) where the fluid thermal conductivity k f is replaced
by fluid effective thermal conductivity computed from
keff = k f +
Prt = 0.85
c p μt
Prt