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Process Modeling for Liquid Composite Molding

presented by

Jinglei Chen

 

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M. S . degree in Chemical Engineering

 

 

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ans-o1

PROCESS MODELING FOR LIQUID COMPOSITE MOLDING
By

Jinglei Chen

A THESIS

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

MASTER OF SCIENCE

Department of Chemical Engineering

1995

ABSTRACT

PROCESS MODELING FOR LIQUID COMPOSITE MOLDING

Jinglei Chen

A process model has been developed for mold filling in the liquid composite
molding process. This model incorporates mass transfer and dissolution of binder from
the fiber preforms into resin, as well as non-isothermal, reactive filling of the preforms,
with heated mold walls. The model was first validated by experimental data Then it was
used to investigate resin viscosity variation due to various factors and to develop design

guidelines for mold filling with soluble binder.

Transverse flow through real bi-directional fiber preforms can be adequately
described by considering flow across orthogonal sets of filaments. This flow problem was
solved by both finite element computations and analytically by applying the lubrication
approximation for higher fiber volume fraction. The analysis of the lubrication equations

was carried out with a decoupling approach developed in this work.

T o my dear wife Min Lu

iii

ACKNOWLEDGEMENT

The author would like to thank Professor Krishnamurthy Jayaraman for his
guidance and encouragement during the span of this Master’s research.

The author wants to express his gartitude to Mr. Prasad Kalyanaraman and Mr.
Douglas James Backes for providing the experimental data. He also wants to thank all

members of the research group for their support and friendship during the past four years.

iv

TABLE OF CONTENTS

List of Tables .............................................................................................................. viii
List of Figures ............................................................................................................... ix
1. Introduction ................................................................................................................ l
2. Binder Dissolution and Process Model ...................................................................... 6
2.1 Binder Dissolution and Resin Viscosity ....................................................... 6

2.2 Resin Viscosity Model ............................................................................... 10

2.3 Binder Dissolution Model .......................................................................... 11

2.4 Process Model ........................................................................................... 14

2.41 Momentum balance ..................................................................... 14

2.42 Energy balance ............................................................................. 14

2.43 Mass balance ................................................................................ 15

2.44 The boundary conditions .............................................................. 15

2.45 The numerical scheme ................................................................... 16

2.5 Model Validation and Prediction of Binder Dissolution ................................... 17

3. Process Simulations and Control ............................................................................. 24
3.1 Resin Viscosity Variations ......................................................................... 24

3.11 RTM simulations without binder .................................................. 24

vi

3.12 S-RIM simulations without binder ................................................ 28

3.13 RTM simulations with binder ........................................................ 31

3.2 Design Guidelines ....................................................................................... 31
4. Transverse Permeability ............................................................................................ 35
4.1 Permeability Prediction ............................................................................... 35
4.2 Finite Element Analysis ............................................................................. 40
4.21 Numerical scheme ........................................................................ 40

4.22 Numerical results ........................................................................... 44

4.3 Transverse Permeability Model .................................................................. 48
4.31 Structural parameters .................................................................... 48

4.32 Decoupling of 3-D flow field ....................................................... 48

4.33 Model prediction and comparisons with FEA results .................... 52

4.4 Experimental Validation ............................................................................. 60
4.41 Experimental ................................................................................. 60

4.42 Comparison with experimental data ............................................... 62

5. In-plane Permeability ................................................................................................ 69
5.1 In-plane Permeability Prediction ................................................................. 69
5.2 Finite Elemnet Analysis ............................................................................. 70
5.21 Numerical scheme ........................................................................ 70

5.22 Numerical results ........................................................................... 73

5.3 In-plane Permeability Model ...................................................................... 81
5.31 Flow through a channel ................................................................ 81

5.32 Model prediction and comparisons with FEA results .................... 84

vii

5.4 Experimental Validation ............................................................................. 84
6. Conclusions and Recommendations .......................................................................... 92
6.1 Conclusions ................................................................................................. 92
6.2 Scope of Future Work ............................................................................... 94

Bibliography ....................................................................................................................... 96

LIST OF TABLES

Table 4.1 : Specifications of bi-directional fiber mats ...................................................... 62

Table 4.2: Transverse permeability for asymmetric pitched bi-directional fiber mats ........ 66

viii

 

Fi

Fi
Fi
Fi;
Fir
Fig

Fig

LIST OF FIGURES

Figure 2.1: Viscosity variation of a vinyl ester resin with % weight binder ....................... 8
Figure 2.2: Causes of non-uniformity during mold filling .................................................. 9
Figure 2.3: Test of binder contribution in viscosity model ......................................... 12
Figure 2.4: The flow scheme for mold filling .............................................................. 18
Figure 2.5: Comparison of predicted and experimental binder washout curves ................ 19
Figure 2.6: Resin temperature transients for non-isothermal radial filling ........................ 21
Figure 2.7: Predicted binder concentration profile for isothermal mold fill ...................... 22
Figure 2.8: Predicted binder concentration profile for non-isothermal mold fill ............. 23
Figure 3.1: Viscosity profile at different initial temperature, RTM without binder .......... 26

Figure 3.2: Viscosity profile at different mold wall temperature, RTM without binder ...27
Figure 3.3: Viscosity profile at different initial temperature, SRIM with out binder ....... 29
Figure 3.4: Viscosity profile at different mold wall temperature, SRIM without binder ..30
Figure 3.5: Viscosity profile at different mold wall temperature, RTM with binder ........ 32

Figure 3.6: Variation of viscosity gradient with ratio og fill time to dissolution time ...... 34

Figure 4.1: Fiber arrangement in an aligned stack of bidirectional fiber mats .................. 39
Figure 4.2: Computational cell for transverse permeation (z-direction) ............................ 41
Figure 4.3: The 3-D mesh for transverse permeation ........................................................ 43

ix

X

Figure 4.4: 3-D velocity field for transverse permeation .................................................. 45
Figure 4.5: Pressure distribution for transverse permeation .............................................. 46
Figure 4.6: Numerical results of transverse permeability .................................................. 47
Figure 4.7: Three-dimensional repeat unit for transverse flow (z-direction) ..................... 49
Figure 4.8: Comparison between numeriCal results and Eq. (4.16), lx=ly ............... 54
Figure 4.9: Comparison between numerical results and Eq. (4.16), I, .='1y .................. 55
Figure 4.10: Comparison between numerical results and Eq. (4.16), 1,( .--—'1y ................ 56
Figure 4.1 1: Comparison between numerical results and Eq. (4.19), lx=ly .............. 57
Figure 4.12: Comparison between numerical results and Eq. (4.19), lx .='ly ................ 58
Figure 4.13: Transverse permeability for bi-directional and unidirectional arrays ............ 59
Figure 4.14: Predicted transverse permeability verse ratio of pitch distances ................... 61
Figure 4.15: Comparison of model prediction and experimental data (cross-plied) .......... 64
Figure 4.16: Comparison of model prediction and experimental data (Cofab) ................. 65
Figure 4.17: Comparion between existing models and the experimental data ................... 67
Figure 5.1 : Computation cell for in-plane permeation ..................................................... 71
Figure 5.2: The 3-D mesh for in-plane permeation ........................................................... 72
Figure 5.3: 3-D velocity field for in-plane permeation ..................................................... 74
Figure 5.4: Pressure distribution for in-plane permeation ................................................. 75
Figure 5.5: Contour of the periodic velocity boundary condition ..................................... 76
Figure 5.6: Numerical results of in-plane permeability ..................................................... 77
Figure 5.7: The anisotropy of the symmetric pitch ........................................................... 78
Figure 5.8: The anisotropy of an asymmetric pitch ........................................................... 79

Figure 5.9: The anisotropy of an asymmetric pitch ........................................................... 80

xi

Figure 5.10: Channel cross section shape of in-plane permeation ........................... 82
Figure 5.1 1: Comparison of numerical results and model prediction, 1x=ly ...................... 85
Figure 5.12: Comparison of numerical results and model prection, lx .=°1y .......................... 86
Figure 5.13: Comparison of model prediction and experimental data (cross-plied) .......... 87
Figure 5.14: Comparison of model prediction and experimental data (Cofab) ................ 88
Figure 5.15: Comparion between expermental data and Carman-Kozeny equation .......... 90

Figure 5.16: Comparion between expermental data and Carman-Kozeny equation .......... 91

Chapter 1 INTRODUCTION

Liquid composite molding process, including Resin Transfer Molding (RTM) and
Structural Reaction Injection Molding (S-RIM), involves injection of reactive resin into
a mold cavity with pre-placed fiber preforms. The resin then polymerizes and solidifies
within the interstices of the fibers to form the final composite structure. It has attracted
much research interest in recent years due to its potential application in automotive
industry.

