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1m cJCIRCJDanOuupfi-nu

 

VISCOUS FINGERING DURING REACTIVE FILLING: A THEORETICAL STUDY
OF THE PHENOMEN ON IN LIQUID MOLDING

By

Chilukuri Nageshwara Satyadev

A DISSERTATION

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

DOCTOR OF PHILOSOPHY

Department of Chemical Engineering

1998

ABSTRACT

VISCOUS FINGERING DURING REACTIVE FILLING: A THEORETICAL STUDY
OF THE PHENOMEN ON IN LIQUID MOLDING

By

Chilukuri Nageshwara Satyadev

During the mold filling stage in liquid molding operations, the resin
flowing through fiber “preforms” is continuously polymerizing, leading to a changing,
adverse viscosity gradient in the filled region. The effects of resin age distribution and the
polymerization kinetics on the stability of mold filling flows are investigated here. When
the extent of reaction is significant, the driving force for resin penetration into the fiber
bundles has been found to increase as the viscosity increases, thus improving the wetting
of fibers. But the adverse viscosity gradient generated has an inherent tendency to generate
fingers. A linear stability analysis of the equations describing the mold filling operation
has been performed to derive criteria for growth/decay of fingers. When the derivative. S,
of the reaction rate with respect to conversion is negative, fingers of large widths are
damped; conversely, when this factor S is positive, disturbances of all sizes grow without
attenuation. Small fingers are not stabilized by the reactive coupling. This is attributed to
the relatively long time scale of chemical reaction, in comparison to the time scale of
finger growth: fingers grow faster than reactive coupling can act to damp them out.
Marginal stability plots have been computed to illustrate the effects of several resin
parameters like gel time, activation energy and fill time/reaction time on the fingering
phenomenon. It has also been found that the combined effect of chemical reaction and

mass dispersion is to stabilize disturbances of both large and small wavelengths; this is in

contrast to the cases where dispersion alone damps out narrow fingers and reaction alone

helps damp out large fingers.

ACKNOWLEDGMENTS

It has been a great experience working with my advisor, Dr. K. Jayaraman of the
Chemical Engineering Department, Michigan State University. I thank him for his help in
every stage of my graduate studies at MSU; his contribution to my doctorate degree is
immeasurable and his immense patience and readiness to assist me and advise me is
gratefully acknowledged.

I thank Dr. Charles Petty of the Chemical Engineering Department at the Michigan
State University for his valuable inputs and insights and for his help whenever
approached. The helpful comments of the other members on the advisory committee are
also acknowledged with thanks: Dr. Lawrence Drzal, CMSC/MSU and Dr. C.Y. Wang,
Department of Mathematics, MSU.

I was fortunate to have been taught by several excellent teachers at MSU and am
grateful to them all for the knowledge they imparted during my interaction with them.

Thanks are due to my colleagues from our research group over the last few years,
for their company and cooperation and helpful comments at the group meetings and
otherwise.

I acknowledge with great respect and gratitude the role played by my parents, C.
Shyamalamba and C. Ramachandra Murty, both before and during this endeavour. They
are two of the greatest human beings I have ever known. My thanks to my brother, C.K.

Mohan, and sister-in-law, Sudha Kailar, for their moral support and guidance throughout.

My wife, Sarada Chilukuri, deserves all praise for her patience and care in dealing
with me during these graduate student years. Thank you, Sarada, for believing in me! And
special thanks to my son, Srirarn Mukunda Chilukuri, for brightening up my environs

during the home stretch.

TABLE OF CONTENTS

LIST OF TABLES ................................................. viii
LIST OF FIGURES ................................................. ix
NOTATION ....................................................... xii
CHAPTER 1 INTRODUCTION ........................................ 1
1.1 Introduction And Motivation .................................. 1

1.2 Background ............................................... 4

1.2.1 Liquid Molding Operations ............................ 4

1.2.2 Effects Of Fluid Rheology In RTM ..................... 4

1.2.3 Viscous Fingering ................................... 5

1.3 Liquid Molding And Viscous Fingering ........................ 16

1.4 Summary ................................................ 17

CHAPTER 2 POLYMER SYSTEMS IN LIQUID MOLDING: A REACTION

KINETICS POINT OF VIEW ............................. 18
2.1 Introduction .............................................. 18
2.2 Polyurethanes ............................................. 23
2.3 Polyamides ............................................... 25
2.4 Epoxies .................................................. 27

CHAPTER 3 VISCOSITY VARIATIONS DURING REACTIVE FILLING OF

FIBER PREFORMS .................................... 30
3.1 Introduction .............................................. 30
3.2 Role Of Resin Rheology In Liquid Molding ..................... 30
3.2.1 Resin Rheology And Apparent Permeability ............. 31
3.2.2 The Mathematical Model ............................ 37
3.2.3 Discussion Of Computed Results ...................... 41
3.2.4 Summary ......................................... 50

CHAPTER 4 VISCOUS FINGERING DURING REACTIVE FILLING OF FIBER

vi

PREFORMS: EFFECT OF CHEMICAL REACTION .......... 52

4.1 Introduction .............................................. 52
4.2 Problem Formulation ....................................... 55
4.2.1 The Governing Equations ............................ 55
4.2.2 Scaling ........................................... 60
4.2.3 The base state equations ............................. 62
4.2.4 The base state solution .............................. 64
4.2.5 Stability analysis ................................... 73
4.3 Numerical Approach ....................................... 80
4.4 Results and Discussion ..................................... 84
4.4.1 Isothermal Reaction ................................ 84
4.4.2 Adiabatic Reaction ................................. 91
4.5 Mechanism of stabilization ................................. 107
4.5.1 Effect of S on the growth rate ........................ 107
4.5.2 Stabilization of large fingers ......................... 109
4.6. Summary ............................................... 1 12

CHAPTER 5 VISCOUS FINGERING DURING REACTIVE FILLING OF FIBER
PREFORMS: EFFECTS OF DISPERSION AND REACTION . . 114

5.1 Introduction ............................................. 114

5.2 Model Problem And Equations .............................. 115

5.3 Scaling And Dimensionless Form Of The Equations ............. 119

5.4 Stability Analysis ......................................... 121

5.5 Initial Growth Rate ....................................... 123

5.5.4 Results .......................................... 130

5.6 Summary ............................................... 140
CHAPTER 6 CONCLUSIONS ....................................... 143
6.1 Summary And Conclusions ................................. 143

6.2 Recommendations for future work ........................... 146

APPENDD( A: Exchange of stabilities analysis for mold filling with chemical

reaction, without dispersion .............................. 147
APPENDIX B: Derivation of component mass balance equations ............ 151
BIBLIOGRAPHY ................................................. 156

vii

LIST OF TABLES

Table 4.1 Thermal, kinetic and rheological data for the resin and glass fibers ..... 59

Table 4.2 Scaling factors for different variables ............................ 61
Table 4.3 Computational results for isothermal reaction ..................... 87
Table 4.4 Computational results for adiabatic reaction ....................... 99
Table 4.5 Cutoff wavenumber data for adiabatic reaction ................... 100
Table 5.1 Computational results from eq. (5-36) .......................... 133
Table 5 .2 Cutoff wavenumber data at several values of 6 .................... 136

viii

LIST OF FIGURES

Figure 1.1 Mechanism of finger growth due to viscosity gradients ............ 7
Figure 1.2 Typical molded plaque: to illustrate the generation of fingers. White -
older resin; Shaded: fresh resin [after Losure, 1994] .............. 11
H - Figure 1.3 Filling with no reaction or dispersion [after I-Iickernell and Yortsos, 19873]
Figure 1.4 Filling with dispersion, no reaction [after Tan and Homsy, 1986] . . . 15
Figure 2.1 Schematic of a RIM Machine [after Macosko, 1989] ............. 19
Figure 2.2 Types of chemical reactions classified by S behavior ............. 21
Figure 2.3 Viscosity changes during a RIM cycle ......................... 22
Figure 2.4 Reaction scheme for an Epoxy system ........................ 28
Figure 3.1 One-dimensional flow of resin, transverse to fibers, in an end-gated “10:21
Figure 3.2 Rheology of Devcon epoxy resin ............................. 33
Figure 3.3 Rheology of Derakane resin .......... z . . :..- .................. 34
Figure 3.4 Apparent transverse permeability Vs viscoelasticity-Hexagonal array 36
Figure 3.5 Resin penetration into fiber bundles .......................... 42
Figure 3.6 Dimensionless superficial velocity at various flow front positions . . . 43
Figure 3.7 Location of y; at various positions of the flow front .............. 44
Figure 3.8 Location of yn at various positions of the flow front .............. 45
Figure 3.9 Velocity reduction factor Vs LMMDR [f,I] ..................... 47
Figure 3.10 LMMDR [f,I] at various positions of the flow front - regime II ..... 48

ix

Figure 3.11

Figure 4.1

Figure 4.2

Figure 4.3

Figure 4.4

Figure 4.5

Figure 4.6

Figure 4.7

Figure 4.8

Figure 4.9

Figure 4.10

Figure 4.11

Figure 4.12

Figure 4.13
Figure 4.14

Figure 4.15

Pressure profiles at various locations of the flow front ............ 49
Schematic of reactive filling, frozen at time “t”, in an end-gated mold 56

Non-dimensional reaction rate for polyurethane resin: isothermal and
adiabatic cases, To = 325°K ................................. 66

Non-dimensional viscosity for polyurethane resin: isothermal and
adiabatic cases, To = 325°K ................................. 68

Base state conversion profiles: isothermal and adiabatic cases, T0 = 325°K
70

Base state dimensionless pressure profiles: adiabatic case, ER = 53, En =
41, (13:1: 0.65, T0 = 325°K ................................. 72

S, V profiles for isothermal filling: ER = 53, En = 41, age] = 0.65, T0 =
325°K ................................................. 74

S, V profiles for adiabatic filling: ER = 5, 5;: = 41, age] = 0.65, T0 =
325°K ................................................. 75

Schematic of the numerical scheme - orthogonal collocation on finite
elements with Legendre polynomials ......................... 82

Growth rate curves for isothermal filling of polyurethane resin: 2" = 0.8,
1.0, 1.2, 1.5; BR = 53, En = 41, use] = 0.65, To = 325°K ........... 86

Marginal stability curve for isothermal filling, data from figure 9:
Damkohler number (Da = flow time I reaction. time) .............. 89

Growth rate curves for S = 0 for isothermal filling for identical base state
viscosity profiles as for fig. 9 ................................ 93

Upper limit of growth rate Vs Damkohler number for isothermal filling,
To = 325°K .............................................. 95

Growth rate curves for varying activation energy during adiabatic fill 97
Marginal stability curve for adiabatic filling: activation energy . . . . 101

Marginal stability curve for adiabatic filling: fill time and varying

temperature; ER = 169, E]; = 131, age! = 0.80, ATM = 38°K ....... 104
Figure 4.16 Marginal stability for adiabatic filling: gel conversion ........... 105
Figure 4.17 Schematic representation of the effect of S on growth rate ........ 108

Figure 4.18 Effect of S on growth rates for identical conversion and viscosity profiles
in the base state ......................................... 111

Figure 5.13 Flow of reacting liquid through a porous medium (moving frame) . 118

Figure 5.1b Profiles of conversion, viscosity and S at t=0 .................. 118
Figure 5.2 Initial growth rate curves - varying 0 ............. ‘ ............ 134
Figure 5.3 Marginal stability: 0 ...................................... 137
Figure 5.4 Initial growth rate curves - positive values of S ................. 141
Figure A.1 Schematic for shell balance - one dimensional flow ............. 151

NOTATION

All variables with an overbar (') indicate base case quantities.
All variables with a hat (A) indicate dimensioned quantities.

All primed (') variables indicate the perturbations.

a length scale used for Deborah number calculation
A amplitude of conversion disturbance

AR, AuArrhenius constants

b defined in eq. (4-22)

b1 KXIKI

b2 Ky’Kl

c concentration of reactant (mol/m3)

co inlet concentration of reactant (mol/m3)

Co, Ro concentration and reaction rate at reference conditions

C volume fraction averaged specific heat capacity (J/kg K)

P

C1, C2 constants in the viscosity‘expression
D isotropic dispersion coefficient (m2/s)
Da Damkohler number, tp/tR

De Deborah number

Eu activation energy for viscosity

Ea activation energy for the reaction (J/mol)

xii

ER dimensionless activation energy for reaction, Ea/(Rg ATad)

G’ shear storage modulus

H,W transverse dimensions of the shell in shell balance

13 unit vector in the flow direction
I initiator concentration
k wavenumber

K permeability tensor

KD dimensionless permeability tensor

k1 initiator rate constant

K, longitudinal permeability

kR rate constant for simple n-th order kinetics

Kt transverse permeability

Km Newtonian transverse permeability (constant)

kx, ky disturbance wavenumbers in the transverse directions
k1, k2 rate constants for non-simple kinetics (e. g. eq. 2-2)
LMMDR[f,I] logarithmic mean mobility difference ratio

m, n order of reaction

ma apparent mobility

mf, m1 apparent mobility for regime I and at the front, respectively
M the initial mobility ratio

p amplitude of pressure perturbation

Po inlet pressure for constant injection process

xiii

P pressure
Pe Peclet number, tD/tp
g velocity in the moving reference frame

q; amplitude of the fi-component of the velocity disturbance

R rate of reaction (mol/m3-s)

Rg gas constant (J/mol °K)

9i reaction rate with no explicit temperature dependence
3 area:

t time (s)

T temperature

tD, zD dispersive scales for time and length

tp, 2}: flow scales for time and length

to, v0, zogeneral scaling factors for time, velocity, length respectively
To inlet temperature

tR reactive scale for time

Tref reference temperature

t1, tn, tmtime bounds for flow regimes I, II, 111 respectively
3; velocity in the fixed frame of reference

Ub interstitial velocity (m/s)

Uo uniform injection (superficial) velocity

v d(ln X)/d§ a —p x (trade)

x, y coordinate directions transverse to the flow

xiv

y A location of the front at time t1

2 length, in the fixed reference frame

Zf position of the flow front

Greek Symbols

0t conversion

ag gel conversion

B defined in eqn.6

-AHR heat of reaction (J/mol)

ATad adiabatic temperature rise (°C)

A2 shell thickness

Au (ll-(12, the initial jump in conversion across the interface
6 porosity

9 (Da) x (Pe) = tD/tR

A 1/

u viscosity = llmobility

11.; initial viscosity

110(T) Arrhenius’ term in the viscosity expression
E length coordinate along the flow direction in the moving reference frame
p d(1n )lfi

pf volume fraction averaged density (kg/m3)
o exponential growth rate

XV

1: relaxation time
\p stream function form of the velocity perturbation, defined in eqn.35

a) frequency (rad/s)

xvi

CHAPTER 1 INTRODUCTION

1.1 INTRODUCTION AND MOTIVATION

This work investigates the flow of reactive, polymeric liquids through fiber bundles
with specific reference to viscous fingering during the mold filling stage in liquid molding
operations. The effect of the polymerization reaction along with the various
thermochemical properties of the resin, and the effect of dispersion during the filling
process, on the growth of fingers, are studied.

The motivation for this study comes from a necessity to understand the flow
processes involved in the manufacture of fiber reinforced polymer composite materials.
The relevant composite manufacturing process is Liquid Molding, of which Resin
Transfer Molding (RTM) and Structural Reaction Injection Molding (S-RIM) are specific
techniques. Flaw-free parts are the major requirement of any composite manufacturing
process. Microstructural defects in the final part may arise from a number of causes --
uneven distribution of the resin, voids embedded within the fiber bed and viscous fingering
of fresh resin with a lower viscosity into aged resin of higher viscosity within the mold.
The scale of the defects is important to determine whether they affect the mechanical
performance of the molded part significantly. Inhomogeneities, caused either by voids or
by an uneven distribution of the resin across the fiber bed, may lead to weak spots in the
final parts. The possible role of viscous fingering during mold filling, in the generation of
these inhomogeneities, is studied.

Mold filling processes involve the flow of polymeric matrix material past

I

2

anisotropic arrays of fibers. Since the dimensions of the fiber tows are extremely small in
comparison to those of the mold itself, the impregnation of the bed of fibers is treated as a
flow through porous media. As a result of the polymerization reaction occurring during
mold filling, a continuous variation of viscosity of the resin develops across the filled
portion of the mold, at any time. Such a viscosity gradient is found to result in the
phenomenon called fingering of more mobile fluid.

This phenomenon of viscous fingering has been observed in solvent-flushing of
packed beds and filters [Hill, 1952], as also in the secondary extraction of crude oil from
rock formations under the surface of the earth, and has been reviewed by Homsy (1985).
During mold filling, one possible mechanism for the formation of inhomogeneities could
be the rejoining of the fingers downstream, as they grow, thus enclosing a region
containing resin with properties drastically different from those of the resin in its
neighborhood. The structure and behavior of the fingers is expected to depend on the
fibrous nature of the medium, due to its effect on the degree of mixing by dispersion, and
also on the thermochemical properties of the resin being injected.

An introduction to liquid molding processes and the effects of fluid rheology
thereon are also discussed briefly in this chapter. The current state of the art and the
background to studies in viscous fingering are presented in the subsequent chapters in this
section. The mechanism of the effect of adverse viscosity gradients on the growth or
attenuation of fingers, is also reviewed.

The focus of this dissertation is on the effect of the polymerization reaction on the
stability of the mold filling process. Chapter 2 discusses the types of polymer resins used

in liquid molding operations, with emphasis on their reaction kinetics. This chapter offers

3

an insight into the factors one needs to look at, to have an a priori inkling of what to
expect would happen to any incipient disturbances during the mold filing process.

The rheological data for an epoxy resin and a vinyl ester resin were taken and a
mathematical model presented for the adiabatic, one-dimensional filling of fiber bundles
with these reacting resins in an end-gated mold. The prediction of the moving front and
the evolving pressure distribution inside the filed portion of the mold at any time are
presented in chapter 3. This analysis and results were presented at the V11 Annual
Advanced Composites Conference and Expo, at Detroit in October, 1991 (Jayararnan and
Satyadev, 1991).

The stability analysis on the mold filling process and the results on its application
to a polyurethane resin, along with the base state results, are presented and discussed in
Chapter 4. The effects of dispersion were not included in this analysis and so this chapter
discusses the influence of polymerization reaction on the finger growth in the absence of
any other stabilizing/destabilizing mechanism. These results were presented at the Annual
AIChE Meeting at Miami in November, 1992. The combined effects of mass dispersion
and chemical reaction are presented in Chapter 5. A model problem is discussed, that
depicts this combined effect on the initial grth rates by means of a simple, analytical
method. This analysis was presented at the V International Conference in Numerical
Methods in Industrial Forming Processes - NUMIFORM ‘95 at Ithaca in June, 1995
(Satyadev and Jayararnan, 1995).

The last chapter in this dissertation presents the conclusions from this work on the

effects of chemical reaction on the stability of mold filling in liquid molding operations.

1.2 BACKGROUND
1.2.1 LIQUID MOLDING OPERATIONS

Resin transfer molding (RTM) and Structural - Reaction Injection Molding (S-
RIM) are techniques that are commonly treated under the category of liquid molding
operations in the polymer composites manufacturing industry. In both these techniques,
the method consists of injection of a low viscosity, unreacted polymer resin inside a closed
mold with a fiber preform in place. The mold must be filled well before the viscosity of the
resin rises to gelation. The main difference between the two techniques lies in the desired
production rate of finished parts; S-RIM is a high speed process, while RTM involves
relatively larger fill times. These differences eventually translate to difference in

operational costs, too.

