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Development of a Microwave Part-Shaped Cavity
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Trent Alan Shidaker

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DEVELOPMENT OF A MICROWAVE PART-SHAPED CAVITY
FOR LIQUID COMPOSITE MOLDING

By

Trent Alan Shidaker

A THESIS

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

MASTER OF SCIENCE

Department of Chemical Engineering

1996

ABSTRACT

DEVELOPMENT OF A MICROWAVE PART-SHAPED CAVITY
FOR RESIN TRANSFER MOLDING

By

Trent Alan Shidaker

A microwave part-shaped applicator was designed to be competitive with conventional
liquid composite molding processes. This thesis presents the developmental work for
this apparatus. Speciﬁcally, the electromagnetic behavior, including modes of
propagation, electric ﬁeld distribution, and cutoff frequencies, in the part-shaped
applicator system is determined theoretically using Maxwell’s equations. Microwave
heating patterns, using ﬁilly cured glass-reinforced polyester plates, are experimentally
characterized in the part-shaped applicator for comparison to theory. Three glass-
reinforced diglycidyl ether of bisphenol A epoxy composites with a diaminodiphenyl
sulfone curative are processed in the part-shaped cavity by microwave heating,
conventional heating, and hybrid heating where both microwave and conduction heating
are employed. Hybrid heating provided superior heating uniformity. Because of the
penetrating nature of microwave energy, the time required to attain the cure temperature
in the composite center was reduced by more than 85% using microwave and hybrid

heating methods.

DEDICATION

To God, through Whom all things are possible (Matthew 19:26), and
to the loving and supportive women in my life:

my mother, Sandra, and my wife, Layne.

iii

ACIG‘IOWLEDGMENTS

I would like to express my sincere appreciation to my advisor Dr. Martin Hawley for
providing the numerous learning experiences to make this thesis a valuable endeavor for
me. Grateful acknowledgment is also due to the committee members, Drs. J es
Asmussen, Jr. and Lawrence Drzal, for their interest in this work. Sincere appreciation
is extended to Scott Hebner, Ann Glenn, Yunchang Qiu, Michael Muczinski, and Dr.
Yong-Icon Lee for their ﬁiendship and valuable research assistance that they so willingly
provided. I would like to thank Rick Delgado, Dr. Valerie Adebite, Dr. K.M. Chen and
Dr. Jianghua Wei for providing many beneﬁcial insights through our discussions. The
professional services of Terry Casey, Brian Rook, Michael Rich, and Barry Tigner are
highly acknowledged. A special appreciation is extended to my wife, Layne, for her
love, encouragement, and support throughout the duration of my program. This
research was made possible by the ﬁnancial support of the National Science Foundation
State/Industry/University Co-operative Research Center on Low-Cost, High Speed

Polymer Composites Processing at Michigan State University.

iv

TABLE OF CONTENTS

List of Tables .............................................................................................................. viii
List of Figures .............................................................................................................. ix
Chapter One: Introduction ............................................................................................ 1
1.1. Research Description and Scope ................................................................. 1

1.2. Microwave Processing of Materials ............................................................ 3

1.2.1. Rationale ..................................................................................... 5

1.2.2. Applications ............................................................................... 12

1.2.2.1. Polymer Processing ..................................................... 12

1.2.2.2. Composite Processing ................................................. 16

1.2.3. Microwave Processing Systems ................................................. 19

1.3. Resin Transfer Molding ............................................................................ 23

1.3.1. Process Cycle ............................................................................ 24

1.3.2. Polymer Matrices ....................................................................... 27

1.3.3. Fiber System .............................................................................. 28

Chapter Two: Theory ................................................................................................. 30
2.1. Maxwell’s Equations ................................................................................ 30

2.2. Wave Propagation .................................................................................... 32

2.2.1. Rectangular Waveguide ............................................................. 33

2.2.2. Rectangular Resonant Cavity ..................................................... 37

2.3. Microwave Energy Absorption Mechanisms ............................................. 39

2.4. Penetration Depth .................................................................................... 42
Chapter Three: Experimental ...................................................................................... 46
3.1. Experimental System ................................................................................ 46

3.1.1. Part-Shaped Cavity .................................................................... 46

3.1.2. External Applicators .................................................................. 49

3.1.3. Experimental Setup and Microwave Circuitry ............................ 51

3.1.4. Therrnometry System ................................................................. 54

3.1.5. Process Material ........................................................................ 55

3.2. Cavity Characterization Methods .............................................................. 57

3 .3. Glass Fiber-Reinforced DGEBA/DDS Processing ..................................... 58