Compared to other manufacturing processes, RTM and S-RIM have a variety of
advantages including design flexibility to mold large and complex parts, rapid production
cycle time, greater reproductivity, and better potential for automation. All these have made
liquid molding process a good candidate for the production of high performance parts.
Hence there is a growing need to fully understand the mold filling process of liquid
composite molding.

Darcy’s law, a macroscale model for flow of incompressible, Newtonian fluids
through porous media, is often employed to model the mold filling process. Based on
Darcy’s law, the superficial velocity is related linearly to the resin pressure gradient by
the ratio of preform permeability to the resin viscosity as shown in Eq. (1.1) for one

dimensional flow.

u=-—Q-=-£-d—P: (1.1)

here P denotes resin pressure, x denotes the coordinate along the flow direction, u is the
superficial velocity, Q is the volumetric flow rate, A is the total area of cross section for
flow, K is the permeability determined by preform architecture, and u is the resin
viscosity. The ratio of K/u is called mobility, which governs the flow patterns and filling
cycle time.

The resin viscosity may vary during the mold filling process. For the S-RIM
process, where a fast reacting resin is used, a viscosity change could be caused by heat
transfer and chemical reactions. For the RTM process, where resin reaction is slow, resin
viscosity will be mainly affected by heat transfer. When the preforms used in the RTM
process have resin soluble polymeric binder on them, the dissolved binder concentration
in resin could become another major factor that affects resin viscosity. Although binder
dissolution can play a crucial role in the RTM process, it has received little attention from
researchers.

Preform permeability is a measure of the ease of penetration of the resin into the
preform. It is governed by the architecture of the preform. The preform permeability is
typically anisotropic. Mathematically it can be expressed as a second order tensor with

nine components.

 

 

iKll K12 K13
g = K21 K22 K23 (1'2)
_K31 K32 K33,

Darcy’s law can be generalized to multidimensional flow as follows.

u= - w (1.3)

1: INN

For layered preforms, three components of the permeability dyadic can be readily

identified along the principal directions.

1K1, o o
0 K2 0 (1.4)
o 0 K

”N
n

33

 

 

Usually these permeability components are classified as the in-plane permeability (KH and
K22) and the transverse permeability (K33). The in-plane permeability governs the
permeation within the plane of the layers, while the transverse permeability controls the
flow through the thickness of the layers.

Since the mat or fabric and the assembling of preforms for thicker, three
dimensional structures are particularly expensive, the injection of resin into these preforms
must be closely controlled to avoid enormous waste. Hence it is essential to understand
the permeation characteristics of a variety of thick and layered preforms. The permeation
in regular arrays of aligned, unidirectional cylinders has been studied extensively with
theoretical analysis, numerical simulations, and experimental measurements. However,

very little work has been done for more complex preform architectures.

4

The purpose of this thesis is to develop reliable and robust models to describe the
resin viscosity variation during mold filling and to predict the permeability based on the
preform architecture. The effects of binder dissolution as well as heat transfer, and
chemical reaction on resin viscosity will be considered. The work on permeability
modeling will focus on bi-directional preforms, which is a key building block for more
complicated preform architecture.

The objectives of this thesis are to:

(1) Analyze the experimental data obtained from our laboratory on RTM with binder
dissolution and develop a mathematical model with the help of this data, for the mass
transfer of binder from the preform to the resin.

(2) Combine the mass transfer model with a viscosity model and other conservation
equations to predict the resin viscosity variation in the mold. The model takes into
account three factors that affect resin viscosity: binder dissolution, heat transfer, and
chemical reaction.

(3) Perform numerical simulations with this model to provide quick estimates of viscosity
variation under various process conditions, as well as design guidelines related to binder
dissolution during mold filling.

(4) Perform numerical simulations of the three dimensional flow in idealized bi-directional
fiber arrays with the help of FIDAP, a finite element fluid dynamics analysis package
developed by Fluid Dynamics International [1], and obtain numerical results for both in-
plane and transverse permeability of bi-directional fiber arrays under various structure
arrangements.

(5) Develop analytical expressions with appropriate simplifications of the 3-D flow

 

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solutions for the in-plane as well as the transverse permeability, and test the model

predictions with experimental data.

 

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Chapter 2 BINDER DISSOL U TI ON

AND PROCESS MODEL

2.1 Binder Dissolution and Resin Viscosity

A chemical binder is used to hold the non-woven continuous strand fiber mats
together during handling. Non-woven fiber mats do not need to be precisely aligned on
top of one another as the directional, woven glass fabric. Hence, the production time can
be greatly reduced. The binder is applied by spray coating it onto the fiber mat. Because
of the spraying method used, the binder is present in small pockets on the fiber bundles
rather then being evenly spread across the fibers. The binder is usually a thermoplastic
polymer which allows thermoforrning of glass fiber mats into preforms. The binder is
different from sizing that is applied on glass fiber to protect the fiber itself from damage
and to improve adhesion between the fiber and the resin.

A typical binder used in thermoformable glass mats is a thermoplastic polyester
of low molecular weight. Such binders are soluble in unsaturated polyester resin or vinyl
ester resin formulations. A binder that dissolves in resin allows faster wet out of the
preform. Dissolved binder in the resin also leads to reduced shrinkage -- see Owen et al.

[2]. The solubility of the binder in resin is varied for different applications;

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thermoformable glass fiber mats contain 4 to 10 weight percent of a thermoplastic
polyester binder. Uneven dissolution of binder in resin within the filled mold may occur
with the lowest concentration near the inlet and the highest near the other end. This will
cause a distribution of properties such as stiffness in the molded part [2]. It may also lead
to uneven shrinkage that generates large thermal stresses in the part. Also, the reaction
rate and exotherm may be decreased by the presence of the thermoplastic binder as
indicated by the work of Lee and coworkers [3] on low profile additives. Hence, it is
important to predict the distribution of binder within the resin during the filling process.

Another consequence of binder dissolution is an increase in the resin viscosity.
Backes et al. [4] have reported that the viscosity of a vinyl ester resin increases nearly two
fold when the binder concentration in the resin reaches about 5 wt%, as shown in Figure
2.1. Also the dissolution time is found to be much less than the gel time of RTM resins,
so that binder dissolution will have significant effects on the viscosity profile during mold
filling.

As shown in Eq. (1.1) increased resin viscosity will require a higher injection
pressure for the same injection rate. This may give rise to preform compaction effects
during mold filling -- see Han et a1. [5] and Mishra and Jayararnan [6] -- which will
hinder wet-out and generate inhomogeneities. Finally, increasing viscosity or decreasing
mobility along the injection flow direction favors the development of "fingering" flow
instabilities, which give rise to inhomogeneities in the molded part. The interlinked
causes of non-uniformities developed during mold filling are displayed in the schematic
of Figure 2.2. This figure displays three causes of viscosity variation -- binder dissolution,

mold wall heating, and cure exotherm during reactive filling. Hence, it is important to

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predict variations in resin viscosity (mobility) during mold filling.

Numerical modeling of mold filling process has been studied by many researchers
in recent years [7]. Computer programs have been developed to predict flow through
complex shaped fiber preforms [8-10]. Some mold filling simulations incorporate both
heat transfer and chemical reaction aspects [11-13]. These address resin flow through
anisotropic fiber preforms, heat transfer between resin and fiber, and resin curing.
However, not enough attention has been given to resin viscosity as an intermediate
variable to control the flow uniformity. No previous model has included the binder
dissolution process. The model described in this thesis combines a mass transfer model
for binder dissolution and other conservation equations to estimate the variation of resin

viscosity due to binder dissolution, mold heating , and resin curing.

2.2 Resin Viscosity Model

In order to take into account the three factors that affect the resin viscosity, the
first step of modeling is to describe the resin viscosity as a function of binder
concentration, temperature, and conversion for mold filling processes coupled with binder
dissolution. For the Derakane vinyl ester resin used in our lab, the following expression

is found to fit experimental measurements [4],

“(Taaacfl = "0(1) Xf1(Cb) xf2(a) (2'1)

where ti

here T d
0,. gel c
\iscosity

conversic

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hereKm.