1.2.2 EFFECTS OF FLUID RHEOLOGY IN RTM

Analysis of the filling process in resin transfer molding is usually done with the
assumption that liquid advancing through the fiber preform maintains a constant viscosity
and remains inelastic (Coulter and Guceri, 1988; Kim et a1, 1991; Pamas and Phelan,
1991; Bruschke and Advani, 1990). This assumption is frequently questionable, in
practice for the following reasons.
a) Often a thermoplastic binder holds the fiber tows or fabric together and allows it to be
shaped before impregnation with resin; this binder usually dissolves in the resin and is
removed from the fiber surface by miscible displacement (Owen et a1, 1989). Dissolution
of binder raises the viscosity by a factor of two or more, as observed by Chen et al (1995).
b) High speed versions of resin transfer molding that are now under development employ

resins that cure very fast (Babbington et a1, 1987). In filling large preforrns with these

5

resins the extent of reaction at various time steps of filling is significant, producing
changes in both shear viscosity and extensional viscosity.
c) In contrast to filling of empty mold cavities where fluid memory effects are not
significant (Wang and Hieber, 1988), viscoelastic effects may be significant in filling of
molds with fiber preforrns. Permeation rates transverse to the fiber axis have been
observed to be significantly lower for viscoelastic fluids, compared to that for inelastic
resins with the same shear viscosity (Chrnielewski and Jayararnan, 1992). The reduction
in permeation rate depends on the extensional viscosity displayed by the fluid as it is
subjected to varying stretch rates in the fiber array.

Hence the changing resin rheology (both inelastic and elastic components) during
the miscible displacement of binder or during the reactive filling process affects the resin

pressure distribution in the mold in a complex way.

1.2.3 VISCOUS FINGERING

Hill (1952) published the first paper on viscous fingering while studying the
displacement of sugar liquors by water (lesser viscosity) during regeneration of packed
beds, where he observed significant channeling of the water through the more viscous
sugar solution. Saffman and Taylor (1958) developed a stability criterion to explain
viscous fingering in immiscible fluids. They showed that the differences in viscosity
between the displacing and displaced fluids, along with the velocity of displacement,
governed the growth rates of disturbances. When the viscosity gradient is unfavorable (the
displacing fluid is lower in viscosity), the flow system is inherently unstable and leads to

growing fingers. This effect of an adverse viscosity gradient on the growth, or otherwise,

6
of disturbances is discussed briefly below. This discussion is equally valid for both

immiscible and miscible fluids.
Efiecr of adverse viscosity gradient: In Figure 1.1(a) is shown a single finger that has just

developed. To the left of its boundary is the displacing fluid I of viscosity It] (unshaded
portion); to its right is the displaced fluid II of viscosity [12 (shaded portion). Let the
pressure at the plane AA’ be Po and the pressure at the plane BB’ be P1. Let the distance

between the two planes AA’ and BB’ be e. We assume, for this discussion, that the
permeability K of the porous medium and the velocity U of the fluid are constant

throughout the flow domain. The pressure P1 at BB’ is determined by Darcy’s law:

P1 = PO—(-I;(—e)-u (1-1)

In the absence of the disturbance, the space between the planes AA’ and BB’ would be

occupied by fluid 11; so the pressure P, is governed by the value of 112 - higher the
viscosity, lower the pressure P1 in comparison to Po (i.e. P1.ii)’ and vice versa (i.e. Phi)-
Once the disturbance has been initiated, however, the fluid inside the finger, at M' has a
viscosity u]; consequently the pressure just inside the boundary, at M', would be PX, as
seen in Figure 1.1(b).

Figure 1.1(b). shows the pressure profiles for the cases of (i) 119112, (ii) u1<u2, and
(iii) u1=tt2 (dashed line). If It] > “2, then the pressure just outside the finger, at M+, would
be Phi which is greater than PX; thus there would be a tendency for flow from the fluid 11
into fluid 1, reducing the finger size; this signifies stable behavior. If u] < 1.12, however, the

pressure just outside the finger,

 

 

 

 

 

 

 

 

 

 

 

 

 

A B
W
Fluid I Fluid II
viscosity 111 viscosity u;
—>
Uo ——> M. W
—>
A’ B’
6
¢ %
W
A
P .
° (I) u1>u2
P1; \
Px \\
(iii) II =
P1, n 1 2
(ii) #1412
>
distance along the mold

Figure 1.1 Mechanism of finger growth due to viscosity gradients

8
at MI”, is Plii which is lower than the pressure Px at M+; this pressure gradient across the
finger boundary would cause fluid I to flow outward into fluid H, indicating finger growth
- instability.

Thus it may be concluded that incipient fingers would grow if the displacing fluid
is of lower viscosity than the displaced fluid, while the fingers would be damped if a
higher viscosity fluid displaces a fluid of lower viscosity. The driving force for finger
growth is the pressure gradient across the finger boundary, which is determined by the
viscosity difference across the two fluids. When a fluid displaces another fluid of higher
viscosity, the unfavorable viscosity difference in the flow direction leads to flow
instability, causing the low-viscosity fluid to channel through the second fluid, yielding
‘fingers’.

Viscous fingering has been studied by researchers in petroleum engineering in the
context of secondary oil recovery, using external fluids to push oil from the reservoirs up
into the wells. Water has been used, with considerable success, for this purpose. Water,
being lower in viscosity than the subterranean oil, pushes the latter through the porous
rock formations. But it was observed that this scheme did not always extract all the oil in
the formation, with water coming out of the wells after a certain period. This has been
seen in laboratories, too, and has been attributed to the phenomenon of viscous fingering.
Fingering in both immiscible and miscible displacement has been studied by various
workers and has been reviewed by Homsy (1987).

Chuoke et al (1959) developed the first mathematical linear stability analysis for
the displacement of two immiscible fluids. They included interfacial tension effects in

their analysis, to obtain the necessary and sufficient conditions for instability of slow

9

liquid-liquid displacement in porous media.

In liquid molding operations, however, the displacing and displaced fluids are both
completely miscible, both comprising the same reactive resin, albeit of different degrees
of polymerization. There is, thus, no interface between the two fluids, and the analyses
done will correspond to those for miscible displacement problems. As will be seen below,
the miscible nature of the two fluids in question provides an opportunity for dispersive
mixing at the edges of fingers, thus smoothing out the viscosity profile. In addition the
fresh resin, that pushes the older resin in the mold, is at a lower degree of polymerization
than the latter, hence of lower viscosity. An adverse viscosity gradient is, thus inevitable in
the liquid molding scenario. Fingers were observed in RTM experiments performed by
Losure (1994). To illustrate the generation of fingers, the mold was first fed with the resin
and left to react for a certain period. A fresh batch of the resin was then injected; this batch
was dyed, to aid visualization of the generated fingers. Figure 1.2 shows a typical liquid
molded part, as an illustration.

Miscible displacement does not involve any interfacial tension. Slobod and
Thomas (1963) performed experiments to show the development of fingers by a fluid
displacing another miscible fluid at different flow velocities. They found that a single,
large finger was generated at small velocities and also saw that the edge of the finger was
quite blurred and attributed this behavior to dispersive mixing of the two fluids across the
edge. Larger flow velocities yielded a number of small fingers that merged, over time, into
fewer fingers, though of considerably smaller widths than those seen in the low velocity
experiments. No rigorous mathematical analysis was, however, presented in this paper.
The type of finger that is produced, i.e. whether it is a single, large finger or comprises

several fingers of smaller widths (finger splitting) is discussed by Park and Homsy (1985),

10

Figure 1.2 Typical molded plaque: to illustrate the generation of fingers. White - older
resin; Shaded: fresh resin [after Losure, 1994]

MULTIPLE FINGER TIPS

  

  
  

w

’ new“
"' ¢
-. v .w,

WW '

 

 

\— VENT PORT INLET PORT

VENT PORT

Figure 1.2

12

where a dimensionless group (Capillary number) is used as a criterion to describe a finger
splitting regime in immiscible flows.

Perkins and Johnston (1963) published a review of the effects of diffusion and
. dispersion in miscible displacement processes in flows through porous media. A summary
of the different empirical equations for longitudinal and transverse dispersion coefficients
is also given. In a later paper, the same authors (Perkins and Johnston, 1965) suggest a
useful way of denoting the size of a finger, as the ratio between the finger width to the
spacing between successive fingers.

Schowalter (1965) formulated stability criteria for steady, constant velocity,
miscible displacement of fluids in an infinite porous medium and discovered that the
criteria for marginal stability are affected by variations in density, viscosity, the
displacement velocity and also the effect of diffusion. Paterson (1985) used the analogy
between flow through porous media and a Hole-Shaw cell and used viscous dissipation of
energy in the displaced fluid as a stabilizing mechanism, to obtain bounds on the
disturbance wavelengths for undarnped growth for miscible fluids.

A linear stability analysis on miscible displacement, in the absence of any
mitigating factors such as dispersive mixing, was performed by Hickemell and Yortsos
(1986). The inherent instability of the miscible displacement process, in the presence of an
adverse viscosity gradient, is observed and appropriate stability criteria are derived in this
theoretical paper. Their results are presented schematically in Figure 1.3. Disturbances of
all widths (wavenumbers) are seen to be unstable.

Tan and Homsy (1986) found that, in the presence of an adverse viscosity gradient
between two miscible fluids in a porous medium, the flow is inherently unstable and that

dispersion tends to stabilize the process by spreading out the mobility profile, more so for

l3

 

 

 

growth rate

 

 

 

 

 

wavenumber

Figure 1.3 Filling with no reaction or dispersion [after Hickemell and Yortsos, 1987]

14

the higher wave number disturbances. They also studied the effect of anisotropic
dispersion on the growth rate; they found that a small transverse dispersion increases the
growth rate drastically in comparison to an isotropic case, resulting in a shift to smaller
finger widths, while a large transverse dispersion tends to stabilize the system at all length
scales. Their results are shown schematically in Figure 1.4, where the growth rates are
plotted against wavenumbers at several times. Fingers of large wavenumbers are seen to
be stabilized by the dispersion process.

Yortsos and Zeybek (1988) emphasized the dependence of dispersion coefficient
on flow rate and concluded that the velocity dependence causes instability at large
wavenumbers through a kind of a feedback mechanism. Nonlinear interactions of viscous
fingers are discussed in later work by Homsy and co-workers (Tan and Homsy (1988),
Zimmerman and Homsy (1991, 1992)). The effect of nonmonotonic viscosity profiles, as
opposed to those considered in earlier literature, is presented by Manickam and Homsy
(1993). In another paper, Tan and Homsy (1992) include permeability heterogeneities in
the porous medium, described by a statistical model, and find a coupling between these
heterogeneities and viscous fingering. Bacri et al (1992) experimentally confirm the
theoretically predicted trends of the effects of various parameters, e.g. viscosity, flow rate
and permeability heterogeneity, on fingering, after including effects of gravity and
nonlinear dispersion in their analysis. Gorell and Homsy (1985) made predictions of the
most dangerous wavenumber, i.e. the wavenumber at which growth rate is the largest,
which would be the fastest growing

Porosity variations can occur in certain situations, e.g. during injection of acid into

the formation, to increase crude oil production from a well. Chadam et a1 (1986)

investigate these reaction-induced porosity variations and their effect on fingering

 

WIN-ate

 

 

 

 

 

Figure 1.4 Filling with dispersion, no reaction [after Tan and Homsy, 1986]

16
instabilities. In a later paper, Chadam et al (1991) looked at the coupling of porosity

variations with viscosity changes. Schincariol et al (1994) present an analysis of the
growth of disturbances in a variable density flow and develop criteria for the growth or

decay of disturbances.

1.3 LIQUID MOLDING AND VISCOUS FINGERING

Since the resin being injected in liquid molding operations is continuously
polymerizing, the fluid in the filled region of the mold at any time has a constantly
changing viscosity; fluid at the flow front has a higher viscosity than the fluid at the inlet to
the mold. Though the question of two fluids does not arise, the flow situation may still be
visualized as one in which the fresh fluid is pushing the older fluid and, thus, analyses
done on miscible displacement of fluids would be applicable. Owing to the adverse
viscosity gradient, in the absence of any stabilizing mechanism, the flow would be
expected to be inherently unstable. Based on existing literature discussed above, it is
easily seen that dispersion across the edge of a finger helps damp out the disturbance.
Also, a higher transverse dispersion (transverse to the bulk flow) increases this smearing
out effect. A brief discussion of the dispersion coefficients in several types of fiber
preforrns may be found in Losure (1994).

As seen in Figure 1.4, the fingers that are most likely to grow are the ones with the
most dangerous wavenumber, seen in Figure 1.4 as corresponding to the peak.
Disturbances at both ends of the spectrum of wavenumbers have comparatively lower
growth rates. The actual dimensions of the liquid molding cell play a major role in

determining the size of the fingers. If the most dangerous wavenumber corresponds to a

17

finger width that is larger than the dimensions of the cell, the instability will not be seen in
experiment.

The mechanical properties of a finished part are affected by viscous fingering. The
time of generation of a finger, with respect to the injection of fresh resin, governs the
strength of the edge of the finger. When the incipient disturbance is closer to the inlet, the
degree of polymerization is still fairly low, and there is adequate opportunity for
subsequent reaction to create crosslinks across the edge of the finger, thus bringing the
finger close enough to the rest of the finished part in mechanical properties. On the other
hand, if a disturbance is generated at a later time, when the conversion is fairly high, there
is lesser scope for crosslinking across the edge of the finger, which makes the region of the
finger behave as an inhomogeneity in the final part, with comparatively poorer mechanical

properties.

1.4 SUMMARY

An introduction to the phenomenon of viscous fingering has been presented in the
preceding sections of this chapter. The appearance of fingers during mold filling, and their
effect on the properties of the final liquid molded parts, was briefly discussed. It was seen
that no work has been done on the effects of reactive coupling on the stability of the flow
through porous media. The following chapters of this dissertation deal with the role of the

polymerization reaction on viscous fingering.

CHAPTER 2 POLYMER SYSTEMS IN LIQUID MOLDING: A REACTION

KINETICS POINT OF VIEW

2.1 INTRODUCTION

Reaction Injection Molding (RIM), the precursor to Resin Transfer
Molding (RTM) and Structural - Reaction Injection Molding (S-RIM), involves
polymerization inside the mold. This is in contrast to conventional thermoplastic injection
molding (TIM), where polymer is injected into the mold and allowed to cool to form a
solid polymer. Monomer casting and thermoset injection molding processes also use
polymerization inside the mold, but they employ heating of the mold walls to activate the
reaction. In RIM, monomer and mold temperatures are usually quite close and the reaction
is activated by impingement mixing of the reactants just before they enter the mold. A
schematic of the RIM machine is shown in Figure 2.1 (Macosko). The two reactants A and
B are mixed in appropriate ratios just before they enter the mold, to initialize the
polymerization reaction. The resin far from the mold inlet has undergone considerable
polymerization reaction compared to resin at the mold inlet.

RIM production began with polyurethanes. Nylon 6, dicyclopentadiene,
acrylamate, epoxies, unsaturated polyesters and phenolic materials are several other

chemical systems currently in use for RIM operations.

18

l9

Inlet Pressure

 

Ratio
Control

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Mixhead

 

 

 

Reactant A Reactant B

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Mold

 

 

 

Figure 2.1 Schematic of a RIM Machine [after Macosko, 1989]

20

As will be seen in chapter 4, the chemical reaction kinetics of the resin
used in liquid molding is an important factor in the determination of flow stability of the
mold filling step; the stability is dependent on the rate of (polymerization) reaction and its
derivatives with respect to conversion and temperature. Thus the form of rate expression
that describes the polymerization reaction has a direct effect on subsequent analyses. For

example, polyurethanes are usually represented by simple n-th order kinetics:
R = kR(1—0t)" (2-1)

while unsaturated polyesters and epoxies have been known to follow more complex rate

expressions of the type:
R = (k,+k2-a"')-(1—a)" (2-2)
Several types of chemical reactions are schematically depicted in Figure
2.2, where the reaction rate, R, is plotted against the conversion, or, showing the different
classes of reaction systems by the trends in variation of (BR/act) E S. Reactions with
kinetics given by equations of the type eqn. (1) have a negative S for all ranges of
conversion, for nonzero orders of reaction, shown in Figure 2.2 as curves (a) and (b), while

reactions with kinetics given by eqn. (2) would depend on the values of the kinetic

constants and the conversion. When the order of reaction is zero, the reactions follow line

(c).

21

S = positive constant “9

 

S>0

a:
.. ((0
5A
8 S 0
fl =
E
“5 (e)
8
E

(b)

S<0

S = negative constant

(a)

 

 

>
extent of reaction, on

Figure 2.2 Types of chemical reactions classified by S behavior

22

The solidification process in all liquid molding operations involves
crosslinking. The viscosity variations arising from this process are linked to the reaction
kinetics of the polymerization process. The rheological changes during a single RIM
cycle, up to gelation, are shown in Figure 2.3. The initial viscosity of the reactant mixture
is low enough to enable rapid mixing. As the monomers react to form high molecular
weight resin, the viscosity rises. This build up must be slow enough for the entire mold to
be filled. Once the mold is filled, injection of reactants is stopped and the viscosity

allowed to increase rapidly to gelation.

 

 
   

 

 

S
t
t
t
u
\
I \
s? t t
g = = Curing
\ h
> t \
\ \
t t
h h
h h
\ t
t
t
t
t t
3 Shut Off \
Gelation
Reaction time

Figure 2.3 Viscosity changes during a RIM cycle

Though fast polymerization reaction is essential for a polymer system to be

successfully used in RIM processes, not all such fast polymerizations are suitable.

23

Polymerizations in solution, emulsion or suspension are not suited for production of a
molded part; polyethylene or polypropylene, which are made only in solution or
suspension, are ruled out. Polyesters, which need extensive condensation of a small
molecule, are poor candidates for liquid molding. Monomers that need to be heated to
their melting temperature for polymerization also are unsuitable since this would need
preheating the mold to high temperatures for the reaction to occur, followed by cooling to
cause crystallization. High initial viscosity of the monomeric liquid makes some polymer
systems unsuitable for RIM.

A brief study of some polymer systems that are amenable to liquid molding
follows. The focus of the discussion in the rest of this chapter will be on the reaction
kinetics of the polymer system under consideration and its effects on the viscosity, since
they are the factors essential for the analysis of the stability of mold filling processes in

later chapters.

2.2 POLYURETHANES

Formation of polyurethanes involves the reaction of isocyanates with
compounds containing an active hydrogen. The active hydrogen compound, e. g. an
alcohol, adds across the carbon-nitrogen double bond of the isocyanate group.

R-NCO+H0—R’—)R—NH-C0—0—R' (2-3)
The { -NH-CO-O-} linkage is called the urethane bond.
Water reacts with isocyanate to form a urea, so careful drying of reactants

and catalysts is done; typical water level in RIM materials is < 0.07%. Reaction of amines

24

with isocyanates also yields ureas. Organometallic compounds can greatly increase the
rate of urethane formation. Isocyanates can also undergo addition or auto-condensation
reactions. Dimerization of isocyanates is reversible at high temperatures, while the
products of trimerization (Isocyanurates) are more stable.The urethane bond is also known
to be reversible at high temperatures and care has to be taken during RIM and postcuring
operations, where temperatures can go to fairly high levels. Urethane reactions give no by-
products. They go to a high degree of conversion and also can be very fast, depending on
the active hydrogen compound. These factors render urethanes well suited to liquid
molding operations.

RIM processes require that a polymerization goes to completion in a few
minutes, preferably less than one. Reaction kinetic data are necessary for modeling the
rheological changes which control mold filling and curing. It has been observed (Macosko
1989) that most urethane reactions proceed according to the following, simple n-th order,
kinetic expression:

_ dINCO]

dt = k, [NCO] (2-4)

Here the reactants are assumed to be in stoichiometric ratios, as is normally the case, and
the concentration of the catalyst is absorbed into the rate constant. The overall order of the

reaction “m” typically has a value of 2. The rate constant, kR, is described by the usual

Arrhenius form:

E a .
kR = AR exp(-fi) (2-5)

In terms of the extent of reaction (conversion), or, the rate expression then becomes:

25
_ do: _ Ea m
R — dt — AR exp( RT) (1 a) (2-6)

From this form of the rate equation, it may be seen that polyurethanes
follow chemical reactions of the type (a) or (b) in Figure 2.2, depending on the order of the
reaction.