3.4. Mechanical Strength Tests ........................................................................ 60
Chapter Four: Results and Discussion ......................................................................... 63
4.1. Cavity Characterization Results ................................................................ 63

4.1.1. Temperature Proﬁles and Heating Patterns ................................. 63

4.1.2. Aperture Effects ........................................................................ 72

4.2. Theoretical Calculations ........................................................................... 79

4.2.1. Electromagnetic Modes in the External Applicators ................... 79

4.2.2. Electromagnetic Modes in the Cavity ......................................... 83

4.3. Processing of Glass-Reinforced DGEBA/DDS ......................................... 90

4.3.1. Complex Permittivity ................................................................. 90

4.3.2. Curing Results ........................................................................... 91

4.4. Mechanical Strength Tests ...................................................................... 101
Chapter Five: Conclusions ........................................................................................ 107
5.1. Part-Shaped Cavity Processing Capabilities and Limitations ..................... 107

5.2. Project Status .......................................................................................... 107
Chapter Six: Recommendations ................................................................................ 110
Appendix A: Vector Wave Equation Development ................................................... 112
Appendix B: Solution to Vector Wave Equations ..................................................... 115
Appendix C: Power Penetration Depth Calculation ................................................... 119
Appendix D: Cutoﬁ' Frequency ................................................................................. 120
Appendix E: Guided Wavelength .............................................................................. 121
Appendix F: Unsteady-State Energy Balance Development ....................................... 123
Appendix G: FORTRAN Code for the Heat Transfer Model ..................................... 127
List of References ...................................................................................................... 133

vii

. run’

Table 1.1.

Table I. 2.

Table 1. 3.

Table 4.1.

Table 4. 2.

Table 4. 3.

LIST OF TABLES

Comparison of Conventional to Microwave Thermoset Processing .............. 6
Microwave Characteristics and Effects. ....................................................... 7
Examples of Components Manufactured by RTM. .................................... 25
Thermal Paper Images for Various Aperture Widths. ................................ 78
Heat Transfer Model Parameters for Conventional, Microwave,

and Hybrid Heating. ................................................................................. 99

Modulus of Elasticity Data for E-glass-Reinforced DGEBA/DDS. .......... 106

viii

Figure 1.1.

Figure 1.2.
Figure 1.3.
Figure 1.4.

Figure 2.1.

Figure 2.2.

Figure 2.3.

Figure 2.4.

Figure 2.5.

Figure 3.1.
Figure 3.2.

Figure 3.3.

Figure 3.4.

LIST OF FIGURES

The Electromagnetic Spectrum and Microwave Frequencies for

Heating. ................................................................................................... 4
Generalized Thermoset Processing Window. ............................................. 9
Typical Microwave System Circuitry ....................................................... 21
RTM Processing Cycle ............................................................................ 26
Instantaneous Vector E-ﬁeld Plots for the TEM Mode in a

Rectangular Waveguide. ......................................................................... 3 5
Cutoff Frequency Versus Dielectric Constant for a Dielectric-

Filled Waveguide with a Cross-Section of 3.8 cm x 8.7 cm. .................... 36
Vector Plots of E-ﬁeld for the TEou Mode in a Rectangular

Resonant Cavity. .................................................................................... 38
Mode Chart for a Rectangular Resonant Cavity with a Cross-

Section of 6.4 cm x 7.6 cm. .................................................................... 40
Power Penetration Depth for the TE." Mode in a 4.2 cm by

8.9 cm Dielectric-Filled Waveguide. ....................................................... 44
Drawing of the Part-Shaped Cavity. ........................................................ 47
Cross-Section of the External Applicator. ............................................... 50
Microwave Circuitry and Cavity Conﬁgurations for Standing

Wave Operation. .................................................................................... 52
Microwave Circuitry and Pumping Conﬁguration for Traveling

Wave Operation. .................................................................................... 53

ix

Figure 3.5.

Figure 3.6.

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.

Chemical Structure of Diglycidyl Ether of Bisphenol A and
Diaminodiphenyl Sulfone. ....................................................................... 56

Fiber Mat Orientation and Sample Locations for Curing
Experiments in Part-Shaped Cavity ......................................................... 61

Glass-Reinforced Polyester Temperature Proﬁles for Heating
Runs With One External Applicator at Port A and an Aperture
Width of4. 19 cm. .................................................................................. 64

Glass-Reinforced Polyester Temperature Proﬁles for Heating
Runs With One External Applicator at Port C and an Aperture
“Width of 4. 19 cm. .................................................................................. 65

Glass-Reinforced Polyester Temperature Proﬁles for Heating
Runs With One External Applicator at Port D and an Aperture
“Width of 4.06 cm. .................................................................................. 66