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flow {hm}
trailsfer Ct

11

where the temperature, binder concentration, and conversion effects are

po(T) =exp(-15.3+§2%£) (mPa—s) (2.2)

2.3
f1(cb) =1-0+0-1841 c,+8.2973 x10'3cg+1.4278 x10"3 cg ( )
a (-3.32x_“.-o.66)
f2(a) =(1-——) a. (2.4)
a

8

here T denotes temperature, cb, binder concentration in resin in wt%, or, conversion, and
org, gel conversion which is 0.015 for the Derakane vinyl ester resin. The experimental
viscosity data and the predicted viscosity from Eq. (2.1) are plotted in Figure 2.3 for zero

conversion case.

2.3 Binder Dissolution Model
An expression for the rate of binder dissolution, R, has been developed first from

the static experiments [4],

Rb=Kmam(bw,-licb) (25)

here K, is the mass transfer coefficient, am is the surface area per unit volume, b is
solubility coefficient, wb is binder concentration on fiber (g/cm3 fiber volume), 0b is binder
concentration in resin (g/cm3 resin volume), and 4) is porosity. The dissolution data for
flow through a rectangular mold [4] is fitted with the following correlation for mass

transfer coefficient between fluid and porous media (see Smith [14]),

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here u is
resin d.
data fittir

given by

Here K is

the mass

Where

Eq. (2.8)

binder dis

13

Kmd)

u

 

dpu
= _p'-'2/3_P_" (2.6)
131(pD) ( p )

here u is superficial velocity, p is density, D is the molecular diffusivity of the binder in
resin, tip is the fiber bundle diameter, [3, and n are constants to be determined through
data fitting. The change of diffusivity with temperature can be found from the correlation
given by Bird et al. [15].

KT

2.7
61tRAu ( )

 

Here K is the Boltzman constant, and RA is the diffusive radius of binder particles. Finally,

the mass transfer rate can be written as

 

 

dpu
R: le-annbw_ 4’ (2.8)
b Bam¢(p1) ( u )( b (1_¢)Cb)
where
B=B< “ )2” (2.9)
161tRA

Eq. (2.8) will then be used with the following component balance equations to predict

binder dissolution during mold filling process.

¢acb+ % 1‘¢R (2.10)

aw
b = 4% (2.11)

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here B i:
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14
2.4 Process Model

2.41 Momentum balance
For resin flow through continuous strand glass mats (isotropic in the flow plane),

Darcy’s law is used instead of the momentum balance equations (see Eq. (1.1)).

2.42 Energy balance

Both heat transfer between resin and fiber and resin and heated mold wall are

assumed to be predominantly due to convection.

pcpcbag +pC,u%;T= -¢QB+¢R,(-A H) +0., (2-12)
6T
(p C,),(1 «Iv-3" =4>Q. (2.13)

Here T is temperature, Cp is specific heat, t is time, it, is the volumetric heat transfer
coefficient between resin and fiber, subscription B stands for fiber property, Rc is reaction

rate, AH is heat of reaction, and Qw is heat flux from mold to the resin and fiber bed,

h
0,. B(T,, T)

here B is mold thickness, Tw is mold wall temperature, T and hw is heat transfer
coefficient between fluid and mold wall for flow through porous media. Bejan [16]

obtained the following expression from boundary layer analysis.

QBistt

here TB

fiber dill

2.43 M:

here R: is

balance €(

2‘44 The i
It i.

15

 

h
w =0.564(—k Ere (2.15)
pCp pCp x

QB is the volumetric heat flux between resin and fiber,
QB=hv(T— TB) (2°16)

here TB is the fiber temperature, by is the volumetric heat transfer coefficient between

fiber and resin ( Lin et a1, [11]),

 

h
(‘3 =0.0917+0.0195u(x10‘2m/5ec) (2.17)
P P

2.43 Mass balance

The continuity equation in one dimensional form is

_ :0 (2.18)

(bigwithw (2.19)

here Rc is the rate of reaction, which depends on reaction kinetics of the resin. And the

balance equations for binder concentration are listed in Section 2.3.

2.44 The boundary conditions

It is assumed in this model that no reaction occurs before resin enters the mold.

Hence one has

where T

atthefl

Mule fc

Other ch;

The aboxr
u Varies I

I"01' radia

0: =0 (2-20)

where Ti is initial resin temperature, and who is initial binder content on the fiber.. While
at the flow front
x =xf , T3 = Tao
We = Wbo (2.21)
P = 0

here TBO is initial fiber preform temperature, and xf is the flow front position,

2.45 The numerical scheme
The above set of equations is solved by the method of characteristic technique[17].

Eq. (2.11) and Eq. (2.13) are numerically integrated along the characteristic line of,

t = constant (2-22)

While for Eq. (2.10), (2.12), and (2.19), numerical integration is performed along the

other characteristic line,

x = g t + constant (233)

The above characteristic lines are for one dimensional rectilinear flow with constant 11. If
It varies with time, a stepwise u profile that is close to the real u variation could be used.
For radial flow with constant flow rate, the characteristic lines are,

t=constant

r = f3 dt + constant (2'24)
4’

17

where r is the radial distance from the center gate and u is a function of r.

2.5 Model Validation and Prediction of Binder Dissolution

The present model is confined to one-dimensional flow with negligible heat
conduction and dispersion, but it takes into account all three factors that will affect resin
viscosity. Such a model can be solved by the method of characteristics [17]. Calculations
with this model thus provide quick estimates of viscosity variation under various process
conditions. The following simulations are done to check the binder dissolution model and
the effects of such simplifications. The preform dimensions for the rectangular mold are
15.56 cm x 56.52 cm x 0.3175 cm (see Figure 2.4).

Three sets of experimental binder concentration data from both isothermal and
non-isothermal filling experiments with the rectangular mold [4] have been used to
determine the mass transfer parameters, [3 and 11 Eq. (2.8). The isothermal filling was done D
at 23°C. For the non-isothermal filling, the resin inlet temperature is still 23°C, while the
mold wall temperature is at 40°C and 60°C, respectively. The variation of the flow rate
during the filling experiments has also been considered by using flow velocities varying
stepwise with time. The model predicted outlet binder concentration change with time is
plotted in Figure 2.5 with the experimental data (from Backes [4]). The empirical
constants, [3 and n are determined by fitting the non-isothermal experimental data with
40°C mold wall temperature. The best fit yields 1.374 x 10 "° for B and -0.95 for n. The
same constants are used to predict the binder concentration transients for the other two
cases. It can be seen from Figure 2.5 that the numerical results match well with the

experimental measurements. The deviation from the experimental data for the isothermal

 

 

 

 

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Heated Mold Wall

Figure 2.4: The flow scheme for mold filling

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numerica
temperar
With an 2
conductic
PTOIides

T.
Simulatio
resin inle
i“creases
“Defiant
Concentra
dirCCilOn
H0n.i50th‘
concentra
Iemperam

1H ICSin. '1

20

nm is less than 5% and the deviation for the non-isothermal run is less than 10%. Hence
the mass transfer model for binder dissolution in Eqs. (2.5) - (2.8) is valid.

In order to explore the effect of ignoring conduction in the energy balance, another
simulation has been carried out for a case studied by Lin et al [11] -- filling of a center-
gated disk shaped mold with palatinol oil. The mold dimensions and process conditions
are the same as those in [11]. Results of the present Simulation are compared with those
given in [11]., where a two-dimensional model with heat conduction is used to generate
numerical results of temperature distribution. Figure 2.6 shows that the comparison of
temperature transients at two different locations in the mold. The same trend is shown
with an average difference less than 15%. This shows that despite the omission of heat
conduction, the present one-dimension model with mold wall heating as a source term
provides good estimates of the temperature profiles.

The binder concentration profile in the mold for a 45 seCond isothermal filling
simulation is shown in Figure 2.7. The simulation was done on the rectangular mold. Both
resin inlet and mold wall temperatures are 23 °C. As can be seen, the binder concentration
increases along the filling direction. The pattern is qualitatively similar as the
experimental results [4]. Since the resin viscosity increases with increased binder
concentration, the resin viscosity in this case will increase along the injection flow
direction which favors the development of flow instability or "fingers". Another case for
non-isothermal filling is shown in Figure 2.8, where resin temperature and binder
concentration profiles are plotted together. As can be seen, the increase of resin
temperature accelerates the dissolution of binder, which yields higher binder concentration

in resin. This is also confirmed by the experiments [4].