Viscosity is dependent on conversion and temperature, both of which are
continuously varying during the molding process. Kinetic results can be combined with
structural relations (probability arguments) to derive expressions for the evolution of the
molecular weight with time and temperature. From these molecular weight expressions,
viscosity is also easily determined. Empirical data on several polyurethane resins

suggested the following expression for the viscosity:

(1 C1 + C 201
u = MT) [Ag—as _ a] (2—7)
where [10(T) is represented by an Arrhenius type of equation:
Eu
uom = A, expL-fi) (2-3)
as in these equations is the gel conversion for the urethane formulation in question due to

formation of a network. Equation 7 suggests that viscosity depends on the reaction

kinetics through the extent of reaction, or.

2.3 POLYAMIDES

Polyamides are the basic unit of Nylon 6 and are the most studied non-

urethane RIM system. They have the advantage of a high modulus, high impact strength

26

and high temperature stability. They are commonly made by anionic polymerization of e-
caprolactum with a suitable initiator and metal catalyst. The mechanism involves a step-
propagation. Acyl lactam is an example of initiator used. Sodium caprolactam and
Magnesium caprolactam are the most commonly used catalysts.

The reaction kinetics for the formation of polyamides has been well studied
(Malkin et al 1982). An autocatalytic model fits a large amount of data for the sodium

catalyzed reaction. The rate equation is found to be of the form:

2
‘1‘}? = kl exp(—§—;-,) [11%] (l - a) (l + 1:172) (2-9)
where “I” is the concentration of the initiator and is the same as the catalyst concentration.
M0 is the initial monomer concentration and or is the extent of reaction (conversion), “k1"
is the rate constant and “k1” is an autocatalytic term. It is seen that the reaction kinetics for
this polymer follows a more complicated rate expression than for the polyurethane system,
due to the presence of the autocatalytic term.

From the above form of rate expression, it is seen that dR/da is negative
for conversions greater than or equal to 50%; for lower conversions, it depends on the
numerical values of the initiationrate constant, “k1”, and the initiator concentration “I”.

The reactant viscosity is low and the reaction rate is relatively slow. This
ensures that molds are easily filled. Gel times are typically over 205. However, the low
viscosity may lead to bubble problems. One disadvantage for this Nylon 6 system is that

of a high operating temperature and require high temperature RIM machines.

27
2.4 EPOXIES

Suitable catalysts and hot molds are needed to adapt epoxy formulations to
liquid molding. Most epoxy formulations are based on bisphenol A and aliphatic
diamines. Highly crosslinked networks result from the reaction since both amino
hydrogens can react with the epoxy groups. A typical reaction scheme for an epoxy
system is shown in Figure 2.4. The reaction is found to follow simple n-th order kinetics,

given by a rate equation of the form:

dd _ Ea) n
37 —- AR exp( RT (1 Ct) (210)

The order of the reaction, “n”, depends on the polymer system being considered. An
amino—ethyl-piperazine formulation is found to have an order n = 2.8 (Osinski 1983),
while a Diglycidyl ether of bisphenol A (DGEBA) / triethylene tetramine ('TETA) system
has an order n=1.64 (Kim and Kim 1987). Like the polyurethanes, epoxies also are seen to

be represented by curves (a) or (b) in Figure 2.2.

The viscosity in typical epoxy resins is high; so the reactants have to be
preheated to enable easy mold filling. The need for hot molds and a typically large
exotherm can cause degradation problems. However, the reactions themselves are
relatively slow compared to typical injection times. Therefore the problems arising from
viscosity buildup and premature gelation are absent.

Kim and Kim (1987) have found that the viscosity rise during reaction, for

the DGEBA/TETA system, follows equation 7 that was presented in the discussion of

polyurethanes. The gel time for a DGEBA/TETA system at 50°C has been found by

28

o
/ on
I
RN H + cnz-CH-R’ —> RN-CHz-CH-R’
H H
(Epoxide) (Secondary Amine)
(Primary Amine)
+
o
CHZ-CH-R’
on OH

I R I
RI-CH-CHZ-N-CHZ-CH-OH-R’

Figure 2.4 Reaction scheme for an Epoxy system

29

Losure (1994) to be about 30 minutes. Thus, in comparison to both polyurethanes and

polyamides, the epoxy resins have a significantly higher gel time.

CHAPTER 3 VISCOSITY VARIATIONS DURING REACTIVE FILLING OF FIBER

PREFORMS

3. 1 INTRODUCTION

Analysis of the mold filling process in liquid molding is usually done with the
assumption that liquid advancing through the fiber preform maintains a constant viscosity
and remains inelastic. This assumption is frequently questionable. During the filling
process in resin transfer molding there is a constant variation of the viscosity of the resin,
and also a significant rise in the elastic nature of the resin, due to various reasons, which
were discussed in chapter 1.

Hence, the changing resin rheology during the miscible displacement of binder or
during the reactive filling process affects the resin pressure distribution within the mold in
a complex way. The next section presents an analysis to show the effect of resin rheology
on several measurable quantities during mold filling in an end-gated mold. Rheological
data taken from experiments conducted on an epoxy resin and a polyester are used to

investigate these effects of changing rheology.

3.2 ROLE OF RESIN RHEOLOGY IN LIQUID MOLDING

The problem addressed in this section is the adiabatic flow of reacting resin
through a fiber preform placed in an end gated rectangular mold. The fibers are aligned
transverse to the flow and the model pertains to one dimensional motion of the resin with

constant inlet pressure P0 at the gate (see Figure 3.1). A complete description of reactive

30

31

filling would couple the changing rheology to the species mass balance and the energy
balance (Garcia et al). The effect of changing rheology is illustrated directly with the help
of rheological data obtained on the resin as it cures within the gap of a rheometer.

The rate of change in rheology with increasing extent of cure, varies with resin
formulation. With some resin formulations for which the conversion at the gel point is low,
this variation is abrupt -- the change is hardly noticeable upto the gel conversion; then
gelation occurs very rapidly. With other formulations, the change in rheology occurs
smoothly -- both the viscosity and the elastic modulus of the resin change over a range of
conversions leading up to the gel point. The effect of such changes in resin rheology upon
the front motion, and upon the resin pressure distribution during the filling process in resin
transfer molding is examined here with a rectangular, end gated mold at constant inlet

pressure.

3.2.1 RESIN RHEOLOGY AND APPARENT PERMEABILITY

The changing rheology of two different resin formulations has been monitored in
the gap between parallel plates in an RMS-800 instrument. Both the Devcon epoxy resin
and the Derakane vinyl ester resin reach the gel point over a time frame of several minutes
in the laboratory. Figures 3.2 and 3.3 show the time evolution of the steady shear viscosity
and of the dynamic shear storage modulus G' at a fixed, low frequency a) of l rad/s, for

these resins. A fluid relaxation time, 1:, may be calculated with the expression

1 = (3-1)

GI
2
(on

32

Region I Region II Region III

 

 

 

IIIIIII

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Fibers .
Yr Tn

Figure 3.1 One-dimensional flow of resin, transverse to fibers, in an end-gated mold

33

H”

“F

5.

 

VIM“, o PG"
<2

6 ,Pa
104

10.1 1 1 1
0 0.5 l 1.5

 

Figure 3.2 Rheology of Devcon epoxy resin

34

a.— . 3.382 unseen

 

 

 

 

 

10‘?

.m...

103
102
10'
10"
10-2

43341133311311“...
.m
................. I... m
.m.
.m
:6 .m m
m
L4 m
cm
a
.2 .m
m
FILED ”

n.9— . £335

35

Different regimes were identified on the basis of these curves. In the first regime (t < t1),
the resin is an inelastic fluid of constant viscosity. In the second regime (t1 < t < tn), the
fluid remains inelastic but displays an increasing viscosity. In the third regime (t > tn), the

fluid develops viscoelasticity in addition to an increasing shear viscosity. The fluid
viscoelasticity developed at various stages is associated with increasing extensional
viscosity of the resin. These figures show that the progression from inelastic resin to gel is
more abrupt for the Derakane resin.

The bulk flow of incompressible, Newtonian fluids through porous media at low
Reynolds numbers is described by Darcy's law. This macroscale representation of the
superficial velocity in anisotropic media requires the specification of two or more
directional permeability coefficients defined by the geometry of the preform. The

permeability Kl transverse to the fiber axis is typically several times lower than the

longitudinal permeability along the fiber axis. The apparent transverse permeability of
fiber arrays with hexagonal packing and a fiber volume fraction of 30 percent, to
viscoelastic liquids, is plotted against a measure of viscoelasticity on Figure 3.4.
Increasing viscoelasticity is represented by increasing values of a dimensionless ratio of

fluid time scale to flow time scale -- the Deborah number, De.

De = 0 (3'2)

 

The permeability Kt (normalized with the constant value for Newtonian liquids,
Kw) starts to drop at an onset Deborah number of 0.4 and continues to decrease with

increasing De. At higher values of the Deborah number, the apparent permeability reaches

an asymptotic value.

36

 

 

‘m . 7 v . Y Y W 7 fi v v v

I
D 0 "
h d
D . G
D C d
I- . d
. d

O
D C d
O
O
s O
0
g I I- \ ‘
‘ p G . d
a: . I
O
D .. . d
_ J
- 4
. d
b d
A A L g L
‘0': AL A L n L_. A A 1

 

 

10" 10° 10‘

(Vblfla

Figure 3.4 Apparent transverse permeability Vs. viscoelasticity - Hexagonal array

37

3.2.2 THE MATHEMATICAL MODEL

The one dimensional flow along the y -direction transverse to the fiber axis in the
fiber array, is described by the following equations. First we define a coordinate moving
with the front

5 E z - 2f ‘ (3-3)

where 2f denotes the location of the front. Then, we obtain

C t' , BUD 0
on Inurty 5: _

U

Permeation 8—13 = .._0 (3-4)
35 ma
M . dzf U0
I _ = _
0 Ion of front dt 8

U0 is the superficial velocity through the fiber array, 8 is the porosity of the array and ma is

the apparent mobility defined by

ma

K‘ (3 5)
u
the apparent permeability K, and the viscosity It.

The boundary conditions for operation at constant inlet pressure are

P=P at éz-Zf

(3'6)
P = O at i = 0

Analysis Of Model Equations: Three different regions are identified on the basis of

apparent resin mobility, which is the ratio of apparent permeability through the fiber

38

preform, to the viscosity of the resin. The mobility of fluid flowing through the first region
close to the inlet is uniform because the shear viscosity is constant and the Deborah
number is lower than the onset value for viscoelastic effects. The mobility of fluid flowing
through the second region decreases towards the front only because of the increasing shear
viscosity. The mobility of fluid flowing through the third region is affected by both
variations in extensional viscosity and increasing shear viscosity.

The analysis of the filling process is presented for two different situations: filling
with constant mobility and filling with the first two types of regions. For the purpose of
this dissertation, the viscoelastic effects are not considered. It was observed by Gonzalez-
Romero and Macosko (1985) that the viscosity of several polyurethane systems is
independent of the shear rate, i.e. Newtonian behavior of the polymeric fluids may be
assumed, up to the gel point. Since the viscoelastic regime does not begin until the
viscosity is quite high, this assumption is made for these resins also.

The solution for the first situation is classic (Collins, 1961). In the other situation,
analytical expressions are developed for the resin pressure distribution and for the velocity
at any given time by assuming an exponential variation of mobility over each subregion.
These expressions are combined with numerical integration over time for the location of
the front and of the crossover planes for onset of shear viscosity increases. The constant
inlet pressure condition leads to progressively lower superficial velocities with time so that
the residence time corresponding to a location in the mold increases with time. In other
words, after the start of the filling operation, fluid elements entering the mold at any time
undergo the same rheology changes at shorter distances from the gate.

Filling With Constant Mobility: The mobility is constant throughout the resin up to time t,

39

from the start of the filling operation and is described by
m E m , = J (3-7)

The front has advanced for a time period less than t1, the solution to the model equations

(eq. 3-4) is readily obtained as

 

 

 

P06 0
P = — - — (Zf-Z)
Zr 1
l
Zf -

8

mP
U0 : I o

Zr

The location of the front at time t1, denoted by y A, may be found from the above equation

as

NI—

 

2mPt
yA=[ 2"? (am

The velocity at the onset of rheology changes is mIPo/yA. This velocity will be used as a

reference velocity for analysis of the second region.

Filling With Shear Viscosity Changes: Over the time period t, to tn, the region behind the
advancing front may be divided into two subregions -- one of uniform mobility m1 and
another where the mobility varies because of the changing shear viscosity alone. The

mobility at the front m, is then given by

K10
m =
f I10

 

for region I] (3-10)

The viscosity is evaluated at the time corresponding to 2f. The location of

40

crossover between the two subregions, termed yl, also varies with time. It is convenient to
fit the variation in mobility from this crossover plane to the front in region 11 with an
exponential at any given front location or a given time. This choice has been validated by
the final converged values of residence times at different locations behind the advancing
front. The velocity U0 at a given front location is then obtained from the following
equation.

mIPo yl
yAUo yA

 

 

+ (Zf_y’)LMMDR[f, 1] (3-11)
VA

This equation involves four dimensionless groups. The group [mIPO/(onAfl represents a
factor by which the velocity is reduced from the velocity at the onset of rheology changes.
The group (Zf/YA) is the current location of the front relative to its location at the onset of
rheology changes. The group (y1/yA) is the current location of the crossover between the
first two regions. The final group termed LMMDR [f,I] is the ratio of a logarithmic mean
mobility difference (between the crossover plane and the front) to the mobility at the front;
this group is given by the following expression.

m f — m I

l
’"f “(’31)
"’1

LMMDR[ f, I] = (342)

If there were no changes in fluid rheology during the filling process (m, = m, = constant),

this group would have a value of l.

The increments in front location and in the crossover location corresponding to

increments in time are determined by integrating eq. (34) and an analogous equation for

y] with the trapezoidal rule.

41

U old + Unew

0 0

2 8 (3-13)

 

AZf = AI

These equations are combined with Eq. (3-1 I) to determine the velocity at the next time

value.
3.2.3 DISCUSSION OF COMPUTED RESULTS

The expressions derived above have been used to compute the motion of the front
and of the different crossover planes, as well as the evolving pressure distribution, for
several cases. These cases are represented by different values of the characteristic velocity
mlpo/y A ranging from 3.59 mm/s to 6.22 mm/s. Scaling all displacements by y A and the
pressure by the inlet pressure Po then yields one curve for each of these quantities that

covers all values of inlet pressure and of Newtonian liquid permeability (Kw) provided the

rheology variation with time is given; something in the nature of a master curve is
obtained.

Penetration: The resin penetration into the fiber bundle is plotted against time on Figure
3.5. The front crosses over at 0.5 min into the region of changing shear viscosity and into

the region of developing modulus at a time of 2 min and a location 1.94yA. Between 0.5

min and 2 min, the change in shear viscosity leads to a 20 percent higher drop in velocity
than would be obtained with unchanging mobility. At longer times -- 4 min, the changing
rheology leads to a 94 percent higher drop in velocity than would be obtained with the

mobility fixed at the value corresponding to 2 min. The dimensionless plot of velocity on

Figure 3.6 shows these trends. It must be kept in mind that the sharper drop at greater

42

 

yr/ YA

 

 

Figure 3.5 Resin penetration into fiber bundles

 

43

 

(Va YA) 1 (mt Po)

 

 

 

 

17/ Wt

Figure 3.6 Dimensionless superficial velocity at various flow front positions

YI/YA

 

 

 

 

Figure 3.7

1‘; 1.4 1.6 1.8 2 2.2

Location of y, at various positions of the flow front

2.4

 

45

 

YII / YA

 

I

1.3

 

 

 

1'2 i 2:1 22 13 2.4

Yr/ YA

Figure 3.8 Location of ya at various positions of the flow front

46

times is caused not just by permeability drops in the third region but also by the expanding
region of changing fluid rheology.

Figures 3.7 and 3.8 show the location of the two crossover planes with time. From these
curves, it is seen that the extent of region with changing fluid rheology expands to the

entire filled region at the end of 4 min. As the front moves forward, the location of y],

which marks the beginning of rheology changes, is seen to move towards the inlet port of
the mold. This is shown in Figure 3.7. Once the front enters the third region, this onset
plane for all rheology changes moves more rapidly toward the inlet; given sufficient time

for this filling operation, y] would be expected to reach the inlet port, depending on the
nature of rheology of the resin. Once the front has reached y“, the third regime begins,

which corresponds to an increasing elastic behavior of the fluid. As the front progresses,

however, the location of yn, i.e., the onset plane for extensional viscosity effects moves
toward the inlet of the mold. Again, with progressing flow front, the rate at which yn

moves toward the inlet increases, as shown in Figure 3.8.

A Unified Correlation: A very simple relation is obtained between the logarithmic mean
mobility difference ratio and a velocity reduction factor. The latter quantity is defined as
follows. The hypothetical velocity that would be obtained for constant mobility
throughout is given by eq. (3-8). The ratio of this quantity to the actual superficial velocity
is termed the velocity reduction factor. The velocity reduction factor is plotted against
LMMDR [f,I] on Figure 3.9. This figure shows that the two quantities are identical over
most of the region (though not exactly same). This may also be seen from equation 3-1 1

after some simplification. Hence, the effect of changing fluid rheology on the filling

47

1.25

 

1.2 -

1.15

I

1.1 -

(ml Po) I (Vb W)

1.05 -

 

 

 

 

LMMDR (fl)

Figure 3.9 Velocity reduction factor Vs LMMDR [f,I]

1 1.05 121 1.15 1.2 125 .

1.3

1.35

1.4

LMMDR (£1)

48

 

1.4 - 1 . . . - . e
1.35
1.3
1.25
1.2
1.15
1.1 '

1.05

 

 

 

 

Yr/ YA

Figure 3.10 LMMDR [f,I] at various positions of the flow front - regime II

 

49

 

Q9-

08-

DJ

05

PlPo

03

I

I

02

01-

 

 

 

 

ylyA

Figure 3.11 Pressure profiles at various locations of the flow front

50

process can be correlated effectively by comparing these two quantities. Figure 3.10

shows the LMMDR's at various front locations.

Pressure Distribution: For the constant inlet pressure process, the pressure profiles at
various instants of filling time in the mold are shown in Figure 3.1 1. When the front is at

locations up to y A, the entire saturated portion contains resin of uniform rheology and so

the pressure profile is linear, as seen in the innermost curve in this figure. As the front
moves further, the rheology changes in the second regime and the pressure in this region is
considerably higher than what it would have been if the rheology had not changed. The
pressure drop is still fairly linear in the first regime. The curve begins to become
increasingly convex, as the front moves forward, and the pressure drop across a small
portion of the fluid, close to the front, is large. That is, the pressure gradients in the bulk of
the fluid are lower than those at the moving front. As the front enters the third regime, the
pressure profile becomes increasingly flatter in the bulk and steeper closer to the front.
This type of pressure distribution with a steep drop near the front is significant for
practical Operation with clusters of filaments making up the preform. The driving force for
longitudinal wetting of individual filaments in a cluster would be uniformly high in the
present instance over the bulk of the region swept by the front. In contrast, without

rheology changes, this driving force decreases linearly.

3.2.4 SUMMARY

The motion of the front, and the pressure distribution, have been tracked in the

51

above analysis, by modeling the measured viscosity profile in a simple fashion. The effect
of rheological changes on the reduction of the flow velocity, in a constant injection
pressure situation, have been represented by a simple relation between two quantities - a
velocity reduction factor and a ratio of the logarithmic—mean mobility difference versus

the mobility at the front.

CHAPTER 4 VISCOUS FINGERING DURING REACTIVE FILLING OF FIBER

PREFORMS: EFFECT OF CHEMICAL REACTION

4.1 INTRODUCTION

It has been found that viscous fingering occurs during mold filling in liquid
molding operations (Losure, 1994). This phenomenon is attributed to growth of
disturbances inside the flow domain, hence to instabilities of the flow process involved in
the mold filling stage. The disturbances themselves arise due to minute variations in any of
the variables, such as velocity, conversion, viscosity, etc., occurring due to
inhomogeneities in the porous medium or in resin mixing.