21

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Chapter 3 PROCESS SIMULATIONS AND CONTROL

3.1 Resin Viscosity Variation

As pointed out in the previous chapter, the resin viscosity plays an important role
during the mold filling process. Resin viscosity, reaction rate, and binder dissolution rate
are affected by the resin temperature. Variation in the resin inlet temperature or in the
mold wall temperature will have effects on all the three factors that will cause resin
viscosity change, which are binder dissolution, heat transfer, and chemical reaction. The
process model developed in the previous chapter can be applied to S-RIM as well as
RTM processes with or without binder dissolution. Calculations with this model provide
quick estimates of viscosity variation under various process conditions. The following set
of simulations are done to investigate the effectiveness of using resin inlet temperature or
mold wall temperature as a control variable to affect the resin viscosity profile. All these

simulations are done on the rectangular mold.

3.11 RTM simulation without binder
Filling simulations for RTM process are performed on the rectangular mold with
a Derakane vinyl ester resin. To isolate the effects of mold wall heating, the simulations

are first done for preforms without binder. The base case has the resin inlet temperature

24

25

at 25°C and the mold wall temperature at 35°C. Then resin inlet and mold wall
temperatures areraised to see its effects on resin viscosity profile. The filling time (45
sec) for all these simulations is much less than resin gel time (about 240 sec at 50°C). The

reaction kinetics of the vinyl ester resin has been provided by Larson and Drzal [18].

 

k. l-
Rcz ,( °) (3.1)
tz-t
where
ki = 5.69x1011 exp (L?) (3.2)
and
_ 8555
2, = 1.7x10 “exp(—T—) (3.3)

Figure 3.1 shows the viscosity profiles for different mold wall temperatures. As can be
seen, the increased mold wall temperature results in a larger decrease in viscosity along
the mold filling direction. This decreased viscosity or increased mobility profile does not
favor the development of flow instability. However, raising the mold wall temperature
might also accelerate the reaction. At a mold wall temperature of 60°C there is a small
increase in viscosity at the flow front as a result of the reaction. However, for the slow
reacting RTM resins, the viscosity increase caused by reaction would be insignificant.
When a fast reacting SRIM resin is used, the increase in the reaction rate due to higher
mold temperatures could be important (see Figure 3.3). Figure 3.2 shows that raising resin

inlet temperature does not affect the viscosity profile to the same extent.

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3.12 S-RIM simulations without binder

Another set of simulations is done for faster reacting polyurethane resins with 25
sec filling time (resin gel time is about 50 sec at 100 °C). The reaction kinetics is given
by Lin et a1 [11],

-5;
R =A1e ”(af—a)"

C

(3.4)

where R is the gas constant, AI , E, , and n are constants determined from experiments.

Here af can be expressed as

“f: 0.406 e 2.49x1o3 (3,5)
The base case is with a resin inlet temperature of 45 °C and a mold wall temperature of
80 °C. The viscosity profiles along the filling direction under various conditions are
plotted in Figures 3.3 and 3.4. It can be seen from Figure 3.3 that for the S-RIM resin a
larger viscosity increase caused by reaction occurs when the mold wall temperature is
raised from 80 °C to 120 °C. In regions away from the front where conversion is small,
higher mold wall temperature still causes resin viscosity to decrease along the flow
direction. Hence, care should be taken while choosing the mold wall temperature and the
filling time for S-RIM. As shown in Figure 3.4, raising the initial resin temperature yields
lower resin viscosity. However, for a 85 °C resin inlet temperature a small viscosity

increase can be observed near the end of the mold due to the acceleration of the reaction.

29

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3.l3 RTM simulat
Experiment
significant resin vi:
of "fingers" and ii
mold wall tempe
ldecrease) along 1
be possible to ea.c
Several simulatio
Show this point.
Wlth a Stepm‘se “
Emperamre (35-
diSSOIV'Cd binder.
(See 40°C and St
local ngion of \

mold wall temp.

32 Dfiign Gui

If uneve
detrimental to 11
either in binder
a window of 01
“5”“? grad-"er,

31
3.13 RTM simulations with binder

Experimental work [4] has shown that binder dissolution in resin will cause a
significant resin viscosity increase along the filling direction, which will lead to formation
of "fingers" and inhomogeneity. It has also been shown through above calculations that
mold wall temperature can be controlled to produce a negative viscosity gradient
(decrease) along the mold. Therefore by controlling the mold wall temperature, it might
be possible to ease the increase in viscosity caused by binder concentration in the resin.
Several simulations for mold filling with preforms with resin soluble binder are done to
show this point. The viscosity profiles with three constant wall temperature levels and
with a stepwise wall temperature are shown in Figure 3.5. It shows that for low mold wall
temperature (35°C) profile, the resin viscosity will increase along the mold due to
dissolved binder. Raising the mold wall temperature will lower the increase in viscosity
(see 40°C and 50°C cases). Figure 3.5 also shows that raising the wall temperature in the
local region of viscosity increase (stepwise Tw) could be more effective than raising the

mold wall temperature uniformly in terms of reducing viscosity gradient.

3.2 Design Guidelines

If uneven binder dissolution and increasing viscosity in the direction of flow are
detrimental to the properties of the molded part, a threshold of gradient may be identified
either in binder concentration or correspondingly in the viscosity. Then one may define
a window of operating conditions for which the binder concentration gradient or the
viscosity gradient will be below the agreed upon threshold. As an example, an important

design parameter is the ratio of fill time, tfill to a characteristic dissolution time tdiss. . This

 

. - c. .E ...C ..... L. . . - . . . . C‘s

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0.. o.- lsu. ........ on.
D: an n. L.

b h b 00‘.

 

32

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ratio may be com?

 

where the front lo:

A small ratio of
enough time for
dissolved in resi
The logo
the fill time to t
Wall temperatup
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at low ratios th.
Wall temperatur.
beat has been In
Iimeto dissolutic
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a Vim“? gradie

Wm 0f the dis.

33

ratio may be computed by the following relation,

L
W =1: Kmamdx (3.6)
diss. ll

 

 

where the front location is denoted by L and mold fill time is given by
t = 1 g: (3.7)

A small ratio of filling time to the characteristic dissolution time means that there is not
enough time for the binder to be dissolved. If the ratio is large, the binder concentration
dissolved in resin will be high.

The logarithmic viscosity gradient at the flow front is plotted against the ratio of
the fill time to the dissolution time in Figure 3.6 for two simulations with different mold
wall temperatures. Figure 3.6 shows that the viscosity gradient increases with increasing
ratio of fill time to the characteristic dissolution time. The gradient increases more rapidly
at low ratios then levels off as the resin nears its solubility limit. Increases in the mold
wall temperature will result in a smaller viscosity gradient at low fill times because more
heat has been transferred resulting in a higher resin temperature. Then as the ratio of fill
time to dissolution time increases (front moving forward), the viscosity gradient eventually
becomes larger for the higher mold wall temperature than for lower mold wall
temperature since the higher temperature resin will dissolve more binder. If for example,
a viscosity gradient of zero is used as a threshold, the fill time should be less than 5

percent of the dissolution time scale at a mold wall temperature of 35 °C.

 

 

 

 

 

 

 

 

 

 

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Chapter 4 TRANS VERSE PERMEABILI T Y

4.1 Permeability Prediction

Permeability is one of the most important parameters in Darcy’s law, a commonly
used macroscale model for the mold filling process of liquid composite molding. Accurate
permeability value is a critical requirement for the success of mold simulations based on
Darcy’s law. Permeability can be obtained through experimental measurements of the
pressure drop versus the flow rate. However, these experiments are very time consuming
due to the wide range of fiber volume fraction and the variety of fiber mats available. A
model for permeability prediction provides a time efficient and cost effective means for
permeability estimation, which is essential for any simulation models based on Darcy’s
law. It can greatly reduce the number of the experiments required to characterize the
permeability of the preforms and further lead to a deeper understanding of the relationship
between permeability and the preform architecture.

The permeation in regular arrays of aligned, unidirectional cylinders has been
studied extensively with theoretical calculations, numerical analyses, and experimental
measurements. ' For these regular arrays the principal directions of the
permeability dyadic can be readily identified. Theoretical values of the transverse

permeability have been tabulated for both square and hexagonal arrays by Sangani and

35

36
Acrivos [19]. Sangani and Yao [20] have later extended this calculation to the randomly

packed arrays. Finite element analysis on both regular and random arrays has been
performed by Berdichevisky and Cai [21-23]. Phelan [24] obtained the permeability for
arrays of porous cylinder also using the finite element method. Results of FEA
simulations on arrays of elliptic cylinders, both solid and porous, have been reported by
Rangnathan and Advani [25]. Flow of generalized Newtonian fluid across periodic,
unidirectional arrays of cylinders has been analyzed by Brushke and Advani [26].
Analytical expressions for the drag force on a cylinder in flow transverse to regular
arrays of cylinders (square packed) at low porosity has been obtained first by Keller [27]

using the lubrication approximation.

lulu.

 

F=9“PU(1-%)' (4.1)

2/5

Here 1 is the pitch distance, and a is the radius of the cylinder (see Figure 4.1). Eq. (4.1)

can be transformed in terms of permeability through [28]

ELM (4.2)
K(1-<l>)
where ‘l’ is the porosity and
2
- — = —la_ 0
o-l V, 1 412 (4 3)

here Vf is the fiber (cylinder) volume fraction. Hence the transverse permeability, K, can

be expressed in term of Vf

“here V,I1
n4 The
Chmielevr
further d
measurem
correspon.
Fo

developed

Be;
expression

final forrm

h .
ere Va Is

the packing

37

 

5
5 , T _
{2.8121152 2&1- V; ’ V; 1 (4.4)
a2 911: l l 911: meax VfWIJ

where Vfim is the maximum attainable Vf for square packed arrays, which equals 0.79 or
rt/4. The validity of these analytical results has been established by the experiments of
Chmielewski et a1 [29] and Skartsis et al [30] on idealized cylinder arrays. Gebart [31]
further discovered that the predicted permeability matches with the experimental
measurements on unidirectional stitched fiber mats if an effective fiber radius
corresponding to a small cluster of filaments is used.

For arrays of cylinders at higher porosity, a cell model with free surfaces has been

developed by Happel [32]. The transverse permeability is predicted by the relation

(4.5)

 

Berdichevisky and Cai [22] have further improved this method to obtain analytical °
expressions for transverse permeability over the entire range of fiber volume fraction. The

final formula derived from an improved self-consistent method is

 

Kz=0.229(1’814 -1]
V

a

(l - W77.) [’2 (4.6)

al./V.)

 

here V, is the maximum packing or available fiber volume fraction, which depends on
the packing patterns of the unidirectional arrays.

In order to gain firrther insight into more complex structures, it is useful to

consider bi-<
of perrneabi
structures. V-
dimensional
principal an
(component:
the unidirect

Ther
[33 -35—] and
penneabiliq
A1118b). I]
fraction ran;

Obtained by

.3

They also F

Here A and
variable in l

38

consider bi-directional fiber mats, which serve as a key building block for the prediction
of permeability of more complicated, three dimensional, woven, knitted or braided
structures. Well aligned bi-directional fiber mats may be represented by periodic three
dimensional arrays with two orthogonal sets of cylinders as shown in Figure 4.1. The
principal axes of the permeability dyadic are along the axes of the two sets of cylinders
(components K, and Ky) and along the thickness direction of the stack, K1. Unlike for
the unidirectional arrays, no analytical model has been reported for this kind of geometry.
There are however some limited experimental data reported by Lee and co-workers
[33 -35] and Morse et al [36]. Lee and co-workers have done extensive measurements of
permeability on various fiber mats including bi-directional stitched fiber mats (Cofab-
A1118b). They reported both in-plane and transverse permeability data for fiber volume
fraction ranging from 0.3 to 0.55. Empirical relations for permeability prediction were

obtained by fitting the experimental data [3 4].

 

_ 3
K =272(1 Vf)

. (4.7)
V}

Kim et al [3 7] investigated the in-plane and the transverse permeation through 0/90 cloth.

They also proposed an empirical relation from experimental measurements.

K=A(BVfim-Vf) (43)

Here A and B are the fitting constants. Since the fiber volume fraction was the only
variable in these relations, they cannot be used to estimate the permeability of different

types of bi-directional fiber mats. As shown in the paper of Morse et al [35], the

39

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40

difference between the transverse permeability of two different types of bi-directional
fiber mats can be as large as 100% even at the same fiber volume fraction. Hence other
structural parameters that affect the permeation should also be considered.

The effects of preform microstructure on the permeation through bi-directional
stitched fiber mats have been studied by Kalyanaraman [3 8]. The main microstructural
characteristics of preforms were identified as (i) the number of filaments per tow, (ii) the
diameter of a single filament, (iii) the number of tows per inch in 0/90 directions, and
(iv) the stitch or weave pattern. All these and the fiber volume fraction can affect the
preform permeability, and should be taken into account in the models for permeability
prediction. This chapter will focus on the prediction of the transverse permeability of bi-
directional fiber mats. First 3-D flow simulations in idealized bi-directional fiber arrays
are performed by using the finite element method (FIDAP). Then an analytical model for
permeability prediction is developed with appropriate simplifications on the 3-D flow
solutions. Finally, this model is tested by comparing with the experimental data obtained

in our own lab [38].

4.2 Finite Element Analysis
4.21 Numerical scheme

For the idealized structure of bi-directional fiber mats shown in Figure 4.1,
periodic repeat patterns or computational cells can be identified with appropriate boundary
conditions. The computational cells for permeation transverse to the stack are depicted in
Figure 4.2. As can be seen, the cell consists of an inlet surface, Fin , the cylinder surfaces,

1"

Cy and an outlet surface, I‘m. Finite element simulations

1 , the symmetric surfaces, F

syma

 

 

 

 

 

 

42

of the flow of a Newtonian fluid through the repeat cell are performed with the help of
FIDAP [1]. The Stokes solution of the interstitial velocity field are obtained by solving

the following equations:

V-a'=o (4.9)

pV2fi=VP (4-10)

Here a is the interstitial velocity. The corresponding boundary conditions are :

17:0, on Fey,
un =0, on PM (4.11)

Pm") -P(rm = - AP

where um is the velocity component in the normal direction of the symmetric surface. A
constant pressure drop, (~AP) is specified between the inlet and outlet surfaces.

A mesh of 4000 brick elements and 4851 nodes is used for the simulations as
shown in Figure 4.3. In each brick element, the velocity is interpolated with trilinear

shape functions, (bk of the form,

¢k=-;—(lir)(1:s)(l¢t), k=1...8 (4.12)

where r, s, and t are normalized coordinates for the element, which vary from -1 to 1. The
pressure is discretized piecewise in the element, where the interpolation function wk is of

the form,

43

  

f/ ’
I ,

l
5:53:0102, , I.

‘ \““ : /"
u." 3:31; a: :+:‘\§-,'{/
.. " ,

‘. I i '. l v .
‘33”. .. ..

.- v” ‘
, _ __.-"s ‘-.———-‘~:.

'\

if
u
'0'
1'
D
§
Figure 4.3: The 3-D mesh for transverse permeation

44

1|! k =1 for r, s, t within the element

4.13
111k =0 eLsewhere ( )

The running time of a typical case is around 5 hours on a SUN 690 MP at Case Center

of Michigan State University.

4.22 Numerical results

Numerical simulations are done for both symmetric and asymmetric pitch cases.
For the symmetric pitch case, the pitches 21,, and 21y are equal. For asymmetric pitch cases
in this thesis, the ratio of (ly-a)/(lx-a) is set to 2 and 4, respectively. The Stokes flow field
are solved for Vf ranging from 0.3 to 0.75. The velocity field is depicted in Figure 4.4
for Vf at 0.6042. As can be seen, the flow is locally parallel for each value of 2 as resin
passes through the clearance of cylinders aligned in x and y-directions. Therefore the
entire 3-D flow field can be decoupled into two regions of one-dimensional flow along
the two filament (cylinder) directions. This pattern remains clear for Vf larger than 0.45.
Furthermore from the corresponding pressure distribution (see Figure 4.5), one can see
that most of the pressure drop occurs near the minimum gap clearance between the two
cylinders, where the flow is essentially one-dimensional. Therefore the above
simplification will not have significant effects on the calculation of the total pressure drop
across the repeat unit cell.