The viscosity of the polymeric resin, coupled with the polymerization reaction, has
an effect on the stability of the miscible displacement process occurring during mold ‘
filling. A linear stability analysis is done to obtain the criteria for flow stability. The onset
of instabilities arising from adverse viscosity gradients of displacement processes without
reactive coupling is compared to that with reaction in this chapter. It is found that, in the
absence of dispersion, the reactive coupling mechanism tends to stabilize the lower wave
number disturbances. The effect of varying permeability is also considered.

Most of the available literature on fingering, cited in chapter 1, refers to
displacement processes occurring in the context of secondary recovery of oil from
subterranean formations. Liquid molding is a means of manufacturing polymer composite

materials, with major applications in the automotive and aviation industry. In this process,

52

53

a polymeric resin flows through a bed of reinforcing fibers placed inside a mold. After
filling, the mold is left to cure, during which time the resin hardens; subsequently the part
is removed from the mold. An important qualitative requirement in these industries is the
avoidance of voids and/or inhomogeneities in the manufactured part, since they would
weaken the structure. Disturbances arising during the filling stage, over a range of
wavenumbers, can grow with time at various growth rates, resulting in fingers. The
properties of the resin in the fingers would be different from those of the resin in its
vicinity, due to possible differences in the degrees of crosslinking from different ages.
Upon curing, then, these fingered structures would be retained in the final part as
inhomogeneities.

Before entering the mold, the resin is in an unreacted (monomeric constituent)
state and the polymerization reaction begins as soon as the fluid enters the mold. The bed
of fibers is a preform and may be random with isotropic permeability or aligned with
anisotropic permeability. The concentration of the fresh fluid entering the mold is different
from that of the older fluid, as the latter has already begun to polymerize. Thus the
incoming fluid, which pushes the older fluid, is less viscous than the latter, creating an
adverse viscosity gradient - a filled region, in the mold, of increasing viscosity in the flow
direction. It is to be noted that the displaced and displacing fluids in liquid molding,
differing only in the degree of polymerization, are completely miscible. Also, the viscosity
gradients in the mold filling problem are continuous across the entire flow domain (due to
a continuous base state concentration profile), while earlier literature considered abrupt
changes in viscosity at an interface (a step profile of base state concentration). Mobility,

defined as a ratio of permeability and viscosity, would thus decrease in the direction of

54

miscible displacement (for uniform permeability in the flow direction), causing fingering
instabilities. The mechanism of generation of fingers under an adverse viscosity gradient
has already been dealt with in chapter 1.

Due to the nature of the fluids and the physical processes considered, earlier
literature has dealt only with the effects of dispersion, surface tension, permeability and
gravity on fingering. Since liquid molding uses reactive flow systems, the stability of the
flow process is also affected by the nature of the polymerization reaction. The effect of
reaction and anisotropic permeation on flow stability during reactive filling, in the absence
of dispersion, is explored in this chapter.

Hickemell and Yortsos (1986) presented the linear stability behavior of miscible
displacement processes in porous media, in infinite domain, in the absence of dispersion.
In their paper, upper and lower bounds on the growth rate were derived when dispersion
was absent, and these bounds were found to be directly dependent on the lowest and
highest values of the viscosity gradients in the domain of the problem. The inherent
instability of the miscible displacement process, in the presence of an adverse viscosity
gradient, is observed and appropriate stability criteria are derived in their theoretical paper.
Disturbances of all wavelengths are seen to be unstable. In our problem, too, dispersion is
absent; however there are two major differences. First, the domain of their problem is
infinite, while the liquid molding problem discussed in this chapter is finite. Secondly, in
the current problem, chemical reaction plays a major role in both generation of the
viscosity gradient and growth/decay of disturbances; the problem investigated by
Hickemell and Yortsos (1986), however, involved no~chemical reaction.

Polymerization reactions may follow various types of kinetics (see chapter 2);

55

reactions having simple n-th order kinetics are considered (e. g. polyurethanes follow
simple second order kinetics). Conditions under which the filling process would be stable
to infinitesimal disturbances will be presented, as marginal stability curves. A frozen
profile approximation, valid when the growth rate of the disturbances is much greater than
the rate of change of the base state (the moving front), is used. The problem is formulated
and described in the next section, where the governing equations are presented, the base
case solution is discussed and the eigen value problem obtained from the stability analysis
is presented and discussed. The solution approach is described in Section 4.3, followed by
a description of the results and their discussion in Section 4.4. Some important

conclusions that emerge from this study are listed in Section 4.5.

4.2 PROBLEM FORMULATION
4.2.1 THE GOVERNING EQUATIONS

Figure 4.1 shows the flow situation being considered. Fiber bundles are placed
inside the mold and the resin enters the mold at an end-gate across the entire cross section.
The figure shows the advancing flow front at some instant of time. A constant injection
rate is considered, which means that the pressure drop along the mold varies as the mold is
filled. This is in contrast to the problem investigated in chapter 3, where a constant
injection pressure was used. Another major difference from the problem discussed in
section 3.2 is the contribution of polymerization kinetics to the resin rheology. In that
section, the effect of changing rheology was included directly with the help of rheological

data obtained on the resin as it cures within the gap of a rheometer. In the following

56

29: vocawéao 5 E .

2:: a comet £55. 9:88 we use—conom 3a 23$

00“..

 

 

:c: 0:5 an ESL 32m
A s v
29: bQEm comma eozfi
as N
\ . o... £82 8.:—
O .
E25 » Alla 5:858:00 ~25
oD 5603/ 32m 88.2.5

 

 

All:

 

57

discussion, however, these kinetic parameters are included explicitly in a component
balance equation. One more distinction from section 3.2 that has to be kept in mind is the
arrangement of fibers inside the mold: in section 3.2 the fibers were all oriented transverse
to the flow of the resin, in ordered arrays, while this section deals with a more realistic
fiber preform. It is assumed, for the purpose of this analysis, that the fiber and resin in the
filled region are at the same temperature, as would occur if the heat transfer coefficient at
the fiber-resin interface were extremely large. The thermal parameters of the resin and
fibers are also lumped as a volume fraction weighted average for the filled region. Another
important assumption is the absence of dispersion; this yields the effect of chemical
reaction on the growth of disturbances, independent of the dispersive effects seen by Tan
and Homsy (1986).

Since the fluid moves with a constant velocity U0 in the z-direction, a Lagrangian frame of
reference is used for convenience, based on the location of the flow front at any time; the

length coordinate in the flow direction, in the moving frame of reference is, then,

Z-U f (4-1)

0

E = 2 — 2,
The impregnation of the fiber arrays is treated as flow through porous media, and is

described by Darcy's law.
. ,. ': 1 A
u = q-I-Uo 13 = {13) (KOVP) (4-2)

where u and q are the velocities in the fixed and moving frames of reference respectively.
The permeation equation (Darcy’s law) involves a permeability tensor, which would be
diagonal if the principal axes are assumed to lie along the coordinate directions. Also, the

permeability is assumed to be equal in both the transverse directions, yielding just two

58

independent terms in the permeability tensor,

a)
II

 

[1,00
02,0
_oofr,

(4-3)

 

The equation of continuity, the reactant mass balance and the energy balance,

lumping the fiber and resin temperatures together and assuming an adiabatic process with

no heat transfer with the surroundings, in the moving frame of reference, would be

Continuity

Reactant Mass Balance

Energy Balance

where

 

 

Vat: = 0
% .__ .
5+9 Vc _ R(c,T)
3§+§wr=3 men) (4‘4)
_—AHR _ ATM
B: pCp - co

The adiabatic temperature rise, ATM, is given by (Boo) and will be the scaling factor for

temperature in this dissertation.

To demonstrate the theory being developed in this chapter, a polyurethane resin

system is chosen. The reaction is of second order kinetics and the viscosity of the resin has

an Arrhenius' type dependence on the temperature. The various kinetic, thermal and

rheological properties are taken from Castro and Macosko (1982) and are listed in Table

4.1. The reaction rate and viscosity for this resin are given by the equations:

Rm.

6:
or

 

n=A,@—

gel

T) = kociu --6t)2 exp[ Ea]

)-(l.5 + (‘1)

 

RT

8

E
J.
exp [R8 T]

(4-5)

59

Table 4.1 Thermal, kinetic and rheological data for the resin and glass fibers

RESIN GLASS
cp (J/kg K) 1,840 790
p (kg/m3) 1000 2,500
k0 (m3/m61 8) 10,560
EaU/mol) 53,200
co (moi/m3) 2,410
arr, (Jlmol) 96,300

Vrscosity described by an Arrhenius’ type equation:

110,61) = Auexp[— 51]

RT
where

Bud/moi) 41,300

A" (Pa-s) 10.3 x 10'8

0tg 0.65

Lumped value of (pCp) for the resin-filled fiber mat = 1.9 x 106 J/m3 K.

 

60
4.2.2 SCALING

All the quantities in the preceding equations are dimensional and appropriate
scales are chosen to make them dimensionless. The chosen scaling factors for the different
variables are listed in Table 4.2. Reaction is the important physical phenomenon in this
problem, so the time and length scales are based on the reaction rate at the inlet
composition and a reference temperature. The scale for temperature is the adiabatic
temperature rise, ATad. Viscosity is scaled with the viscosity at the reference temperature.

Using these scales, the equation of continuity, the component mass balance and the energy

balance, in dimensionless form, yield

Continuity Voq = 0
8a

Reactant Mass Balance at +q0 0t R(or, T) (4-6)
3T

Energy Balance 5 + q 0 VT = R(a, T)

while the permeation equation becomes

q = 4. K-VP—i3 (4-7)
where
b, 0 0
K = 0 b2 0
O 0 1
1 (4‘8)
71.5-
p.
k, K
K: K:

In these equations b, and b2 are measures of the anisotropic nature of the medium

permeability, while It is the dimensionless mobility, whose gradient will be seen to have

61

Table 4.2 Scaling factors for different variables

Concentration:

Permeability:

Pressure:

Velocity:

Reaction:

Temperature:

Time:

Length:

Viscosity:

Activation energy:

c=c0-(1—01)

N’
ll

K,-K

P = (”if") ' [11:24)]?

 

R(&, T) = R(co, Tref)~R
T = ATad- 1

Co

= R(CO: Tref) . t

H)

UOCO
- = _— ' Z
R(co’ Tref)

N)

 

 

fl=u¢u
E
agar“, agar“,

62

considerable importance later in this chapter.

4.2.3 THE BASE STATE EQUATIONS

(NOTE: All base state quantities are identified by an overbar.)

Since the only velocity component is Uo in the z-direction (one-dimensional flow),
the base state solution for the velocity vector, in the moving frame of reference (see
equation 4-2), is

(I = 0 (4-9)
The base state solutions for the conversion, temperature and pressure profiles are obtained
by solving the following simplified forms of equations 4-6 and 4-7.

36!.

E = Rut, T)
37 _ _ —
E — R(0t, T) (4-10)
- _ - 81"
~74“, T) 5E - 1
The boundary conditions to be used in conjunction with the above equations are:
P = 0 at E, = 0 front
('1 = 0; '— = T0 at E = -zf inlet (4-11)
bi = 0; T = T0 at t = O

For an isothermal process, the equations are independent of temperature and only the first
and third equations of (4-10) need to be solved. For an adiabatic reaction, the first two
equations of (4-10), when combined using the inlet conversion and temperature as the

initial conditions, yield

"11
1|
9|
.1.
"I

(4- 12)

63
Thus, taking

91(6r)aR(6t, T) Ema, 70+ 6:) (4-13)
we can rewrite the first equation of (4-10) as:

— = arm) (4-14)

The rate of reaction and viscosity, after scaling , is given by

are. T) = (1 - 602 “4&6 ‘ 'rl‘ll

_ a, —(i.5+a) 5,, 1 1
(0%. T) = (1- ) ex (———)
” age, p[RgATad T To

(4-15)

 

 

Incorporating Eq. 12 for an adiabatic process and using definitions for the dimensionless

activation energies for reaction and viscosity growth, ER and E,l respectively, shown in

Table 4.2, we get the following equations for the dimensionless reaction rate and viscosity:

ERG”:
10(10 + an]

- -(1.5 a) -E 0'1
11 = (l- or ) + exp[——P—:—]
age, T0(T0+a)

9i =(1—6t)2 exp[
(4-16)

 

The rate of reaction is plotted as a function of the conversion, for a range of values of the

activation energy ER, in Figure 4.2. Setting ER = 0 yields the isothermal case. For low
values of ER (up to a value of about 14.26), the derivative (dR/dor), described in later
sections, is negative, while at higher values the plot goes through a peak, corresponding to
a change in sign of the derivative. The effect of this derivative on the flow stability will be
discussed after the stability analysis has been presented.

Figure 4.3 shows the viscosity profile as a function of conversion, for different

64

values of EII’ the dimensionless activation energy for viscosity. E,,=0 corresponds to the
isothermal case, since the thermal dependence vanishes. For E,,<16.45, the viscosity is
always increasing with conversion. As 15,, is increased, the viscosity corresponding to a
given conversion becomes lower. Also, for higher values of E,,, the viscosity gradient

changes sign for increasing conversion. This implies that V, the gradient of the viscosity
defined later in this chapter, changes sign somewhere in the filled region, affecting the

stability of the flow process.

4.2.4 THE BASE STATE SOLUTION

To obtain the base state conversion profiles, the initial value problem described by

equation 4-14 is solved for different positions of the flow front, Zf, with the initial

condition given in (4-11). The independent variable, time, corresponds to a location (2) in

the filled portion of the mold, since

2 = U02
(4-17)
=> 2 = t

The base case is a plug flow for which every location in the filled portion of the mold
would correspond to a unique conversion and temperature with given inlet conditions.
Since it is a constant injection process, all fluid elements spend the same amount of time to
reach a given location in the mold and, therefore, would attain the same conversion when
they reach this location.

For an isothermal process the initial value problem can be solved analytically,
since the exponential term in equation 4-16 vanishes. For an adiabatic process, due to this

exponential factor, a numerical approach is needed. Once the conversion profile is

65

Figure 4.2 N on-dimensional reaction rate for polyurethane resin: isothermal and adiabatic
cases, To = 325°K

 

7.0

reaction rate

66

 

 

ER=53

 

Figure 4.2

conversion

 

 

67

Figure 4.3 N on-dimensional viscosity for polyurethane resin: isothermal and adiabatic
cases, To = 325°K

 

68

 

 

 

 

 

 

14.0 I I I I .
.4
To = 2.67
12.0 r "
10.0 r E ,,=16.45 7
8.0 - 5
>. 13,, =41
.15.
8 )-
°§ E,,=0
6.0 '- ..
4.0 '- "
gel
2.0 r ‘
0 0 l I 1 L g
0.0 0.2 0.4 0.6 0.8
conversion

Figure 4.3

1.0

69

available, the temperature profile is easily computed in the adiabatic case using equation
4-12.

The solution to equation 4-14 is shown in Figure 4.4, for several values of the
dimensionless activation energy and for a fixed inlet temperature, as a plot of conversion

versus the dimensionless distance from the mold inlet. The range of values for ER is
obtained by varying the activation energy E3. The case of ER = 0 corresponds to

“isothermal” filling, since the temperature dependence of the reaction rate vanishes. For
this case the fluid travels a larger distance in the mold before reaching the gel point,

compared to a case of high ER. Thus, as the process moves away from the "isothermal"

state, gel conversion is reached much faster and the length of mold that can be filled by the
resin is much smaller.

The solution to Darcy’s law, given in equation 4-10, is now obtained using the
conversion profiles of Figure 4.4. The pressure profiles are shown in Figure 4.5, for the

adiabatic case under the following conditions: ER = 53; E,, = 41; gel conversion = 0.65;

Inlet temperature = 325°K. The base state pressure is plotted against the dimensionless

distance, 2, from the mold inlet, for several flow front positions, Zf. As the mold gets filled,

the pressure at the inlet rises due to the increasing resin viscosity. This means that the
pressure at which the resin is pumped into the mold is continuously increasing, for a
constant injection rate to be maintained. For isothermal filling, however, the conversion
profile indicates a slower approach to gelation. The corresponding pressure profile for the
isothermal case would be expected to show lower injection pressures for the same extent
of mold filling.

The derivative of the reaction rate w.r.t. conversion, (dR/dor), is termed S and is

70

1.0‘1

0.9 -

' as“: 0.65
0.8 -

O.7_ tgu=o.23 13,411.93 18:,=1.36

05- Baa-53.0 53:14.26
0.5 -

 

RI
7:

0.4 _ 53:00 (isomer-trial)

0.3 -

 

O .2 _ To=2.67

0.1-

 

0.0

 

0.0 0.4 0.8 1.2 l 1.6 . 2.0

Figure 4.4 Base state conversion profiles: isothermal and adiabatic cases, To = 325°K

71

Figure 4.5 Base state dimensionless pressure profiles: adiabatic case, ER = 53, E" = 41,
age, = 0.65, T0 = 325°K

 

 

72

:1.
E1

"Dl

 

 

 

I l v Q
0.00 0.04 0.08 0.12 0.16 0.20 0.24 .

Figure 4.5

 

73

found to have a significant role on the stability of the flow process. Figure 4.6 shows S and
the mobility gradient V, (defined as d[ln X]/d§), for the polyurethane resin up to a
conversion of about 0.6, under isothermal conditions. By this time the viscosity has risen
to a fairly large value (gel conversion is 0.65). As will be explained in a later section, S < 0
is a necessary condition for stability and V at 0 in the domain of the problem is a sufficient
condition for real growth rates (from an exchange of stabilities analysis). It is seen that
both S and V are negative throughout the domain for the isothermal case, which means
that the growth rate is real and negative for at least some wave numbers (in the small

wavenumber region). The quantities S and V are calculated and plotted for the adiabatic
case with ER=52.67 and an inlet temperature of 325°K in Figure 4.7. For this case while S

is always positive, V changes sign in the flow domain considered. Hence it is expected that

at all lengths of filling (2f), the flow would be unstable to disturbances. The behavior of S,
for different values of ER, may also be seen from an inspection of the plot of rate versus

conversion (Figure 4.2). For small values of the dimensionless activation energy, i.e. for

ER less than about 14.26, the reaction rate curves have a negative slope at all conversions.

This suggests stability of the flow process under these conditions.

4.2.5 STABILITY ANALYSIS

In order to study the growth behavior of infinitesimal disturbances that the filling
process may be subjected to, a linear stability analysis is to be performed on the model
equations in (4—6). Perturbations on each of the variables are introduced into the problem,

described by:

74

 

 

 

 

 

0.0 - . . . .

, .

.21) .

I l

:3 4.0 - ’ ' _ .
-6.0 b «

: ‘
's'qis ‘ i0 ‘ D .05 A 0.0

§

Figure 4.6 S, V profiles for isothermal filling: ER = 53, 13,, = 41, age, = 0.65, T0 = 325°K

75

 

5.0

-5.0

 

40.012

 

Figure 4.7 S, V profiles for adiabatic filling: ER = 5, E,, = 41, age, = 0.65, To = 325°K

76

P: P (i, t) +P'(x, y, g, t)
it; t) + Nor. y, t. t)
= a(§, t) + a'(x, y, g, t) (4'18)
q(§. t) + q’(x. y. 5. t)
= T(§. t) + T(x. y. 6 t)

NIQQP

Here the base state variables are dependent only on one space variable (E) and time, while
the disturbances (primed quantities) are three dimensional and time dependent. All the
variables in equations 4-6 and 4-7 are replaced by the representation given in (4-18) and
the equations are then linearized. For an adiabatic reaction, the lumped energy balance
may be combined with the component mass balance and equation 4.12 for the base state
solution, to yield the following relation between the temperature and conversion
perturbations: T’ = or’. This is valid for reactions 'that are much faster than the rate at
which heat transfer takes place. Linearization of the continuity equation incorporating
Darcy's law yields

- 32 32 a(- _d__'p ).
l b — ' b — ' +— l. -—)= 0 4-19
[‘ .2“ 2a,”) a: 8‘5 1 ‘ ’

while linearization of the reactant mass balance results in

aa' +(k_’_ 7-). $31)?) 3&_ dSR or’ (44,»

a: a: at:

Frozen profile approximation: It may be noted that the base state in the filling problem is
time dependent. It is assumed that the growth rate of disturbances is much faster than the
rate of change of base state. That is, a'quasi-steady state is assumed and the stability

analysis is done with time frozen at to. The rate of change of the base state, dE/dt, may be

77

represented by the reaction rate R, which provides a bound for the time of applicability of
the quasi-steady state approximation. For times of the order of about ln(R) and lower, the

QSSA is inapplicable. From the conversion profiles in Figure 4.4, at large values of ER,
d-dldt is much higher than at low ER. This suggests that the quasi-steady state
approximation may possibly be invalid at large values of ER.

In the limit of applicability of the QSSA, the disturbances are, decomposed into

Fourier components:
01' ~ A(5) t”=XPIi(k,.X + kyYI + 000”]
12' ~ p(§) eXP[i(k,x + kyy) + 600m (421)
1.":- 37-; 01' ~ 5% A(§) exp[i(kxx + kyy) + o(t0)t]

Here, it is noted that N is tied to the conversion disturbance 11' since viscosity is directly

related to conversion. In these equations kx and It). are the wavenumbers of the

disturbances in the respective directions and are, later, combined into a single

wavenumber, k. 0(t0) is the quasistatic growth rate of the disturbance, evaluated at time t0

and is a complex quantity. If 0' has a negative real part the disturbances would decay with
time, while they would grow if it has a positive real part. If the imaginary part of the

growth rate is non-zero, then it indicates an oscillatory trajectory for the disturbances with
time. It may be shown that it is a real quantity for this eigenvalue problem (Appendix A).

Using the above definitions of the disturbances, we get from eq. 4-19,

8 -8p 1 d - - 2
AF— “:11 A = A bk
34 at Ma (§)] ( )p (442)

where bk2 = b,k§+b,k§

A relation obtained by substituting the Fourier expansions of the disturbances into

78
eq. (4-20) is

- d a -

(4-23)
This equation relates the amplitudes of conversion and pressure disturbances and is a

useful relation in simplifying the equations at a later stage.

Combining eqs. 4-22 and 4-23 we get:

a - dp (O-S) _ 2 " -
'a'EP‘ dE (o—S+V)] ' bk 1” (4 24)
where S, V are defined by
d9i
55-1
1 d dim d d (4.25)

In equations 4-25, S and V are the quantities referred to during the discussion of the base
state solution. The former is the derivative of the reaction rate w.r.t. concentration
(conversion) and the latter is a viscosity gradient. Both these quantities are seen to be
associated with the growth rate 0', which is also the eigen value in equation 4-23, and so
determine the stability characteristics of the filling process. An analysis of the exchange of
stabilities for the eigen value problem indicates that a sufficient condition for the growth
rate to be a real quantity is a non-zero V, or that V does not change sign in the entire
domain. It may be shown that a sufficient condition for the quasistatic growth rate, 0', to be
a real quantity is that V does not change sign in the entire domain, and is non-zero.

The boundary conditions used in solving the eigen value problem are

79
dp _ _ _
d—fi - at g — Zf (4’26)
p = 0 at E = 0
The first condition is obtained by taking the disturbance in conversion to be zero (i.e. A=0)
at the mold inlet and using eq. (4-22), while the second boundary condition implies an

undisturbed pressure at the flow front.

The transformation

" _ 4W
A p ’ 275 (4-27)
with I]! = 0 at the inlet
leads to the following simplified eigenvalue problem:
d2 div S + V
J—V—-b k2["——" ] =0
4:2 dz; o-S ‘1’ (4-28)
with the boundary conditions
I)! = 0 at E = —zf
(4-29)

§§V=0 at §=0

Equation 4—28 may be compared to the eigenvalue problem developed by Hickemell and
Yortsos (1986) in eq. (2.19) in their paper, rewritten as:
illLv $-13 [11!] 1|! = 0 (4-30)
(15,2 dé o
This equation is subject to decay of the disturbance at infinite distances on either side of
the interface. It is easily seen that the main difference between equations 4-28 and 4-30 is
the factor {-8) added to the growth rate in equation 4-28. The boundary conditions in our

problem .ply refer to decay of the pressure and conversion disturbances at the inlet and

80

front respectively. When we take a chemical reaction with S = 0, i.e. a zeroth order
reaction, then, the behavior of the fingers would be predicted by the results in their paper,

i.e. the growth rate is bound by the values of the viscosity gradient:

inf V<O<sup V
i i

That is, the growth rates are bound by the infemum and supremum values of V in the

(4-31)

domain of the problem. If V is positive everywhere in the domain, the flow process is
unstable for all wavenumbers.

When the chemical reaction is of a higher order, however, a direct comparison
cannot be made. Even though S is being subtracted from the growth rate (in equation 4-
28), the fact that S, unlike the growth rate, is dependent on the location, a, complicates the
problem; if S were independent of the location, (0 - S) could be combined into a new,

reduced, growth rate and the results of Hickemell and Yortsos (1986) applied.

4.3 NUMERICAL APPROACH

The results of the stability calculations are presented as growth rate data and
marginal stability curves, solving the eigen value problem (equation 428), using the
boundary conditions in equation 4-29. This equation is solved using an orthogonal
collocation scheme on finite elements (Finlayson, 1980).

The numerical procedure is illustrated in the schematic shown in Figure 4.8. The
flow domain is divided into two elements based on the viscosity profile. The interface

between the two elements, depicted by z, in the figure, is the location in the flow domain

where the viscosity gradient increases sharply. The number of collocation points in each

81

Figure 4.8 Schematic of the numerical scheme - orthogonal collocation on finite elements
with Legendre polynomials

82

i -+ 1 2 N+l IF N+3 2N+2 F

Z _> 0 21 If

E —> -zf 0
Elementl 7 Element 11

Figure 4.8

83

"i"

element is N. The index in this figure denotes each collocation point. Legendre'
polynomials are used to approximate the solution in each element. The domain in each
element is then transformed into the Legendre’ domain where the boundary locations are
at 0 and 1. Thus we have 2N collocation points at which the eigenvalue problem (equation

4-28) is applied. The boundary conditions in equation 4-29 are applied at the left end of

the first element (2 = 0, the inlet) and at the right end of the second element (2 = Zf, the

front). The solution from each element is matched at the interface between the two

elements (location 21); that is, the solution I]! and its first derivative from each element are

equated). We obtain a system 0f linear, homogeneous, algebraic equations with the

unknowns being the values of w at each collocation point, as well as the eigen value, 0.

”(2N+2)x(2~+2)"1’(2~+2)x1 = 9(2~+2)x1 (4'32)
That is, we have (2N+2) equations with (2N+2) unknowns. There are two other
parameters: the wavenumber, k, and the quasistatic growth rate, 0'. A nontrivial solution
for this system of equations requires that the coefficient matrix, M, of the linear system be
singular. The resulting characteristic equation for 0' solved at a range of chosen values for
the parameter, k and the most dominant root is taken as the growth rate. The eigenvalue
problem is solved for several values of N, the number of collocation points in each
element and convergence was observed for N=6.

The cutoff wavenumber is defined as the wavenumber at which the growth rate

changes its sign. This is computed directly from the characteristic equation by setting a =
0 and solving the resulting polynomial in (k). The characteristic equation has also been

used to obtain marginal stability plots on a variety of parameter planes.

84
4.4 RESULTS AND DISCUSSION

The base state solution, as obtained for a polyurethane resin with second order
kinetics, was discussed in section 4.3. The solution of the eigen value problem is presented
and discussed in this section as growth rate and marginal stability curves. Some of the
operating conditions and properties of the resin are then varied and their effect on the flow

stability of each of these new formulations is examined.
4.4.1 ISOTHERMAL REACTION

It is instructive to consider the ER = 0 limit of isothermal reactive filling at first.

This limit is more representative of therrnosetting polyester resins, for which the gel
conversion is very low and the heat liberated by reaction is low upto the gel point.

The growth rate 0’ is evaluated for this case at a few locations (Zf) of the flow front

and plotted against the wave number (represented by bmk) in Figure 4.9. It is seen that,

for any 2f, the growth rate is negative at low wavenumbers and becomes positive for larger

values of (k). This implies that the process is stable for disturbances of large wavelength.
The results for isothermal filling are shown in Table 4.3 where the cutoff

wavenumbers are tabulated for varying filling time and gel conversion. The range of

' wavenumbers over which the flow is stable may be shown on a plot of the cutoff

wavenumbers. Figure 4.10 is one such plot, in which the cutoff wavenumbers are plotted

as a function of the Damkohler number, which is the ratio between flow and reaction time

scales. It can easily be shown that the Damkohler number (Da) is equivalent to the length

of the mold that has been filled by the resin. It is seen that the flow becomes unstable for a

85

Figure 4.9 Growth rate curves for isothermal filling of polyurethane resin: 2, = 0.8, 1.0,
1.2, 1.5; ER = 53, E,l = 41, age, = 0.65, T0 = 325°K

86

 

 

 

 

 

 

 

 

 

 

 

’ o—o z,= 1.0 '
3.0 ' H zf = 0'8 -I
H 2" = 1.2
’ Ia—a zf= 1.5 1
2.0 - -
G 1.0 " -
0.0
1 r
.1 o __ ————— .
-2.o , - ' - . -
0.0 0 2 0.4 0.6 0.8
k «lb

Figure 4.9

Table 4.3 Computational results for isothermal reaction

87

To = 325°K; ER = 52.64; E,, = 40.68

—AHR = 96,300 J/mol-°K; (pop) = 1909221106 J/m3-°K

Arm, = 121.56°C

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

“8 I If I “f I (018), (b=kI.0) (0:01)
VARYING EXTENT OF FILLING (2,)
0.65 05 0.3333 6.1760 2.4852 7.8588
0.65 0.7 0.4118 1.6992 1.2920 4.0856
0.65 0.9 0.4737 0.5449 0.7382 2.3343
0.65 1.2 0.5455 0.1016 0.3187 1.0080
0.65 1.5 0.6000 0.0132 0.1150 0.3637
VARYING GEL CONVERSION

0.65 1.5 0.6000 0.0132 0.1 150 0.3637
0.73 1.5 0.6000 0.0986 0.3140 0.9930
0.8 1.5 0.6000 0.2597 0.5096 1.6115
0.9 1.5 0.6000 0.7700 0.8775 2.7749

 

 

88

Figure 4.10 Marginal stability curve for isothermal filling, data from figure 9:
Damkohler number (Da = flow time I reaction time)

89

 

2.0

   

k‘lb

1.0 '

 

Unstable

 

 

 

 

Figure 4.10

90

wider range of wave numbers, with increasing Zf. This implies that, as the mold gets filled

to a greater length, the flow becomes increasingly unstable. At the larger filling times, the
degree of conversion is larger than at small filling times; from Figure 4.6, it may be seen
that at these larger conversions, the magnitude of S is lower (sign of S is negative). Since
the flow is more stable when S is more negative, the results of Figure 4.10 are
understandable. In terms of the ratio of the time scales, a small Da indicates a slower

chemical reaction in comparison to the flow rate.

Efi’ect of anisotropic permeability: The effect of anisotropic permeability is seen by
comparing the ordinate in Figure 4.10 for several values of the degree of anisotropy, b. It is
clear from this figure that when the transverse permeability is smaller than the
permeability in the longitudinal direction (b < 1), the cutoff wavenumber, kc, is larger than
it is for the isotropic case (b = 1). That is, the range of wavenumbers for which the flow
becomes stable is greater when the transverse permeability is small. This may be
understood from an inspection of equation 4—19 which is a combination of the equation of
continuity and Darcy’s law for disturbances. When the transverse permeability is small,
i.e. when b, and b2 are small, conservation of mass dictates that for a disturbance to
survive or propagate, the spatial pressure gradients in the disturbances have to be very
large. This means that only sharp fingers (disturbances of large wavenumbers) would
grow, while those of small wavenumbers are eliminated. Thus anisotropy with low

transverse permeability helps in elimination of large fingers.

Efiect of order of reaction and S: The dependence of reaction rate on conversion, i.e. the

91

order of reaction, governs the stability of the flow process. To understand the effect of
reaction on the stability of the system, we compare the growth rates of the second order
reaction system with the solution to the ei gen value problem with S=0. The latter situation
corresponds to a reaction with zeroth order kinetics with the same inlet and front
conversions, and the same viscosity profile. Figure 4.11 is a plot of the growth rate curves
a resin with the same viscosity behavior but with 8:0. It is seen that for this case the
growth rate is always positive, while for S < 0 it was negative for large disturbances. Thus,
if the same converison profile is obtained with a zeroth order reaction, there is no stable
region. Only when S<0 (for the second order reaction), do we get a stable region.

In Figures 4.9 and 4.11, the plots approach a finite growth rate asymptotically, for
large wavenumbers. These asymptotic limits are plotted for various flow front positions
for both S=0 and S<0, termed ultimate growth rate (on), in Figure 4.12. This ultimate
growth rate is seen to be higher for the no-reaction case, again indicating the dependence

of flow stability on the order of reaction.

The results from an analysis of the isothermal mold filling, thus, provide us an idea
of the behavior of the flow process governing mold filling, and it is expected that adiabatic

operation of liquid molding would lead to similar results.

4.4.2 ADIABATIC REACTION

Some liquid molding operations can be carried out under adiabatic conditions.
Mathematically this is slightly more complex, due to the involvement of an additional

balance equation (energy balance); however the adiabatic nature simplifies the equations

92

Figure 4.1 1 Growth rate curves for S = 0 for isothermal filling for identical base state
viscosity profiles as for Figure 9

93

 

 

 

 

 

 

 

500 T ' I ' U I
z, = 1.5
P If = 1.0 I
z, = 0.8
4.0 P If: 1.2 I
3.0 " ‘
6 b I
2.0 " "
1.0 I- 5
0.0 .. l A I A l j
0.1 I 0.10 0.20 0.30 0.40
k ‘lb

Figure 4.1 1

94

Figure 4.12 Upper limit of growth rate Vs Damkohler number for isothermal filling, To
= 325°K

95

 

8.0
7.0 I
6.0 .

5.0 '

3.0 '
2.0 "

1.0 "

 

 

 

 

S<0

 

 

Figure 4.12

1.6

96

to some extent, as seen in section 4.3. To be even more realistic, fully non-isothermal
equations would have to be solved, which render the model equations and their stability
analyses, and consequently their solution, more complicated.

The results of the computations are now presented for mold filling under adiabatic

conditions. Figure 4.13 shows the growth rate curves for several values of ER. ER may be
altered by modifying the resin formulation to one with a different activation energy or a
different heat of reaction. It is seen that for small values of ER, there exists a range of

wavenumbers with a negative growth rate, corresponding to a stable regime. This leads us
to examine the stability characteristics of other formulations of the polyurethane resin,
obtained by altering the kinetic and rheological parameters. The stability of the filling
process may be affected by the thermochemical properties of the resin and the operating
conditions. In the following analysis, several parameters are varied independently and the
flow stability examined. In each of these cases, except for the case of varying Zf, the mold
is filled to the same length enable comparison. The effect of varying the inlet temperature

is also presented. The results are presented in the form of marginal stability plots.

Dependence of stable regime on various parameters:
For the adiabatic case, the parameters that may be varied are the fill time (which is

equivalent to the Damkohler number), adiabatic temperature rise (ATad) (which may be

changed by choosing appropriate values of the inlet concentration (c0) and the heat of
reaction (-AHR)), the kinetic activation energy (15,), the inlet temperature (T o) of the resin

and the gel conversion (age,) of the polymer system. Most of these parametric studies have

been done for a resin with the following properties:

97

 

 

 

 

 

 

 

 

 

k‘lb

Figure 4.13 Growth rate curves for varying activation energy during adiabatic fill

98
IE;m = 131.17; ER = 168.96; ATad = 37.87°K; age, = 0.8; reference temperature = 370°K.

The activation energy and initial concentration of the resin were not changed from the

values taken for earlier calculations but the heat of reaction was altered from 96,300 J/mol
to 30,000 I/mol, which changed the adiabatic activation energy. For this lower exotherm, it
was found that the viscosity gradient, V, did not change sign, thus satisfying the condition
for exchange of stabilities. The results of the computations are presented in Tables 4.4 and

4.5.

a) The effect of changing the dimensionless activation energy, ER, on the stability at a

fixed inlet temperature, noted earlier in Figure 4.13, is shown in Figure 4.14 as a marginal

stability plot. As observed earlier, a decrease in ER leads to an expansion of the range of
stable wavenumbers, in the low-wavenumber region. This change in ER may be
accomplished by decreasing the kinetic activation energy, 13,. A higher activation energy

means that more energy is required for the polymerization reaction to occur, which makes
it more difficult for the viscosity on either side of the edge of an incipient finger to rise.
This, in turn, means that the driving force available for finger growth (pressure drop across

the edge of the disturbance) does not reduce as fast as it would for a lower Ea case, making

it more stable than the case of a higher Ea.

b) Figure 4.15 depicts marginal stability plots for changing Da (= tF/tR) and varying inlet
temperature, in adiabatic mold filling using a resin with ER = 14.26, which does have a

stable regime as seen in Figure 4.14. The plot indicates, as in the isothermal case, that the

Table 4.4 Computational results for adiabatic reaction

99

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

No. (AT)ad ER To age, 15,, t
= : _ :-
Varying ER with fixed To

1. 121.56 52.67 2.67 0.65 40.86 0.206
2. 121.56 25.00 2.67 0.65 40.86 0.527
3. 121.56 14.26 2.67 0.65 40.86 0.801
4. 121.56 5.00 2.67 0.65 40.86 1.194

Varying 05.1
5. 37.87 168.96 9.242 0.65 131.17 0.85
6. 37.87 168.96 9.242 0.80 131.17 0.85
7. 37.87 168.96 9.242 0.90 131.17 0.85
Varying T.
8. 37.87 168.96 8.58 0.80 131.17 0.85
9. 37.87 168.96 8.98 0.80 131.17 0.85
10. 37.87 168.96 9.242 0.80 131.17 0.85
11. 37.87 168.96 9.51 0.80 131.17 0.85

 

Varying Exotherm, which alters both (A'T).d and ER, hence To

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

12. 25.25 253.45 13.86 0.80 196.73 0.85
13 37.87 168.96 9.24 0.80 131.17 0.85
14. 50.50 126.72 6.93 0.80 98.36 0.85
Varying filling time
15. 37.87 168.96 9.242 0.8 131.17 0.535
16. 37.87 168.96 9.242 0.8 131.17 0.750
17. 37 .87 168.96 9.242 0.8 131.17 0.850
18. 37.87 168.96 9.242 0.8 131.17 0.950

 

 

 

100

Table 4.5 Cutoff wavenumber data for adiabatic reaction

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

No. I 011‘)“, BR TO I 13,l 1 I (bk2)c
fl VaryingT—o—_—__—
1. 37.87 168.96 8.58 131.17 0.85 0
2. 37.87 168.96 8.98 131.17 0.85 0
3. 37.87 168.96 9.242 131.17 0.85 0.47096
4. 37.87 168.96 9.51 131.17 0.85 0.5435
Varying filling time
5. 37.87 168.96 9.242 131.17 0.535 3.2122
6. 37.87 168.96 9.242 131.17 0.750 0.9041
7. 37.87 168.96 9.242 131.17 0.850 0.47096
8. 37.87 168.96 9.242 131.17 0.950 0.2281

 

 

 

101

 

 

 

 

3.0 - . - . .
2.0 - I
k «lb . I
1.0 - 4
p I
0'000 5.0 10.0 15.0
ER

Figure 4.14 Marginal stability curve for adiabatic filling: activation energy

102

region of stability becomes smaller with increasing fill time. The reason for this behavior
follows exactly from that discussed in the isothermal case. Also, an increase in the inlet
temperature is found to expand the range of stable wavenumbers of fingers. The higher
temperature would make the reaction go faster, thus making the viscosity profile less
steep. This reduces the pressure drop across the edge of the disturbance, making the

process less unstable. This effect is opposite to that of the activation energy.

c) The effect of anisotropic permeability in adiabatic filling is seen, as in the isothermal
case, by looking at the ordinate in Figure 4.15 for various values of the degree of
anisotropy, b. The range of wavenumbers over which fingers are damped is much larger

for the case with lower transverse permeability.

c) Figure 4.16 depicts the effect of gel conversion on the stability of the mold filling
process. A larger gel conversion implies stability over a wider range of wavenumbers. At
lower gel conversions, the viscosity increase with conversion is faster, and makes the
molecules less mobile. The number of sites available for further reaction, as a
consequence, is much lower and this causes the driving force for finger growth to decrease

at a slower rate, making the process more unstable.