Once the flow field is solved, the values of the permeability can be obtained from
Darcy’s law (Eq. (1 .1)). All the permeability results are presented in dimensionless form,
scaled by d2, where d is the diameter of the cylinder. The transverse permeability for both

symmetric and asymmetric pitch are compared in Figure 4.6. As can be seen, Kz for

45

 

 
   

    

\ \h\\\\i\\

0‘5“ \ \ \ \\\ \
r \\\\

 

\

Figure 4.4: 3-D velocity field for transverse permeation

46

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47

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.

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_

c u .m.._........_. a. .V. .......

.u.~-..:.m.._. 5. .V.

d

   

mod
-

1AA: A . A

lALLALA 4A

 

 

m
a.»

0—

48

asymmetric pitch is lower than KZ of symmetric pitch. For 0.50 Vf, the transverse
permeability of asymmetric pitch is 30% lower than Kz of symmetric pitch. The difference
is larger at lower fiber volume fraction. At 0.50 fiber volume fraction, Kz for the

asymmetric case is 60% less than Kz of symmetric pitch.

4.3 Transverse Permeability Model
4.31 Structure parameters

As shown in Figure 4.7, Vf of bi-directional fiber arrays can be calculated from

__ it (a: lx+ay2 I?)

(4.14)
f 41x ly (aJr +ay)

 

where a,( and ay are radius of the cylinder aligned in x or y-direction. For maximum

packing, (lx=ay , and Iy = ax)

V =

M (4.15)

.rsln

As shown in the next section, the transverse permeability of bi-directional fiber arrays can

be determined by the following group of structural parameters: ax, V/me, lxlly, and a,/ax.

4.32 Decoupling of the 3-D flow field

As indicated by the numerical results, the 3-D transverse flow field can be
decoupled into two regions of one-dimensional flow mathematically. At high Vf, Eq.
(4.10) can be broken into the following two equations for 1-D flow in x-z and y-z planes

using the lubrication approximation (see Figure 4.7).

49

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mi}

“N

 

 

 

4L

 

 

 

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50

 

 

azu dP
z- _
p. 2—— for ay<z<ay
(4.16)
azu
1 =5? for a <z<a +2aJr
3y. dz y y

Here uz is the z-component of interstitial velocity. Eq. (4.16) can be easily solved to
obtain 112

u z(x) = i 95 (xz—hfl) for —ay< 2:<ay

2
“ dz (4.17)

1 dP 2
uz(y) = 2; E (yz-hzz) for ay<z<ay+20x

Here h21 and h22 are half of the gaps clearance, which vary with z.

r
- _ 2_ 23
hzr‘lx (“y Z) 1 (4.18)

hz, =1, - [af-(z -ax-ay)2]3

Upon integrating Eqs. (4.17), one can obtain two components of the pressure drop

1R
AP é- 9fl(Q/21’)pay for -a <z<a
1 5,2 y y
WWI“) (4.19)
9 l a”2
_ Tl:(Q/2 x)“ x for ay<z<ay+2ax

AP2 a 5
8 J2 (1y -ax) ’2

where Q is the volumetric flow rate

I: h:1 I: ”:2
Q= f f u,(x)dxdy= f f uz(_y)dydx (4.20)

-IN -h,, -r,r 41,,

51

Hence the total pressure drop along one repeat cell becomes

 

 

 

 

 

 

 

 

 

 

 

 

 

91: U 21 a 1/2 21 1’2
_Ap=_(AP1+AP2)= “ z x y . ,a. (4.21)
sfi (Ix—4,)”2 (1,195”
where U2 is the superficial velocity,
U = Q 4.22
2 (21.25) ( )
Hence the transverse permeability, Kz can be calculated from Darcy’s law:
* 1/2 1/2 -1
K: = - I-l U22(ax+ay) = 8fi<ax+ap [lay + lyax (4.23)
AP 911; (Ix-095,2 (ly_ax)5/2
Using Eqs. (4.14) and (4.15), one obtains
-1
(__Vf_.M)3/2 ( Vf .“+°‘ l3)3l2
é: 81/2-(1+l3) Vfinax 132*“ + meax 132+“ (4.24)
a: 91!: 13(1' Vf .BZ+B)5/2 (1- Vf .“+afi)5/2
VM 32+“ Vfinax 62+“ l
where
lJr a
a =__ and B=_)’ (4.25)
I), aJr

Hence one can determine the transverse permeability by knowing ax, V/me, l,,/1y , and

a/ax-

52

If the radius of the cylinders are the same (ax=ay=a , [3:1)

 

 

 

-1
( f 2 )3/2 ( f 2“ )3/2

5;: mfl meax 1+0: + mex 1+0: (4.26)

a 9n (1- Vf . 2 )5/2 (1- Vf . 2a )5/‘2

VM 1+ot meax 1+0: ]

Further for symmetric pitch (1x = ly = l, or = l ),
_Z_ K=__\/:(1_VV fV)5/2(_ Vf ) 3/2 (4.27)
02 97‘ meax Vfiuax

This expression is very similar to that of unidirectional arrays ( square packed ) as shown

_2 81.0 -l—VLW _Vr_)-1 (4.28)
a2 91tVfiw Vim,

4.33 Model predictions and comparisons with numerical results

by rearranging Eq. (4.4)

The transverse permeability predictions for the symmetric pitch case are plotted
in Figure 4.8 against Vf with the computed results from FEA analysis. The analytical
solutions match well with the numerical results for fiber volume fraction greater than 0.55
(< 10%). The difference between the two increases at lower V. This trend is expected
since the decoupling of the 3-D flow is only valid at high fiber volume fractions. Hence
the analytical solutions of transverse permeability provide a check on the computed results

from the numerical simulation. The comparison of model predictions and numerical results

53

for the asymmetric cases are plotted in Figures 4.9 and 4.10. The same trend is observed.
A modified model is obtained by changing the power index 3/2 in Eq. (4.26) to

3/4 in order to match the numerical results at lower Vf.

 

F (_Vf_._Z_)3/4 (i.£)3l4 -l
§=16¢§ VM 1+a + me 1+... (429)
a 911: (1_£ .Lffl (1 “i ._2_“_)5l2

L Vfim 1+0: Vfiw 1+ot

 

 

The predictions obtained from this modified model are shown in Figure 4.11 with the
numerical results for the symmetric pitch case. As can be seen, the model prediction
agrees well with the F EA results for the entire range of fiber volume fraction. The
comparison for an asymmetric case is plotted in Figure 4.12. The same trend is observed.
The transverse permeability of bi-directional fiber arrays (symmetric pitch) is
compared with that of unidirectional arrays (square packing) in Figure 4.13. At the same
fiber volume fraction bi-directional arrays have higher transverse permeability than
unidirectional arrays. This result is reasonable since for bi-directional arrays

IL (4.30)

Vfinax

Nlh
It

And for the square packed unidirectional arrays, one has

Vr (4.31)
Vfim

NIQ

Hence for the same Vf, the gap is larger for bi-directional arrays than for unidirectional
arrays. Thus the transverse permeability of bi-directional arrays is higher. Note the

maximum attainable fiber volume fraction is rt/4 for both cases. As Vf approaches to its

54

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maximum value, transverse permeability in both cases goes to zero. However the decay
rate of the transverse permeability for bi-directional arrays is faster.

Finally the effects of pitch ratio, lfly is investigated in Figure 4.14, which shows

the variation of transverse permeability with lx/ly for several different fiber volume

fractions. The model shows that the transverse permeability is lowered significantly as the

pitches along the two cylinder axes are made asymmetric. This trend is also observed in

the FEA results (see Figure 4.6).

4-4 Experimental Validation
The theoretical and numerical results of permeability presented in previous sections
are obtained for a perfectly aligned stack of bi-directional fiber arrays. However, the real
Structure of the bi-directional stitched fiber mats is different from this idealized structure.
In bi-directional fiber mats, the filaments are grouped in fiber tows, which are porous and
generally do not have a circular cross section. Compressed inside the mold, the fiber tows
are usually in elliptic shape. Hence the validity of the prediction model based on idealized

bi-directional cylinder arrays needs to be tested.

4.41 Experimental
Measurements of in-plane as well as transverse permeability of bi-directional
stitched fiber mats have been carried out in our own lab [3 3]. Three type of bi-directional
mats were used, two of which were assembled from unidirectional plies by stacking them
in the 0/90 arrangements. The third one is a bi-directional stitched fiber mats. The

specification of these materials are listed in the table below.

61

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62

Table 4.1 Specifications of bi-directional fiber mats

 

 

 

 

 

_ Mats Cross-plied UA8 Cross-plied UB6 Cofab
Tow weight 450 yards/lb 225 yards/lb
Tows/inch 13.72 9.5 6.5
Filaments/tow 1600 3200 2000
Filament diameter 19 microns 19 microns 15.5 microns

 

 

 

 

 

 

The impregnation fluids was a mixture of glycerol and water, with a viscosity range from
66 mPa-s to 230 Pa-s at room temperature. Dow Coming’s Silicone sealant and vacuum
grease were used to provide effective sealing. Compressed nitrogen was used to force the
liquid through the permeation cells. The fiber preform is cut from a large roll of fiber
mats, and the stacked preform is then transferred to the Wabash instrumental press to be
compressed. The details of the flow loop set-up, the preforming technique, as well as the
permeation cells assemblies for in-plane and transverse permeability measurements can

be found in [38].

4.42 Comparisons with experimental data
The results of the transverse permeability measurements for bi-directional fiber
mats consisting of cross plied UA8 and UB6 mats are shown in Figure 4.15 with the

model predictions (Eq.(4.19)). As can be seen, the model prediction matches fairly well

63

with the experimental data for fiber volume fraction ranging from 0.45 to 0.65. The
comparison of the analytical solution with the experimental data obtained for the Cofab
mats is plotted in Figure 4.16. Same trend is observed. Since for both cases the pitch is
symmetric, Eq. (4.27) is used to generate analytical results. The parameter a in Eq. (4.27)
is set to be the radius of the single filament, which are 9.5 microns for cross-plied mats
and 7.75 microns for the Cofab stitched mats.

The choice of using the radius of a single filament can be explained by the
microstructure of the bi-directional fiber mats. Before being placed into the mold, the
fiber mats were consolidated in order to attain high fiber volume fraction. In the
compressed fiber mats, the fiber tows are flattened and the large pore channels between
the tows are eliminated. This has been confirmed by the micrographs obtained from
Scanning Electron Microscopy [3 8]. Since fiber tows become indistinguishable as the
preforms are compressed, the flow through the preform network take place only in gaps
between the filaments. Therefore the suitable length scale in the permeability model
should be the radius of the single filament rather than the radius of the whole fiber tow.

The permeability values of the Cofab is lower than that of the cross-plied UA8 and
UB6 since Cofab mats have lower filament radius which leads to smaller gaps between
them. Furthermore the predicted permeability values are slightly lower than the
experimental measurements at Vf below 0.55. This is because at lower fiber volume
fraction, the fiber tows are not completely flattened. The gaps between the fiber tows are
greater than the average pore size between the individual filaments. Hence more flow
occurs through the preform than predicted by the model, resulting in higher permeability.

The experimental data are compared with the some of the existing permeability

64

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65

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66

models as shown in Figure 4.17. The empirical constants in these models are determined
by best fitting the experimental data. As can be seen, none of these models can provide
a close match with the experimental data. The success of the proposed model is not
surprising since this model is specific to preforms with bi-directional architecture, and
thus taken into account the preform geometry without lumping the them into model
parameters which may vary for different type of fiber mats. In addition to preform
geometry, the model also includes the effect of the preform microstructure, which is
through the filament radius, a relevant unit in the compressed preforms.

Finally, the model is used to predict the transverse permeability of bi-directional
mats with asymmetric pitch. The experimental data for bi-directional mats with
asymmetric pitch are reported by Morse et a1 [36]. The model prediction and the
experimental data are shown in Table 4.2. The fiber volume fraction is 0.37 for both mats.

The ratio of the pitch distance, a is calculated from the ration of the tow ends in 0/90

 

    
 
 
   

 

directions.
Table 4.2 Transverse permeability for asymmetric pitched
bi-directional fiber mats
a ( lx/ly) Kz experimental
Mat 1 3.48 16 darcy ll darcy

 

32 darcy

 

 

 

 

One can see from Table 4.2 that the trends predicted by the model is supported by the

 

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68

experimental data. However due to the flattening of the fiber tow in compressed preform,
the definition of pitch distance is not as clear as that of the idealized regular arrays. More
work has to be done so that proper structure parameters can be included in the model.
In conclusion, an analytical model based on the lubrication approximation is
developed to predict the transverse permeability of bi-directional fiber mats. The model
prediction is first confirmed by the numerical results at high Vf. Furthermore the
transverse permeability predictions match well with the experimental measurements on bi-
directional fiber mats when the filament radius is used as the length scale. The usage of
the filament radius rather than the fiber tow radius is supported by the study of the
microstructure of the compressed fiber preforms. Hence the analytical model developed

for the transverse permeability is valid for the real fiber preforms.

Chapter 5 IN-PLANE PERMEABILI TY

5.1 In-plane Permeability Prediction

As for the transverse permeability, the longitudinal permeability of unidirectional

 

fiber arrays has also been studied by many researchers. The theoretical values of this

quantity for square and hexagonal packed unidirectional arrays are obtained by Drummond

“an. "m “ ' \m'l- I:

and Tahir [39]. An analytical solution for the longitudinal permeability at low fiber
volume fraction was obtained by Happel [32].
K
_1=_1_ 4Vf_VfZ_3+21n_l. (5.1)
02 3 Vf Vf

Berdichevisky and Cai [22,23] have later obtained an improved analytical
expressions by using the improved self-consistent method (see Eq. (5.2)). This expression

is also valid at high Vf.

Kx=0.211(Va-0.605)[0.9077f] (roam—Vi]

a

(5.2)

a

+0.292(o.907 - Va) ( Vf)‘1'57(1 - Vf)155

For bi-directional fiber mats, Lee and co-workers [34, 40] proposed the following

empirical relation by fitting the experimental data.

69

70

592(1-

Kx=47e Vf)+[2018(1-Vf)—592]u

(5.3)
K=5K
5

y I

This empirical relation is only valid for the type of fiber mats used in their experiments.
In the following sections, an analytical expression is proposed for in-plane
permeability of bi-directional fiber mats. The model is developed based on the
simplifications of the 3-D flow through the idealized structure (see Figure 4.1). The model

prediction is compared with the experimental data obtained in our own lab [3 8].

 

5.2 Finite Element Analysis
5.21 Numerical scheme

The FHA analysis of the in-plane permeation is also carried out on repeat cells
with appropriate boundary conditions as for the transverse permeation. The computational
cells for in-plane permeation is depicted in Figure 5.1. In this case, periodic velocity
boundary conditions are applied at the inlet and outlet surfaces. Eqs. (4.9) and (4.10) are

solved with the boundary conditions listed below.

0, on I‘m
0, on I‘M

Pa‘m) —P(I‘m = -AP

:7
u
" (5.4)

As shown in Figure 5.2, a mesh of 8000 brick elements and 9451 nodes is used

to perform numerical simulations using F IDAP. The running time of a typical case is

72

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5.22 Numerical results

The Stokes flow field are solved for Vf ranging from 0.3 to 0.75. The velocity
distribution of a typical case of in-plane permeation is shown in Figure 5.3. As can be
seen, the flow occurs mainly along the fiber axes aligned in x-direction, along which the
pressure drop is specified. The corresponding pressure contour is plotted in Figure 5 .4.
Finally, the contour of the velocity at inlet and outlet surfaces are shown in Figure 5.5.
The periodic boundary condition is observed.

The in-plane permeability of asymmetric pitch arrays, K, and Ky are shown in
Figure 5.6, along with the in-plane permeability of the symmetric pitch, Kp . Figure 5.6
shows significant difference between the in-pane permeability of symmetric and
asymmetric pitch cases. At 0.70 V,, K, is 70% higher than K1, and Ky is 14% less than
KP. The difference is even bigger at lower fiber volume fraction. For 0.50 fiber volume
fraction , K,( is three times as large as KD and Ky is 30% less than KP.

The numerical results of the in-plane and the transverse permeability for symmetric
pitch arrays are shown together in Figure 5.7. The two curves intersect at 0.60 fiber
volume fraction. The transverse permeability is higher than the in-plane permeability at
fiber volume fractions less than 0.60. The ratio K/Kp is of order one. For fiber volume
fraction higher than 0.60, the in-plane permeability is much larger than the transverse
permeability. The anisotropy (Kt/Kp) is much stronger. For asymmetric pitch, the in-plane
permeability IQ and Ky are compared with the transverse permeability, K2, in Figure 5.8

and Figure 5.9, respectively. Figure 5.8 shows that IQ is always higher than Kz for fiber

74

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Figure 5.5: Contour of the periodic velocity boundaxy condition

 

 

 

 

 

 

 

 

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81

volume fractions ranging from 0.50 to 0.70. However, K2 is higher than Ky for Vf lower
than 0.57. The variation of the anisotr0py (K/Kwnm, where Kp.m,n=Ky) follows the same

trend as symmetric pitch cases.