Geometric limitations: The wavenumbers (k) referred to above may be converted to
appropriate wavelengths of the fingers (1“,), using the relation (k = 211: 15c] Aw), where lsc is
the length scale for the problem. To illustrate the significance of these results, the finger

wavelengths corresponding to the cutoff wavenumbers are compared to the actual physical

103

Figure 4.15 Marginal stability curve for adiabatic filling: fill time and varying
temperature; ER = 169, E,, = 131, age, = 0.80, ATM = 38°K

104

 

 
    

1'8‘ -III—To=350K

+TO=360K
+TO=380K

1.6 5

1.4 4

12*

1111)

085
C

0.4 ~

0.2:

 

 

 

0.5 0.6 0.7 0.8 0.9
till time I reaction time

Figure 4.15

105

1.4

 

1.25

08.
e

klb

0.6 -

0.4 -

0.2~

 

0 . . a

 

 

0.6 0.65 0.7 0.75 0.8 0.85 0.9
gel conversion

Figure 4.16 Marginal stability for adiabatic filling: gel conversion

 

106

dimensions of the mold. The thickness of the mold, B, represents the largest possible

finger size, while the diameter of the fiber tow, d,, is the smallest possible finger width.
Representative values are chosen as below:

Mold thickness, B: 10mm

Fiber tow diameter, d,: 0.1mm

Injection velocity, U0: 0.5 cm/s

The data for polyurethane resins given in Table 4.1 are used, to obtain a length scale (15,)

of 0.0637 mm.

The results of Figure 4.16 are now transformed into wavelengths. From Table 4.5,
the cutoff wavenumber when 2, is 0.535 is kc = 1.79. When converted to dimensional
finger widths, this corresponds to 2., = 0.22 mm. All fingers narrower than 0.22 mm will
grow; conversely, all fingers wider than this are eliminated. This limiting finger width is
now compared to the physical dimensions B and d,. It is seen that the tow diameter is
much less than kc; similarly, the mold thickness is much larger than this cutoff value.
While very large fingers are eliminated, small fingers are undamped. A similar analysis
may be done for other values of Zf, and it is clear that, as the mold gets filled, even fingers
of large sins tend to grow.

The above discussion was done for the case of isotropic permeability (b=1). As
dicussed earlier, if the transverse permeability is less than the longitudinal permeability (b
< 1), the cutoff wavenumbers are larger, which means that a wider range of disturbance

wavelengths is stabilized. For example, when b=0.l, we get kc = 5.67, which corresponds

to a wavelength of 71,, = 0.07mm. All disturbances wider than this are eliminated. Since

107

this is less than the fiber tow diameter, it may be concluded that any disturbance that is
generated by the adverse viscosity gradient is damped out due to the low transverse

permeability.
4.5 MECHANISM OF STABILIZATION
4.5.1 EFFECT OF S ON THE GROWTH RATE

Eq. 4-23 may also be expressed, using Eq. 4—27, as

o-S - dp __ 2
[o—-S+V] 2. FE _ b k u (4.33)

From this equation it is seen that when the wavenumber is zero, for non-zero amplitudes
of the disturbances, the growth rate is equal to S. The necessary condition for dampening
of disturbances is, then,
S < 0 (4-34)

This means that there is a possibility of stabilizing disturbances of small wavenumbers
(large finger widths) when the criterion in equation (4-34) is satisfied.

The effect of S on the growth rate is demonstrated by the schematic shown in
Figure 4.17, which represents an incipient disturbance at some instant. The line AA’ is the
base state solution. Before the onset of the disturbance (t<0), the points M-, M-I- are at the
same, base state, conversion. The deviation from the base state, shown in the figure,
corresponds to an increase in conversion (positive amplitude, A(E)); at this time (t=0), the
conversions at M-, M+ are in an increasing order.

Ifthere were no disturbance, then the fluid particles at the two points would react

at the same rate and remain at the base state conversion at t > 0. In the presence of the

 

108

 

Fluid I
viscosity 111

 

 

 

Figure 4.17 Schematic representation of the effect of S on growth rate

109

disturbance, however, several possibilities exist for the behavior of the disturbance at t > 0,
depending on the shape of the reaction rate curve (Figure 4.2).
i) If S < 0, since the conversion at M- is lesser than that at M+; thus the polymerization
rate at M- is higher than that at M+, bringing the conversions at the two points closer. This
decreases the amplitude of the disturbance at t > 0.
ii) If S > 0, the higher conversion at M+ results in a higher rate of polymerization than at
M-, widening the gap in conversion between the two points at longer times, t > 0. Thus the
disturbance grows with time for positive values of S.

The above discussion was presented by taking k = 0, to demonstrate the effect of S
on the growth rate. However, it has to be remembered that the viscosity gradient, V, also

dictates the stability behavior of fingers.

4.5.2 STABILIZATION OF LARGE FINGERS

It was seen that chemical reaction, in addition to creating an adverse viscosity
gradient that causes fingers, can couple with the flow process to stabilize large fingers but
not small fingers. The mechanism for this inability to eliminate small fingers may be
explained by plotting the growth rate against the ratio, It, between a reference growth rate
and the reaction rate. The former is the growth rate, at a given wavenumber, for a
hypothetical fluid with the same viscosity and conversion profiles as the resin, but with S =
0, i.e. a zeroth order reaction. A low value of the ratio K, then, corresponds to a fast
reaction in comparison to disturbance growth rate. In addition the reference growth rate, as
seen in Figure 4.11, increases with an increasing wavenumber; thus a small value of 1C

implies small wavenumbers and vice versa. To illustrate the mechanism, the growth rate vs

110

Figure 4.18 Effect of S on growth rates for identical conversion and viscosity profiles in
the base state

4.0

111

 

3.0 '

2.0 '

growth rate

 

0.0

-l.0
0.0

Figure 4.18

A

S<0

1.0 2.0 3.0

reference growth rate / reaction rate

 

4.0

112

K plot for a specific case is shown in Figure 4.18. It is seen that the growth rate is negative
for small It and becomes positive at large K. At large values of K (short wavelengths), the
time scale for finger growth is much smaller than the reaction time scale. This means that
the growth of fingers occurs faster than chemical reaction can damp them out. Since large
values of 1: correspond to small finger widths, we may conclude that small fingers cannot
be eliminated by the reactive coupling mechanism. On the other hand, at low It (large
wavelength), the reaction time scale is small, which means that the chemical reaction is
faster than the rate at which the finger would grow; this enables damping of these large

fingers.

4.6. SUMMARY

Flow of reacting resin through a fiber preform located in a a mold produces a
continuous distribution of resin age from the inlet to the advancing resin front. The effect
of this age distribution on fluid mixing within the fille dregion has been examined with a
linear stability analysis. The results of this analysis for isothermal and adiabatic reactive
filling indicates that the role of reaction is two fold. In the first place, it generates an
adverse viscosity gradient, which leads to generation of fingers. Secondly, it contributes to
stabilization of these fingers at smaller wavenumbers.

The effect of reactive coupling on the fingering is derived as an eigen value
problem, the largest eigen values of which are the growth rates used to determine the
stability behavior of the filling process. Reactive coupling occurs in the form of a quantity

S, defined as a gradient of the reaction rate with conversion, and V, the viscosity gradient.

113

For positive values of S, the process is always unstable and disturbances of all
wavelengths would grow; for polymeric systems with a negative S throughout the flow
domain, mold filling can be stable and large disturbances can be damped. Thus the
kinetics of the polymerization reaction determines the stability of the mold filling process.
The stability characteristics of a polyurethane resin have been investigated for a
variety of parameter planes. It is seen that the resin may be reformulated suitably, altering
the kinetics or rheological properties, to reduce the amount of fingering during mold
filling. The gel conversion, adiabatic temperature rise and transverse permeability may be
increased to help eliminate fingers, while the process becomes less stable as the mold gets

filled. Stabilization may also be achieved by raising the inlet temperature.

 

CHAPTER 5 VISCOUS FINGERING DURING REACTIVE FILLING OF FIBER

PREFORMS: EFFECTS OF DISPERSION AND REACTION

5.1 INTRODUCTION

Mold filling in liquid composite molding processes involves the flow of a
polymerizing liquid mixture through an anisotropic porous medium comprised of fiber
reinforcements. The work of Homsy and co-workers has established that dispersion within
the porous medium is a stabilizing influence at higher wavenumbers or narrower finger
widths. In the present application, spatial variations in monomer concentration are
brought about not only by dispersion but also by the polymerization reaction within the
porous medium. Hence it is important to understand the influence of these two factors on
the finger widths most likely to grow during the mold filling process.

The fiber bed is, in general, anisotropic both for permeation of the resin through the
preform and dispersion of the monomer in the reaction mixture. The permeability of the
preform is determined by its physical structure and can vary according to the kind of fiber
arrangement chosen, viz. random or aligned. The effective dispersion coefficient is
dependent on the characteristics of both the porous medium and the resin. Due to the
polymerization reaction, the molecular weight of the resin increases with time; this affects
the dispersion of the monomeric resin (of low molecular weight) through the polymerized
resin. The bulk dispersion depends on the injection velocity as well as the length scale of

the fiber bed.

114

115

The object of this chapter is to analyze viscous fingering during reactive filling of an
end gated mold containing a pre-located porous fiber bed. The mechanism for growth or
attenuation of fingers in flow of a reacting liquid mixture through a porous medium is
explored here. This mechanism is directly related to the reaction kinetics of the resin and
dispersion characteristics of the monomer through the polymerized resin and will also lead

to criteria for attenuation of fingers in the presence of reactive and dispersive effects.

5.2 MODEL PROBLEM AND EQUATIONS

In order to study the combined effect of dispersion and reaction on the generation of
fingers, the flow situation is described by a simplified model problem. The physical
problem is posed in the context of multiple shot injection of two-component resins, mixed
in-line, into a mold. A batch of fluid is injected into the mold and allowed to react for
some time, during which it undergoes polymerization, with a rise in viscosity. At a later
time (t=0), another shot of resin is injected. Due to the difference in ages of the two shots
of fluid, a step jump A01 in the extent of reaction is observed at the interface of the two
shots. This multiple shot approach was used by Losure (1994) to prepare molded
composites with a variety of fingered rnicrostructure; the mechanical properties of such
composite specimens were found to be affected significantly by the details of the
microstructure. It may also be noted that the step jump in the extent of reaction is quite
different from the continuous conversion profile seen in actual liquid molding operations,
as discussed in chapter 4. The purpose of this chapter is to present a simplified analysis of

the combined effects of dispersion and chemical reaction on the stability of the mold

116

filling process. A more realistic problem would involve a continuous conversion profile
(hence, viscosity profile), which will not be part of this dissertation.

In order to delineate clearly the effect of chemical reaction on fingering,
mathematically, we pose the problem in an infinite porous medium, as in the work of Tan
& Homsy (1986) which addressed dispersion alone. The locations far from an incipient
disturbance, in this analysis, are assumed to be unaffected by it and, therefore, may be
assumed to correspond to infinity with no significant loss of accuracy. Figure 5.1a depicts
the flow domain at the initial time t=0. Region I in this figure is resin that has been injected
and allowed to react till an extent of reaction or, (corresponding to viscosity Ill). The
instant of time when fresh resin with an extent of reaction 01/2 (viscosity 112) is injected is
termed (t = 0) for the purposes of this analysis. The velocity of injection is a constant, U0.

The defining equations for this model problem are the equation of continuity,
Darcy's law (for flow through porous media), an equation for the location of the moving
flow front and a component mass balance written for the monomer concentration. The
bulk flow is, as in chapter 4, in the z-direction. The differential balance equations
governing this problem, expressing the velocity as a three dimensional vector since it is

required for the later stability analysis, are

Continuity V013 = 0
Darcy’s Law 12 = 35—171 KOVP (5_1)
Reactant Mass Balance %%+ 13 0 V6 = — R + V0[D 0 V6]

The velocity vector in Darcy’s Law represents the local velocity inside the porous

medium. In the reactant mass balance equation, the change in concentration both due to

117

Figure 5.1a Flow of reacting liquid through a porous medium (moving frame)

Figure 5. lb Profiles of conversion, viscosity and S at t=0

 

118

velocity U0

——>
Regionr

—’ RegionH V
112. S2. 00 11151.00

      

—_>

1’20:
Figure 5.1a

0‘1, 111

0‘2, H2

 

 

SEER/BE 8:82

Figure 5.1b

 

119

mass dispersion and chemical reaction are included; the former was missing in the balance
equations dealt with in chapter 4.

A moving frame of reference is defined, using the uniform flow velocity U0 and the
location of the interface zI, introducing the new coordinate E. In this moving reference

frame, the x- and y- coordinates are unaltered due to the one-dimensional nature of the

bulk flow.

2:24,st 2‘ (5-2)

0

The velocity vector in the moving frame of reference has a corresponding change.

= ill-U0 i3 (5-3)

IQ>

The model equations may then be rewritten, in the moving reference frame, as

Continuity Voq = 0
Darcy’s Law Q+Uo i3 = 11:17, KOVP (54)
Reactant Mass Balance $+§0VE = — R+VO[DO V8]

Figure 5.1b schematically shows the profiles of conversion and viscosity at the
initial time, t=0, when the second shot of fluid has been injected. Both these quantities see
a step jump at E =0. (The quantity 8 on both Figures 5.1a and 5.1b will be discussed in

section 5.4.)

5.3 SCALING AND DIMENSIONLESS FORM OF THE EQUATIONS

Appropriate scales are used to make these equations dimensionless. The problem

discussed in chapter 4 had chemical reaction as the only important physical phenomenon,

 

 

 

120

which made the choice of scales for length and time rather easy. In this problem, however,
both mass dispersion and chemical reaction are dominant mechanisms. As a result we
have two possible choices for the length and time scales: one based on the reaction
kinetics and another based on dispersion. In keeping with the work of Tan and Homsy
(1986) and others, who solved similar problems but with no chemical reaction, the length

and time scales for this problem are based on the dispersion characteristics of the porous

medium and are taken to be, respectively, D/Uo and D/Uoz; this choice eliminates the

Peclet number from the equations. (A brief discussion on the choice of scaling factors is
given in Appendix B.) In dimensionless quantities, then, eq. (54) may be rewritten as

V0q=0

80:
E'PQ'VCX

0 R(01, T) + V20:

In these equations, the permeability has been normalized by the longitudinal permeability
K], with K = KD/e = (1/8) x (K/Kl). In addition, as in chapter 4, X = NH.
The component mass balance, then, is left with only one non-dimensional

parameter, 0, which is the product of the Damkohler number and the Peclet number. The

Damkohler number (Da = tf/tR) is the ratio of flow time scale to reaction time scale. The
Peclet number (Pe = tD/tf) is the ratio of the dispersive time scale to the flow time scale.

The product of these two, 0, represents a ratio of the dispersive time scale to the reaction

time scale (0 = tD/tR). A large value of 0 corresponds to a small value of tR in comparison
to tD, which implies that the chemical reaction is much faster than the rate at which

dispersion takes place; in other words, the effect of a large 0 is a smothering of dispersion

 

121

effects by chemical reaction. The choice of dispersive scales has, thus combined the Peclet

and Damkohler numbers into a single dimensionless quantity.

5.4 STABILITY ANALYSIS

Linear stability analysis leads to the following eigenvalue problem with two
coupled equations. The analysis is similar to that presented in chapter 4 for the reaction-
only case, with the same forms of base state and perturbation variables (eq. 4-18),
linearization of the perturbed equations and decomposition of the disturbances into

Fourier components (eq. 4-21).

2 (E __ .2 _ k'2 p(E)A

8R 2 2 _ 861
[CS—9 E H: —D ]A — - (A01) 52- q,

(5-6)

qg is the amplitude of the velocity disturbance and A is the amplitude of the conversion

disturbance. It must be noted that, in these equations, the base state conversion has been
normalized by (Art), the initial step change in conversion across the interface. That is, E =
(E°/A0t). This yields d5 =(1/A0t) dE", where 5° is the un-normalized base state
conversion. This ensures that the limits of the conversion are 0 - 1 and enables direct
comparisons to the results of Tan and Homsy (1986) as a limiting case.

The two coupled equations in eq. (5-6) are combined to obtain

86L
__ _1__ 2 a- _ .2 _
where L=[p(§) (D +p(§) Ea D k )] (5 7)
and p(§) = — é: j—EX

L is an operator as defined above, while p(§) is related to the quantity V defined in chapter
4 (eq. 4-25).

. In this problem, 0 is the exponential growth rate of disturbances and is the
eigenvalue, and k is the wavenumber. The boundary conditions associated with the
eigenvalue problem are the decay conditions at 00 (the effect of the disturbances is not felt
by the fluid at locations far from the interface), and continuity conditions at the interface,
on disturbances in velocity, conversion and pressure (Tan and Homsy, 1986). These
boundary and continuity conditions are discussed in greater detail in section 5.5.2

Other groups arising in the scaled eigenvalue problem are S = (BEBE) - the
incremental rate of reaction with increasing extent, discussed in chapter4; and p = -d(lnX)/
d5, where (—p) x (daldfi) = V represents the gradient of mobility (X) along the flow
direction (also discussed in chapter 4). As explained in chapter 2, the type of resin
polymerization affects the stability of the mold filling process, through the parameter S.
Figure 5.1b shows a sample profile of S at time t=0; given the step jump in conversion
profile, S also follows a similar pattern. Some polymerization reactions involve
consumption of inhibitor almost up to the gel point, the rate of which is independent of the
monomer concentration (Gonzalez-Romero & Macosko, 1985). For this zeroth order
reaction, 8:0, and the stability of the mold filling process is unaffected by the chemical

reaction. In other polymerization reactions, the initial stage of the network forming

 

123

process is of first order kinetics, which leads to a constant value of S. For many
polymerization reactions 8 is negative and it becomes increasingly negative as the reaction
progresses. For example, polyurethanes follow second order kinetics (Castro & Macosko
1982). Figure 2-2 shows the behavior of S for different types of chemical reactions,

schematically, and has been discussed in more detail in chapter 2.

5.5 INITIAL GROWTH RATE

Thu purpose of this study is to describe the combined effect of dispersion and
chemical reaction on the stability of the process. The solution of the complete form of the
model equations for the base state (from eq. 5-5) and that of the eigenvalue problem (eq.
5-7) is quite involved and would require extensive numerical computations. The problem
is more involved than that of Tan and Homsy (1986) and that presented in chapter 4, where
only one of dispersion and reaction was present. In the current problem, both dispersion
and chemical reaction terms, and the corresponding Damkohler number, in the component
balance equations. However, a description of the disturbances in their initial period of
existence is very important, and governs their subsequent behavior. A study of the initial
growth rates is important and tells us a great deal about the stability of the flow process. In
addition, the initial growth rates may be studied relatively easily; some simplifications
may be made that make the equations amenable to mathematical analysis with
comparatively less, simple, numerical computations. Therefore, the following sections of
this dissertation chapter will present a semi-analytical solution to the eigenvalue problem

at small times after the two shots of fluid are brought into contact.

124
5.5.1 DERIVATION OF THE PROBLEM

This section presents the steps involved in the derivation of eq. (5-7) at t = O to
yield the eigen value problem for initial growth rate analysis. The most important feature
of this initial growth rates analysis is the representation of the conversion profile in the
domain of the problem as a Heaviside step function (H(F,)), shown schematically in Figure
5.1b.

Using the initial (step) conversion profile, i.e. 0t(§) = (AOL) H(§), we get BEIGE =
(AOL) 5(§). Also, the definition p(§) = -(1/X) (deH) was used. This yields, from equation

(5.6),

.2
[ +p<§> ( a) 6(a) 1 Q ——B—(Aa) (5-3)

[Dz—o-k2+9 S]A = (AOL) (AOL) 8(§) qg

Eliminating A from these equations, we get

 

[Dz-049w S]{ 2““ [02+p(§) (Au) sew-I621 45,}
k' p(§) (5-9)
= (Aa) (Am) 5(a) Q
Since Aa and k’ are independent of 5,, this gives
[Dis-13w Sl{-'— [02+p<§) (Au) men-1621 4,}
Pg) (5-10)
= k'2 (Au) 5(a) q,
01'
[DZ—o—k2+6 S] ng = k'2 (Act) 6(§) qg (5-11)

where the operator L is given by

125

= _1_
p(§)

Tan and Homsy (1986) solved a similar problem where they assumed a viscosity

L [02+ 9(a) (Au) 8(§)D-k'21 (5-12)

that varied exponentially with concentration. This means that p is independent of 5, hence
é. (Note that the viscosity itself does depend on g; it is only the first derivative of the
natural logarithm of viscosity, i.e. the mobility gradient, that is invariant.) Mobility is
defined as the ratio between permeability and viscosity. The mobility ratio, given by mR, is
then mR = mobzlmobl = 111/112 = exp(p) (where it, is the viscosity of the fluid to the right
of the interface).

In this problem, too, an exponential viscosity profile is chosen as a representative
example. This makes equations (5-11), (5-12) amenable to analytical solution at t=0; if the
fi-dependence of p were retained, these equations would be far more complicated and
would require numerical solution for the initial growth rate as well. This choice of
viscosity profile facilitates comparison of limiting case (h = 0) results of this analysis with
those obtained by Tan and Homsy (1986).

Thus, taking p to be independent of 5,, the eigenvalue problem becomes

[02+ka S][D2+P(§) (Au) 6(§)D-k’21q§ = k'2 p (Au) mg) 4, (5-13)

5.5.2 BOUNDARY CONDITIONS
Due to the discontinuity at the interface, this fourth order o.d.e. has to be solved

separately on either side of the interface (E < O and g > 0) and the two solutions matched at

the interface. Two of the conditions are given by the decay of the q: disturbance as E —)

 

126

:00. That leaves us with two more conditions, needed to solve the fourth order o.d.e. In the

absence of any other simple, independent boundary conditions on qg, we use four

matching conditions on the disturbances at the interface. All these conditions are now

presented.
I. The decay of disturbances is given by:

At E = too qg = 0 (5-14)
I]. The matching conditions at the interface are given by:

3) Matching of the velocity disturbance.

At §= 0 qgw“) = q§(0') (5-15)

b) Matching of the conversion disturbance.

At g = 0 Arm”) = A(o’ ) (5-16)
c) Matching of the pressure disturbance.
Using the equation of continuity, Darcy’s law and the Fourier expansions
for disturbances, the pressure disturbance p(§) may be expressed in terms of the velocity

disturbance qg. This relation is obtained as

 

1 dq
p(§) = - 2 _ 712; (5-17)
(bk ) 7»
Using the continuity of pressure at the interface, p(0+) = p(0"), we get
d4: 43

 

 

where am is the mobility ratio defined earlier.

111. The final condition involves integration of the o.d.e. (equation 5-13) across the

 

127

interface, i.e. from 0’ to 0+.

0+
1 [Dz-u-k2+e SllDz+P(§) (Au) 6(:)D-k'21q¢ d:
0' + (5-19)
0
= k'2 9 (Au) J 6(a) q, d:
O-

The eigenvalue problem to be solved, thus, consists of the o.d.e. in equation (5-13), the

boundary conditions (5-14) and the matching conditions (5-15), (5-16), (5-18), (5-19).

5.5.3 ANALYTICAL SOLUTION

The initial growth rate for an exponential viscosity-concentration relationship, with
a step jump in concentration at time t=0, was analyzed by Tan & Homsy (1986). A similar
approach has been used in this work to obtain the initial growth rates (at t=0) for the model
problem with combined reaction and dispersion. The particular case in which viscosity
varies exponentially with conversion, i.e. p=constant, is chosen for the following analysis.

The first step is the representation of the solution for g at 0, using the decay

conditions. Since the base state profile is a step function, for § at 0, the Dirac delta function

in eq. (5-13) vanishes and the equation may be simplified to

[D2 —m2] [Dz-k'2]q§ = 0

(5-20)
where m2(§) = o + k2-95(§)
The solution to equation (5-20), on either side of the interface, is given by
qg = A1 exp[m_ :1 +31 exp[k'fi] for §<O (5 21)
qg = A2 exp[-m+§] + 32 exp [—k'§] for g > O

The constants A1, A2, B1, B2 are to be determined using the matching conditions.

 

128
Applying eq. (5-15), i.e. matching of the velocity disturbance, on eq. (5-21), we

get
A1 = A2+Bz--B1 (5-22)
Applying eq. (5-18), i.e. matching of the pressure disturbance, on eq. (5-21), and
using the relation (5-22), we get

1
k' —m_

 

A1 = [(k' + m+mR)A2 + (k' + k'mR)Bz] (5-23)

In applying eq. (5-16), i.e. matching of the conversion disturbance, since A 5 Mg,

we get
2 .2 0“
[D + P (Act) 5(5)!) - ’6 lqglo_ = 0 (5-24)
In this equation, using eq. (5-21), the first term (taking the second derivatives of q:) yields
2 0” 2 2 .2
D q§|0_ = (A2m+-Alm_ )+(Bz—Bl)k (5-25)

The second term, due to the presence of the dirac delta function 5(§), vanishes. The third

term also gives, using eq. (5-22),
43%|: = 0 (5-26)

Thus (5-24) may be simplified, using (5-22) and (5-23), as

 

2
m +m m k'+m +m k'
Bz=- [* * R< ‘) ‘] (5-27)

k’2 + k'mR(k' + m_ )+ m_k'

Three of the matching conditions have been used above, to obtain a simple relation
between two of the constants. The integral boundary condition (5-19) is now applied on

eq. (5-21) to get another relation between A2 and Bz. The integral equation is rewritten as

 

129

0+ 0‘
[Ibis-13% .9qu, d§= k'2 9 (Au) 1 6(a) q, d:
0‘ 0’

a) The first term in the integral on the left hand side yields

0+
ID2(ng)d§ = D(ng)|:_
0-

Using the definition of Mg, this gives

D(Lqé) :_ = 0345': +p(Aa)5'(§)DQg|:_ +P(Aa)5(§)D24gl:_ 4’04); 2

Now, using the properties of the dirac delta function

==()

5| =5,
0

0-
6" _ = 8'
o

 

0+

 

the second and third terms in eq. (5-30) vanish. We are now left with

+ O

0* 3 o 2 o
D(L ) = D - k' D
4: lo- ‘1: 0- ‘1: 0-
Using eq. (5-21), then, we get
D(ng) :_ = A2m+[mi + k'z] — A1m_ [m3 — k'z]
b) The second and third terms on the l.h.s. of eq. (5-28) vanish.
c) The right hand side of eq. (5-28) yields
0+

.2 '2
k p (Au) j8(§) qg 61E = k P (Au) q§|§=o
0-

= k'2 p (Aa)[A2+Bz]

(5-28)

(5-29)

(5-30)

(5—3 1)

(5-32)

(5-33)

' (5-34)

Equations (5-33) and (5-34) are now plugged into eq. (5-28) to give, after simplifying,

 

130

A
192 = —2-3—[—mi + m+k'2 -— k'2p(Aa) + m__ k'2 — m_ mi] (535)
k' D(Aa)

(5-27) and (5-35) are now combined to eliminate the constants A2 and Bz, yielding a

simple, algebraic equation involving the eigenvalue, 0', of the problem.

 

 

(m++m_) (Ic'2-—mfi)—k'2 p Au k'(m_ +m+ M)+m+(m++m_ M) 5 36
k' p Au ‘ ' (k'+m_) (1+M) (' )
where
MECXMP)

.2 2 2 2

k 5% = blkx+b2ky
2 2 (5-37)

m+ = o+k —9 S1

The parameters in the growth rate curves are 9, S] and 82. The cutoff wavenumbers
are defined as the wavenumbers at which the growth rate changes sign, implying a
transition between stable and unstable regimes. Eq. (5-36) may be solved for the cutoff
wavenumbers kc by setting 0:0. The roots of the algebraic equations in either case are
obtained by Newton-Raphson iterations. The initial guesses must be close to the roots, to
obtain convergence of the numerical scheme. Suitable guesses for the lower cutoff
wavenumber are obtained from the results of two different analyses. One was performed
by Hickemell & Yortsos (1986) for non-dispersive displacement through porous media,

and another was done for reactive filling without dispersion (Jayaraman et a1 1992).

5.5.4 RESULTS

For consistency with the data used in chapter 4, a second order reaction is chosen to

 

13 I
investigate the behavior of the initial growth rate profiles. For any n-th order reaction, the

first derivative of the reaction rate R w.r.t. 0t, i.e. S = dR/da, is S = - k n (l-a)“. Since
both k (reaction rate constant) and n (order of the reaction) are positive quantities, and so
is ( 1 -a), it follows that S is always negative and that at larger conversions the magnitude of

S is smaller. That is, lSll < ISZI (see Figure 5.1b). In addition, for a second order reaction

(as in polyurethanes), the range of possible values of S is {-2k to O}, which in
dimensionless terms is {-2 to O}. The choice of Act is also important. When AOL = 1.0 and
in the limiting case of 6 = O, the results would correspond to those obtained by Tan &
Homsy, 1986. In the physical problem under consideration, however, this is not practical
since it would require the fluid on the right of the interface to be completely reacted; hence
this particular case is only of academic interest.

For the purpose of illustration of this analysis, Aa has been chosen to be 0.2.
Specifically, the first shot of fluid is allowed to react till a conversion of 0.2 is reached and
then the second shot is injected, at which time we take t=0. The corresponding values of S
are 82 = -2.0 and S] = -1.6 (see Figure. 5.1b). Thus the conversion and S follow step
profiles and possess a discontinuity at §=0. As discussed in section 5.5.1, an exponential
viscosity profile has been chosen, and the value of p, the gradient of viscosity w.r.t.
conversion, has been taken to be 3.0. The parameter 9 is then varied and equation (5-36)
solved for the growth rate 0 as a function of the wavenumber k. The parameters used in.
this computation are, then:

time instant, t = O
p = 3:

Au: 1.0

132

Table 5.1 Computational results from eq. (5-36); Parameters: Aa = 0.2; p = 3.0; 81 =
-l.6; 82 = -2.0

Table 5.1

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

9 = 0.0
0 k1 k2
0 0 0
0.001 0.0038 0.1448
0.002 0.0081 0.1393
0.003 0.0130 0.1333
0.004 0.0186 0.1268
0.006 0.0328 0.1107
0.007 0.0429 0.0996
0.008 0.0617 0.0800
9 = 0.003
6 k1 11‘2
0 0.0172 0.1268
0.001 0.0237 0.1194
0.002 0.0314 0.1108
0.003 0.0415 0.0999
0.004 0.0589 0.0818
0.0041 0.0626 0.0779

 

 

133

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0 = 0.002
6 k1 k2
=
0 0.0105 0.1353
0.001 0.0159 0.1290
0.002 0.0220 0.1219
0.003 0.0293 0.1138
0.004 0.0384 0.1038
0.005 0.0520 0.0894
0.0052 0.0563 0.0849
0.0054 0.0628 0.0782
9 = 0.004
0' [(1 k2
W
0.001 0.0337 0.1078
0.002 0.0449 0.0958
0.0025 0.0533 0.0870
0.0027 0.0583 0.0818
0.0028 0.0620 0.0780

 

 

 

 

growth rate

134

0.009

 

0.008 .

0.007 - -

0.006 1

0.005 .

g

0.003 .

0.002 ~

0.001 -

 

*9=0

 

0 0.05 0.1
wavenumber

Figure 5.2 Initial growth rate curves - varying 9

 

 

135

Table 5.2 Cutoff wavenumber data at several values of 9; Parameters: Ant = 0.2; p =
3-0951 = -1.6; 32 = -2.0

ctcrs: infill

Table 5 .2

136

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

9 kcJ kuz

0 0 (55:00:
0.001 0.0048 0.1430
0.002 0.0105 0.1353
0.003 0.0172 0.1266
0.004 0.0254 0.1169
0.005 0.0360 0.1045
0.006 0.0539 0.0851
0.0062 0.0617 0.0769

 

 

cutoff wavenumber

137

 

0.14 .

0.12 ‘

0.1 .

0.08 .

om.
.

0.04 .

0.02 -

 

 

 

Figure 5.3

0.001 0.002 0.003 0.004 0.005 0.006 0.007
0

Marginal stability: 9

138
S] = 'l.6;

82 = -2.0

The values of 9 are raised from 9:0 and the two solutions for the wavenumber k are
evaluated at increasing 6. These results are tabulated in Table 5.1. The values of the initial
growth rate, a, are plotted against the wave number, k, for the different values of 9 as
shown in Figure 5.2. The curve for 9:0 corresponds to the flow process dominated by
dispersion. It matches up well with the result of Tan & Homsy (1986). For this case, small
fingers get smeared out due to dispersion, while large fingers remain undamped. This is in
contrast to the case of S>0, where two different non-zero cutoff wavenumbers may be
identified. As the rate of chemical reaction is increased relative to the rate of dispersion, by
increasing 6, it is seen that only a small range of intermediate finger sizes are unstable.
The polymerization reaction combined with dispersion thus stabilizes a broader range of
fingers. For the chosen values of S], 82 and p (-1.6, -2.0 and 3.0 respectively) it is seen
that, when the rate of reaction is about two hundred times the rate of dispersion (6 >
0.005), disturbances of all sizes decay.

The marginal stability curve is plotted in Figure5.3. The cutoff wavenumbers k,: at

the different values of 0 are shown in Table 5.2. For a given 6, fingers of wavenumbers
within the envelope would grow, while the rest are damped out. The width of the unstable
region decreases with a rise in 6. These results may be compared to results reported earlier
for miscible displacement without reaction or dispersion. Hickemell and Yortsos (1986)
have reported that without dispersion, any incipient fingers would invariably grow.

The wavenumbers (k) referred to above may be converted to appropriate finger

widths (A), using the relation (k = 2n lsc/ 2.), where lSC is the length scale for the problem.

 

139
To illustrate the significance of these results, the finger widths corresponding to the cutoff

wavenumbers are compared to the actual physical dimensions of the mold. The thickness
of the mold, B, represents the largest possible finger size, while the diameter of the fiber

tow, d,. is the smallest possible finger width. Representative values are chosen as below:

Mold thickness, B: 10mm

Fiber tow diameter, d,: 0.1mm
Dispersion coefficient, D: 0.001 cmzls (after Losure, 1994)
Injection velocity, U0: 0.5 cm/s
This yields a length scale (15c) of 0.0002 mm.

From Table 5.2, the cutoff wavenumbers when 0 is 0.004 are kc] = 0.0254 and kc2 =
0.1169. When converted to dimensional finger widths, these correspond to A] = 4.95 mm
and 2.1 = 1.075 mm. All fingers of sizes between these values will grow; conversely, all

fingers of widths outside this range are eliminated. These limiting finger widths are now

compared to the physical dimensions B and d,. It is seen that the tow diameter is much
lesser than the lower limit M. This means that all fingers with widths of the order of tow

diameter will be eliminated. Similarly, the mold thickness is much larger than the upper

limit, A]. While very large fingers are eliminated, there is an intermediate range of finger

widths that are undamped.

For the case of 0 = 0.001, Table 5.2 lists kc] = 0.0048 and kc2 = 0.1430. These
correspond to 2.1 = 26.25 mm and 702 = 0.88 mm. It is clearly seen for this case that the
mold thickness is much lesser than the upper limit on unstable finger widths, 7t], and the

tow diameter is smaller than the lower limit, L2. This means that fingers of sizes of the

140

order of tow diameter will be eliminated, while all fingers larger than M will grow. A

similar analysis may be done for other values of 0, and it is clear that, as chemical reaction
becomes more significant in comparison to the dispersion mechanism, the range of sizes
of growing fingers becomes smaller.

For the hypothetical case where the parameters SI and 82 are both positive, chosen

as +3 and +1 respectively, which corresponds to a reaction with autocatalytic kinetics, the
unstable range of wavenumbers is larger with reaction than without reaction as seen in
Figure 5.4. This computation has been performed to confirm the necessary condition for
stabilization, S < 0, obtained in chapter 4. Compared to the no-reaction case (0:0), the
upper cutoff wavenumber is greater for higher 6. Also, for any given wavenumber of the
disturbance in the unstable range, the growth rate is higher for the 0>0 case. This means

that positive SI and 82 destabilize the flow process.

5.6 SUMMARY

The combined effects of chemical reaction and dispersion on the growth of flow
instabilities in reactive flow through a fiber bed are strikingly different from the effect of
dispersion alone, or that of reaction alone, on fingering. The model problem to make this
study was inspired by a multiple-shot injection experiment done by Losure (1994). Since
the initial growth rates govern subsequent behavior of any incipient disturbances, the
stability analysis of the model problem was simplified to obtain the initial growth rates,
and their behavior investigated. An analytical solution of the eigen value problem was

possible, under the conditions chosen.

 

141

 

0.2 -

0.1 -

 

0.0
0.0

Figure 5.4

 

' *0s0
X08002

 

 

 

0.2 1 0.4 0.6

Initial growth rate curves - positive values of S

0.8

 

 

142

It is seen that the result depends on the type of reaction as seen in the incremental
rate of reaction with increasing extent of reaction, S. As the rate of chemical reaction is
increased relative to the rate of dispersion, the process becomes increasingly more stable.
If S is increasingly negative along the flow direction, and the reaction rate is several times
the rate of dispersion, it is possible to eliminate fingers entirely. When S is positive, the
unstable range of wavenumbers is increased with reaction. These effects are more
pronounced when the spatial variation in S is steeper. The wavenumbers may also be
converted to finger widths, and predictions on the growth or elimination of fingers may be
made by comparing the cutoff finger widths to the dimensions of the mold and fiber tows

used.

 

CHAPTER 6 CONCLUSIONS

6.1 SUMMARY AND CONCLUSIONS

A study of the flow of reactive, polymeric liquids through fiber bundles has been
presented in this dissertation. Following the observation of fingers during the mold filling
process and subsequent investigation of their deleterious effects on some of the
mechanical properties of the final parts (Losure, 1994), this work attempts to obtain
theoretical criteria that enable prediction of the generation and growth of these fingering
instabilities.

A simple model has been developed to track the motion of the flow front and the
pressure distribution inside the filled region of the mold at any time. The effect of
rheological changes on the reduction in flow velocity, in a constant pressure scenario, have
been represented by two quantities - a velocity reduction factor and the log-mean mobility
difference ratio (LMMDR). These analyses were done using representative rheological
data for an epoxy resin and a vinyl ester resin. The data enabled division of the entire flow
domain into three regimes - one with an invariant viscosity, another with a changing
viscosity, and a third regime where elastic effects come into play. The pressure
distributions, after the viscosity increase significantly, are steep very near the flow front
and provide greater driving force for fiber wet out, i.e. penetration of the resin into the
fiber tows to wet individual fibers is facilitated.

Since the resin in liquid molding operations is continuously undergoing
polymerization reaction, the viscosity of the resin also increases along the flow direction

towards the advancing front. An adverse viscosity gradient is known to generate fingers.