5.3 In-plane Permeability Model
5.31 Flow through a channel

As one can see from Figure 5.3, the flow occurs mainly along the fiber axes for
in-plane permeation. Hence one can consider the flow is along a channel, the cross section
of which is shown in Figure 5.10, where the cylinders aligned transverse to the flow

direction are considered as a solid wall. The friction loss in this case can be expressed as

 

Pvt-105-1: —AP 20;.

(5.5)
Lx p U:

f=C(

where Dh denotes the hydraulic diameter, f, the friction factor, c, the shape factor, Lx, the
distance over which the pressure drop applied, and U”, , average interstitial velocity

component in flow direction. Ui,‘ can be obtained from Eq. (5.5) as

.1 -AP

 

U. "(203) (5.6)

u
1

Using Darcy’s law (see Eq. (1.1) ), one can solve for the in-plane permeability

 

=1; .l_fi_ JV” 2 5.
K,‘ _APqu cl+[3(1 fi2+a Vf)(ZD,,) (7)

82

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83
Note that

 

Ux=i(l-a+aBVf)U.

5.8
1+3 B,” ( )

From the definition, the hydraulic diameter is

 

2a,(1- “a ‘3 V)
52*“
Db:
(0.5+ ““5 V)
13%: f

(5.9)

 

Hence the in-plane permeability can be expressed in terms of Vf as

(5.10)

Here the in-plane permeability, K, is determined by a,, c, lx/ly, g/ax, and Vf. If the radius

of the cylinders are the same ( a,( = ay = a, [3 =1 ),

 

 

 

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a2 02 C(0.5+Vf)2

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Kozeny constant.

84

5.32 Model predictions and comparisons with the FEA results
To obtain the analytical solutions for in-plane permeability, the shape factor, c
needs to be determined (see Eq. (5.7)). In theory the shape factor can be calculated once
the velocity field is solved. In order to provide quick estimates of the permeability, c is
determined by fitting the numerical results from FIDAP for the symmetric pitch case. For
=20, the analytical solutions agree well with the numerical results for fiber volume
fraction varying from 0.3 to 0.75 as shown in Figure 5.11. The same value of c is used
to yield in-plane permeability prediction for an asymmetric case. Both the analytical and
numerical solutions are plotted in Figure 5.12 against fiber volume fraction. As can be

seen, the agreement between the two is good.

5.4 Experimental Validation

The analytical solutions of the in-plane permeability for bi-directional fiber mats
are checked against the experimental data provided by Kalyanaraman [3 8]. When the
radius of the filament is used in the model as for the transverse permeation case, the
predicted in-plane permeability is substantially lower than the experimental measurements.
This can be explained by the microstructure of the compressed fiber preforms. As shown
by Scanning Electron Microscopy, there exists considerable dislocations and
inhomogeneities in the filament packing for the compressed fiber preform [3 8]. The non-
uniformities in the pore size create continuous channels of diameter larger than the
average filament diameter. Hence the real pore size should be represented by a length
scale larger than the single filament radius. The model predictions are plotted with the

experimental data in Figures 5.13 and 5.14 with an effective fiber radius about eight times

 

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larger than the filament radius. As can be seen, the predicted in-plane permeability values
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this case are calculated based on the shape of the flow channel (see Figure 5.10). This
result is supported by the preform microstructure. For real fiber mats, the structure of bi-
directional fiber mats is composed of layers of unidirectional mats aligned along and/or
transverse to the flow direction. For Vf larger than 0.45, more than 90% of flow occurs
in layers where filaments aligned along the flow direction. Hence, the relation deve10ped
for unidirectional fiber arrays is more suitable in this case. However the determination of
the effective fiber radius still depends on the experimental data. Therefore more work

needs to be done to link the effective fiber radius to the preform architecture.

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Chapter 6 CON CL USIONS AND

RE C OMMENDA T I ONS

6.1 Conclusions
The conclusions of this thesis are:

(1) An easily solved process model has been developed for controlling the resin viscosity
and mobility gradients during mold filling of the liquid molding processes (S-RIM and
RTM). The model predicts resin viscosity variations due to dissolved binder content, mold
heating, and chemical reaction. The model predictions on binder dissolution matches well
with experimental data under several different mold wall heating conditions. This means
that the model describes the interaction between non-isothermal effects and binder
dissolution effects well. Finally, design limits on the ratio of filling time to a characteristic
dissolution time have been developed for a case with this model.

(2) Raising the mold wall temperature is found to be an effective control means to reduce
the extent of viscosity increase along the filling direction as long as the resin is still far
from its gel state. However, raising the resin inlet temperature may cause an increasing
viscosity in the direction of flow when the resin inlet temperature is higher than the mold

wall temperature. Simulations also show that a variable mold wall temperature profile

92

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93

could be more effective in reducing the extent of resin viscosity increase than a uniform
wall temperature profile when binder dissolution is coupled with the mold filling process.
(3) The effect of preform architecture on permeation through bi—directional fiber mats has
been investigated with finite element analysis (FIDAP). The variation of permeability
with fiber volume fraction ranging from 0.3 to 0.75 has been computed along different
directions of periodic, repeat cells of bi-directional cylinder arrays. Arrangement with
symmetric pitch as well as asymmetric pitch have been compared for permeability. The
FEA analysis shows that the transverse permeability is lowered significantly as the pitches
along the two cylinder axes are made asymmetric. This is supported by the experimental
data reported by Morse et al. The anisotropy of permeability for these symmetric arrays
( K; K, ) is of order 1 until a fiber volume fraction of 0.65 is reached; after this, the
anisotropy is strong. As the array is made asymmetric (different pitches along the two
axes directions), similar trends are seen for the ratio K/ KW"; it is of order one until a
fiber volume fraction of 0.65. It must be emphasized that the predictions are for perfectly
aligned stacks. Fiber mats should be aligned exactly on top of each other in order to test
these predictions.

(4) For transverse permeability prediction, an analytical model based on the lubrication
approximation is developed to predict the transverse permeability of bi-directional fiber
mats. The model prediction is first confirmed by the numerical results at high Vf. A
simple modification of the analytical expression yields prediction that matches the
numerical results over the entire range of fiber volume fraction. Furthermore the
transverse permeability predictions match well with the experimental measurements on bi-

directional fiber mats when the filament radius is used as the length scale. The usage of

94

the filament radius rather than the fiber tow radius is supported by the study of the
microstructure of the compressed fiber preforms. Hence the analytical model developed
for the transverse permeability is valid for the real fiber preforms.

(5) A modified Carman-Kozeny type of equation is also derived for the prediction of in-
plane permeability of the bi-directional arrays. It has been found that a constant shape
factor can be used to fit the numerical results obtained from FEA analysis for symmetric
pitch case over the entire range of fiber volume fraction. Using the same constant, the
model prediction for a asymmetric pitch case also agrees well with the numerical results.
However, the analytical solutions only agree well with the experimental data for Vf larger
than 0.575. Furthermore the classical Carman-Kozeny equation provide better match
between the analytical solutions and the experimental data due to its closer representation
of the real fiber structure. For both cases an effective fiber radius corresponding to a small
cluster of filaments is used. The study on the preform microstructure reveals that
dislocations and non-uniforrnities in the filament packing create continuous channels
which are wider than the average gaps between the filaments. More work needs to be

done to link the effective fiber radius to the preform architecture.

6.2 Scope of Future Work

(1) To develop models for the prediction of permeability at low fiber volume fraction
(<0.45).

(2) To study the effects of fabric stretching during the preforming process.

(3) To develop models for the permeability of woven fabrics.

(4) To perform mold filling analysis using the information gained from the permeability

 

 

95

study as well as the resin viscosity behavior and its reaction kinetics.
(5) To conduct molding runs to confirm the results from the numerical simulations and
to develop guidelines on proper locations of gate(s) and vent(s), and appropriate resin

injection rates.

 

 

BIBLIOGRAPHY

 

 

BIBLIOGRAPHY

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99
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