143

144

However, the coupling between the evolving reaction and the viscosity gradient
complicates the mechanism and produces some stabilization at larger length scales. In the
moving frame of reference, the cutoff wavenumbers when the viscosity gradient is steady
are different from those when the viscosity gradient is produced by an evolving chemical
reaction. The study of the generation and subsequent behavior of disturbances is done by
performing a linear stability analysis on the equations describing the system. It is found
that a necessary condition for damping of fingers is given by equation 4-31, i.e. S < 0. The
effect of various parameters on finger growth has also been investigated. The
thermochemical properties of the resin, like the activation energy, gel conversion and
reaction exotherm, may suitably be altered to eliminate fingers during mold filling. In
addition, the operating conditions (resin inlet temperature, injection velocity, etc.) may be
changed to affect finger growth. The ranges of stable wavenumbers in each case are
presented in the form of marginal stability curves. The effect of anisotropy of permeability
has also been presented.

Disturbances of small wavenumbers, i.e. large finger widths, have been found to be
stabilized by the coupling of chemical reaction with the flow process. This is in contrast to
the stabilizing effect of dispersion for fingers of small widths (Tan and Homsy, 1986). The
complementary nature of the two mechanisms of damping of fingers is utilized in chapter
5 to obtain the ranges of stable wavenumbers in the presence of both mechanisms. The
combined effect of mass dispersion and chemical reaction, on the fingering process, was
examined in the context of a multiple-shot injection where the interface between resins of
different ages is examined. It has been found that the individual effects of chemical

reaction alone (chapter 4) and dispersion alone (Tan and Homsy, 1986) are both seen in

 

145

this case. That is, disturbances at both ends of the spectrum of wavenumbers are
stabilized. In addition, it is found that an increase in chemical reaction rate increases the
range of stable wavenumbers; so much so, that at certain levels of reaction rates in
comparison with dispersion rates, all disturbances die out. The necessary condition
discussed earlier, i.e. S < 0, for stabilization is also confirmed by this analysis.

The sizes of fingers generated are limited by the geometric dimensions of the mold
and of the fiber tows used in the preform. In the case of an end-gated mold, the thickness
of the mold represents the largest finger size possible, while the diameter of the fiber tows
corresponds to the smallest finger size. The upper and lower cutoff wavenumbers in Table
5 .2 and figure 5.2, for any 6, correspond to a lower and an upper limit on the widths of
growing fingers, respectively; all fingers with widths outside this range will be eliminated.
The relation between finger size (A) and wavenumber (k) is (k = 21: l“! 3.), where lSC is the
length scale. Using representative numbers for the various parameters (dispersion
coefficient, injection velocity, mold thickness and fiber tow diameters), the following
conclusions are arrived at (see figure 5.2).

(a) In general, the diameter of the fiber tows is less than the lower limit on finger
widths for growing fingers. This means that fingers of very small sizes, of the order of tow
diameter are eliminated, while those of intermediate sizes grow.

(b) When dispersion mechanism is more dominant, the mold thickness is lesser
than the upper limit of growing finger widths; this means that all large fingers in this
situation are unstable. As reaction becomes more significant in comparison to dispersion
(larger values of 9), some of the large fingers are eliminated.

A mechanism has been suggested for the stabilization of large fingers by reactive

 

146

coupling. For resins with S < 0, small fingers grow at a faster rate than chemical reaction
can act to stabilize them; in the case of large fingers, the rate of growth of the fingers is

slow enough to enable the reactive coupling mechanism to damp them out.

6.2 RECOMMENDATIONS FOR FUTURE WORK

The main focus of this dissertation has been on the role of the polymerization
reaction on the stability of the flow process. In the discussion of chapter 3, where the flow
of resin past cylinder arrays was studied, the effects of viscoelastic properties of the resin
have not been looked into (in the third regime). A more detailed investigation of the elastic
effects would provide better understanding of the phenomenon.

Another area that deserves further attention is the combined effect of chemical
reaction and dispersion. In chapter 5, only the initial growth rates were considered. A
complete numerical computation of the eigen value problem, at all times, would be useful.
Though the presence of dispersion and reaction terms in the model equations makes the
numerical procedure more involved, the actual behavior of the disturbances at large times

would be an interesting topic for further research.

APPENDICES

APPENDDI A: EXCHANGE OF STABILITIES ANALYSIS FOR MOLD FILLING

WITH CHEMICAL REACTION, WITHOUT DISPERSION

In the stability analysis of Chapter 4, all disturbances are expressed as Fourier type
expansions. For example, disturbances in pressure are given by equations of the type:

I” = p(§) - e2913([z'{l~=,1r+k,y}l-9209031)) (A-l)
where p(§) is the amplitude of the disturbance, kx and ky are the wavenumbers in the
respective directions and 0' is the growth rate. This follows the resolution of the
disturbances by the method of normal modes into various components, each satisfying the
linear system (eigenvalue problem). In general, the growth rate 0' is complex, described as

o = oR + io, (A-2)
where OR is the real part and 0'; is the imaginary part. When the real part of the growth rate
is positive, the disturbance in unstable; when it is negative, the disturbance attenuates with
time and is termed stable; when it is zero, it is said to be neutrally stable. The imaginary
part of the growth rate corresponds to an oscillatory variation in the disturbance.

The cutoff wavenumber is defined as the wavenumber of the disturbance at which
the real part, OR, of its growth rate, 0', is zero. This means that, at the cutoff point, the

disturbance neither grows nor decays. However, if the imaginary part, 01, of the growth

rate is non-zero, the disturbance retains an oscillatory variation with time and is termed

overstable. The principle of exchange of stabilities is said to be satisfied when both the

147

148
real and imaginary parts of the growth rate vanish at the cutoff point. In this case, at the

onset of instability a stationary pattern of motions prevails, in contrast to the occurrence of
oscillatory motions at the onset of instability in case of overstability.
For the general eigenvalue problem
Lu = Au (A-3)

exchange of stabilities is examined by performing the following mathematical operation:
(v, Lu) = (Lev, u) (A-4)

where L is a linear operator, L9 is the complex conjugate of the operator L, A. is the
eigenvalue and <..> is the inner product operator.
For the problem discussed in chapter 4, in the absence of dispersion, the

eigenvalue problem was obtained as (equation 4-23)

In this eigenvalue problem, then, both sides of equation (A-S) are multiplied by the

complex conjugate of “p” to give, taking {(G-S) / (O-S+V) E h},

p*§E[X % 4:] = kapr“ (A-6)
This may be expanded to give
3858“”? Yd 'h'] 1(d_§)(:§)'h= “21W (A-7)

In differential form, this becomes

d[kh(p*dp d_§ )]- Ah(d—£)(Zg)* d§ = bk zippfllfi (A-8)

149
This equation may now be integrated in the limits (1'; = -zf) to (§ = 0), yielding

[mej—g) )]l: —J' (Xh)(d—g)(Z—g) d: = j bk zippwg (A-9)

‘1; ’21
The boundary conditions at g = -zf and l; = 0 (equation 4-26) on the pressure disturbance
and its first derivative lead to the first term on the left hand side of equation (A-9) being

null. This results in

—j (Xh)(d—§)(d—g) dz; = j bk 2pr’talg (A-lO)

“31“!
Both the products (pp*) and (dp/d§)(dp*ld§) are positive. In addition, (bkz) is positive.
This means that {pr*} and {X (dp/d§)(dp*/d§)} are both of the same sign. So “h” has to
be negative in the domain of the problem, for the above equation to be satisfied. Also,
since the right hand side of the equation is real, “h” has to be real, i.e. the imaginary part hI
is zero, in the equation

h = hR-r-ih, (A-ll)

Now, from this definition of “h”, we obtain

 

 

h +'h — ORHOFS (A12)
R 1’ - oR+io,—S+V -
whichyields
o,V

(oR—S+ V)2+o,
For h to be zero, we need V = 0 or 0'1 = 0. A sufficient condition that arises for real growth

rate, 1 e for 01: -0, is that V must not equal zero anywhere 1n the flow domain, 1 e V

 

 

 

 

150

should not change sign.

 

APPENDIX B: DERIVATION OF COMPONENT MASS BALANCE EQUATIONS

The schematic for the shell balance for one-dimensional flow past porous media is

shown:

 

 

 

 

Figure A] Schematic for shell balance

Let the change in concentration of the reactant, between the planes at {z} and {z+Az } , be
Ac.

The volume of the shell is [WHAz]. Uo here is the interstitial velocity and E is the
porosity.

Net accumulation (A) of the reactant, in time At is = [WHAz] 8 Ac.

151

152
Reactant entering the shell by FLOW (Fin) in time At is = (At) [WH(ch)Iz]

Reactant leaving the shell by FLOW (Four) in time At is = At [WH(ch)Iz+AZ]

Reactant consumed by CHEMICAL REACTION (RR) in time (At) is = R At [WHAZ] E

Reactant entering the shell by DISPERSION (Din) in time At is = (At) [WH] [-D(ac/
(32)] | z e

Reactant leaving the shell by DISPERSION (Dout) in time At is = At [WH] [-D(3c/

82)] I Z+AZ E

The overall shell balance is now written by combining the individual terms
A = F. -F0u,—RR+Din—Dom (B-l)

m

That is,

[WHAz]e(AC) = (At)[WH[(UbC)l —(UbC)| AJ]-(At)[WHAz]£R
Z 2+

 

 

 

 

 

a a (B-2)
+ (At)eWH -D—C +D—C—j
32 32
z z+Az
Dividing through by [WHAZ] At, we get:
8C BC
—D— + D—
sA—C = (1,612— U,,C|2+ A: _eR + 8 az z 32 2+ AZ (133)
At A2 A2

Now, dividing through by 8 and taking limits as At, Az -*0, we get, using the definition of

interstitial velocity (Ub = U0 8):

ac _ a a a
5; _ 3E(-U0C)-R+§Z-(D§—f) (BA)

Eq. (B-4) may be rewritten for a three dimensional flow field as

153

gg+40Va—VO[DOVC] = -R (3'5)

Using the equation of continuity (dU/dz = 0) and assuming D to be constant, we get, for a
second order reaction,

2
Ba

- — - — 2— — -
_Co-a—t— .. UOCOaz R0(1 0t) DCOBZZ (B 6)
This yields:
a a 82 R
or 0t 0t _ o _ 2 -
”a: + U ”—32 —D—azz — (—C0)(1 0t) (B 7)

This is the final, component balance equation, in DIMENSIONAL quantities.
Scaling;
Equation (B-7) may be rewritten in dimensionless terms using appropriate scaling

factors. For the general scaling factors for time, length and velocity, given by to, 20,
UO(=zo/to) respectively, the following dimensionless form of the component balance

equation is obtained:

2
3a 80L ‘0 3a_ I0 2
.37 472—; g - tR(1—u) (B-8)

where t1) 5 202/1) is the dispersion time scale and tR .=_ co/Ro is the reaction time scale.
Also, let the flow time scale be t1; The time scale t0 may now be selected from the three

choices tD, tR, or tr.

We now define dimensionless quantities as below:

Peclet Number, Pe = tD/tF = (ono/D).

 

154
Damkohler number, Da = tF/tR = (zo/Uo) / (co/R0).

9 = (Pe) x (Da) = tD/tR

.‘L . I16 '1' . 01' 1t 111' . -‘0 1 e ~-t01 11‘ L' .11...
time-to=tR=co/Ro;
length-z(,=zR=Uo co/Ro;

velocity - v0 = U0;

we obtain, in dimensionless quantities:

2
9939-2 a_a -

(1 - u)2 (B-9)

These scaling factors were used in chapter 4, where the problem of mold filling with
chemical reaction in the absence of dispersion was studied. In that problem, due to the

absence of dispersion, tD —) 0° and the third term on the l.h.s. of this equation vanishes.

A E . e -n: . dtime nits ar caled - .1- .__ .- ' _1~ cl - 2(1).
time - to: tD = zole;
length - 20 = 21);
velocity - v0 = U0;
Using these definitions, the time scale may also be expressed as D/Uoz; the length scale as
D/Uo.

Applying these scales we obtain, in dimensionless quantities:

2 .
30. au au_tp 2
E 3.2-g-a (l-a) (B-IO)

155

These scaling factors were used in the paper by Tan & Homsy (1986), where there was no
chemical reaction. This means that tR -—) 0° and the term on the r.h.s. of the equation
vanishes. In chapter 5 however, where both dispersion and chemical reaction are present,

this equation is rewritten using the definition 0 E (Pe) x (Da) = tD/tR.

2
Ba Ba Ba _ 2

BIBLIOGRAPHY

Babbington, D.A., J.H. Enos and J.M. Cox, "High speed resin transfer molding of vinyl

ester resins," AUTOCOM '87, Dearbom, Michigan, Paper no. EM87-351 (1987).

Bacri, J.C., Rakotomalala, N ., Salin, D. and Woumeni, R., "Miscible viscous fingering:

Experiments versus continuum approach", Phys. Fluids A 4(8), 1611(1992).

Bruschke, M.V. and S.G. Advani, "A finite element approach to mold filling in anisotropic

porous media," Polymer Composites 11, 398 (1990).

Castro, J.M. and Macosko, C.W., "Studies of Mold Filling and Curing in the Reaction

Injection Molding Process", AIChE .1 28(2), 250(1982).

Chadam, J ., D. Hoff, E. Merino, P. Ortoleva and A. Sen, "Reactive infiltration

instabilities", [MA J. Appl. Math., 36, 207(1986).

Chadam, J., A. Pierce and P. Ortoleva, "Stability of reactive flows in porous media:

Coupled porosity and viscosity changes:, SIAM J. Appl. Math., 51(3), 684(1991).

Chmielewski, C., C.A. Petty and K. Jayararnan, "Crossflow of elastic liquids through

cylinder arrays" J. Non-Newtonian Fluid Mech. 35, 309 (1990).

Chuoke, R.L., van Meurs, P. and van der Poel, C., "The Instability of Slow, Irnmiscible

Viscous Liquid-Liquid Displacements in Permeable Media", Trans. AIME 216, 188(1959).

RE. Collins, "Flow of fluids through porous materials", Reinhold Publishing, (1961).

156

157

Coulter, LP. and 8.1. Guceri, "Resin transfer molding: process review, modeling and
research opportunities," Proc. Manufacturing International '88, ed. T.G. Gutowski, ASME,

NY, p. 79 (1988).

Finlayson, B.A., "Nonlinear Analysis in Chemical Engineering", McGraw-I-Iill Inc., p.73

and p.113 (1980).

Garcia, M.A., C.W. Macosko, S. Subbiah, and S. I. Guceri, "Modeling of reaCtive filling in

complex cavities," Intern. Polymer Processing XI, p.73, (1991).
Gonzalez-Romero, V.M. & C.W.Macosko, J. Rheol., 29(3), 259(1985).

Gorell, SB. and G.M. Homsy, "A theory for the most stable variable viscosity profile in

graded mobility displacement processes", AIChE 1., 31(9), 1498(1985).

Hickemell, E]. and Yortsos, Y.C., "Linear Stability of Miscible Displacement Processes in

Porous Media in the Absence of Dispersion", Studies in Appl. Math. 74, 93(1986).
Hill, S., "Channelling in Packed Columns", Chem. Eng. Sci. 1, 247(1952).

Homsy, G.M., "Viscous wagering in Porous Media", Ann. Rev. Fluid Mech. 19,

271(1987).

Jayararnan, K. and C.N. Satyadev, ”The Role of Fluid Rheology in Resin Transfer
Molding", Advanced Composite Materials: New Developments and Applications

Conference Proceedings, Detroit, Michigan, USA, pp. 225-232 (1991).

Kim, DH. and Kim, S.C., “Engineering Analysis of Reaction Injection Molding of Epoxy

158
Resin”, Poly. Comp., 8, 208(1987).

Kim, Y.R., S.P. McCarthy, J.R. Fanucci, S.C. Nolet and C. Koppemaes, "Resin flow
through fiber reinforcements during composite processing," SAMPE Quarterly, p. 16,

(1991).

Losure, N.S., “Viscous Fingering Flows in Reactive Resin Systems”, Ph.D. Thesis,

Michigan State University, 1994.

Macosko, Christopher W., “RIM, Fundamentals of Reaction Injection Molding”, Hanser

Publishers, New York, 1989.

Malkin, A.Y., Ivanova, S.L., Frolov, L.G., Ivanova, AN. and Andrianova, Z.S., “Kinetics
of Anionic Polymerization of Lactarns. (Solution of non-isothermal kinetic problems by

the inverse method)”, Polymer, 23, 1791(1982).

Manickam, O. and G.M. Homsy, "Stability of miscible displacements in porous media

with nonmonotonic viscosity profiles", Phys. Fluids A5(6), 1356(1993).

Osinski, 1.8., “Characterization of Fast-One Resins for Reaction Injection Molding”,

Polym. Eng. Sci., 23, 756(1983).

Owen, MJ., V. Middleton, C.D. Rudd, and ID. Revill, "Fiber wet-out in resin transfer
molding” Proc. of Conference on Interfacial Phenomena in Composite Materials,

University of Sheffield, p. 208 (1989).

Park, CW. and G.M. Homsy, "The instability of long fingers in Hele-Shaw Flows", Phys.

159
Fluids, 28(6), 1583(1985).

Parnas, RS. and ER. Phelan, "The effect of heterogeneous porous media on mold filling

in resin transfer molding," SAMPE Quarterly p. 53, (1991).

Paterson, L., "Fingering with miscible fluids in a Hele Shaw cell", Phys. Fluids 28(1),

26(1985).

Perkins, T.K. and QC. Johnston, "Mechanics of viscous fingering in miscible systems",

Soc. Pet. Eng. J., March 1962, pp. 70—84.

Perkins, T.K. and QC. Johnston, "A review of diffusion and dispersion in porous media",

Soc. Pet. Eng. J., December 1965, pp. 301-317.

Saffman, PG. and Taylor, Sir G., "The penetration of a fluid into a porous medium or
Hele-Shaw cell containing a more viscous liquid", Proc. Roy. Soc. (London), Series A,

245, 120(1958).

Satyadev, C.N. and K. Jayararnan, ”Analysis of finger growth during the flow of reacting
liquid through porous fiber preforrns", Simulation of Materials Processingz.'I'heory,
Methods and Applications - NUMIFORM-95, pp. 333-338, Eds. Shan-Fu Shen and Paul

Dawson (1995).

Schincariol, R.A., F.W. Schwartz and C.A. Mendoza, "On the generation of instabilities in

variable density flow", Water. Resour. Res., 30(4), 913(1994).

Schowalter, W.R., "Stability Criteria for Miscible Displacement of Fluids from a Porous

160
Medium", A.I.Ch.E. J. 11(1), 99(1965).

Slobod, R.L. and Thomas, R.A., "Effect of Transverse Diffusion on Fingering in Miscible-

Phase Displacement", Soc. Pet. Eng. J. 3, 9(1963).

Tan, CT. and Homsy, G.M., "Stability of miscible displacements in porous media:

Rectilinear flow", Phys. Fluids 29(11), 3549(1986).

Tan,.C.T._ and G.M. Homsy, "Simulation of nonlinear viscous fingering in miscible

displacement", Phys. Fluids 31, 1330(1988).

Tan, CT. and Homsy, G.M., "Viscous fingering with permeability heterogeneity", Phys.

Fluids, A4(6), 1099(1992).

Wang,K.K. and C.A. Hieber, ”Injection. Molding Simulation," Proc. Manufacturing

International '88, ed. T.G. Gutowski, ASME, p.87 (1988).

Yortsos, Y.C. and M. Zeybek, "Dispersion driven instability in miscible displacement in

porous media", Phys. Fluids 31(12), 3511(1988).

Zimmerman, W.B. and G.M. Homsy, "Nonlinear viscous fingering in miscible

displacement with anisotropic dispersion", Phys. Fluids A3, 1859(1991).

Zimmerman, W.B. and G.M. Homsy, "Three dimensional viscous fingering: A numerical

study", Phys. Fluids A4(9), 1901(1992).