Thermal Paper Images for Polyester Plates in the Part-Shaped
Cavity: Standing Wave Operation. ......................................................... 68

Glass-Reinforced Polyester Temperature Proﬁles for Heating

Runs “With an Active External Applicator at Port A and Non-

Active External Applicators at Ports B and C With an Aperture

Width of 3.61 cm. .................................................................................. 69

Glass-Reinforced Polyester Temperature Proﬁles for Heating
Runs With an Active External Applicator at Port C and Non-
Active External Applicators at Ports A and B With an Aperture

Width of 3.61 cm. .................................................................................. 70
Thermal Paper Images for Polyester Plates in the Part-Shaped

Cavity: Traveling Wave Operation. ........................................................ 71
Investigated Aperture Characteristics. .................................................... 73

Initial Temperature Proﬁle as a Function of Aperture “Width at
Port A. ................................................................................................... 75

Figure 4.10. Initial Temperature Proﬁle as a Function of Aperture Width at

Port C. .................................................................................................. 76

Figure 4.11. Initial Temperature Proﬁle as a Function of Aperture “Width at

Port D ................................................................................................... 77

Figure 4.12.

Figure 4.13.

Figure 4.14.

Figure 4.15.

Figure 4.16.

Figure 4.17.

Figure 4.18.

Figure 4.19.

Figure 4.20.

Figure 4.21.

Figure 4.22.

Figure 4.23.

Figure 4.24.

Theoretical Mode Chart for a 6.4 cm x 7.6 cm Rectangular

Cavity. .................................................................................................. 81
Theoretical Vector Plots of E-ﬁeld for the TEou Mode and the

Aperture Conﬁguration. ........................................................................ 82
Cutoﬁ‘ Frequency in the Cavity as a Function of the Process

Material Dielectric Constant. ................................................................. 84

Theoretical E-ﬁeld Strength Plots and Thermal Paper Image for
the Mid-Plane (i.e., y = 0.5 b) of a 4.2 cm x 8.9 cm Waveguide
Filled With Dielectric Material (6 = 4.3040049). ................................. 86

X-Y Vector Plots for the mm. mm, and TE); Modes. ........................ 87

Theoretical Temperature Proﬁles at Various Times for a
Constant Microwave Power Input of 0.1 15 W/cm3 in an Inﬁnite

Slab of Polymer Between Brass Boundaries. ......................................... 89
Complex Permittivity of Uncured DGEBA/DDS at 25°C as a
Function of Fiber Content. .................................................................... 93

Temperature Proﬁles for Conventional Curing of E-glass-

Reinforced DGEBA/DDS in the Part-Shaped Cavity. ............................ 94
Temperature Proﬁles for Microwave Curing of E-glass-

Reinforced DGEBA/DDS in the Part-Shaped Cavity. ............................ 95
Temperature Proﬁles for Hybrid Curing of E-glass-Reinforced
DGEBA/DDS in the Part-Shaped Cavity. .............................................. 96

Average Center-to-Wall Temperature Gradient for Microwave,
Conventional, and Hybrid Curing of E-glass-Reinforced
DGEBA/DDS in the Part-Shaped Cavity. .............................................. 97

Theoretical Temperature Proﬁles at Various Times for
Conventionally Heating an Inﬁnite Slab of Polymer Between
Brass Boundaries. ............................................................................... 100

Theoretical Temperature Proﬁles at Various Times for a
Constant Microwave Power Input of 2.0 W/cm3 in an Inﬁnite
Slab of Polymer Between Brass Boundaries. ....................................... 102

Figure 4.25. Theoretical Temperature Proﬁles at Various Times for Hybrid
Heating With a Constant Microwave Power Input of 2.0 W/cm3
in an Inﬁnite Slab of Polymer Between Brass Boundaries. ................... 103

Figure 4.26. Load-Deﬂection Data “With Initial Slope for Sample 7H (Hybrid
Cured E-glass-Reinforced DGEBA/DDS). .......................................... 104

Figure F.l. Boundary Conditions of the One-Dimensional Heat Transfer
Model. ................................................................................................. 125

xii

CHAPTER ONE

INTRODUCTION

1.1. Research Description and Scope

Increasing demand for composites and polymers in the automotive, aerospace,
construction, and sporting goods industries has invited efﬁcient and novel techniques for
processing polymeric materials. Most conventional closed mold processing techniques
using thermosets rely on conduction heating from the hot mold surfaces to induce the
crosslinking. Large temperature gradients inherently develop in resins and composite
systems with thick cross-sections or with low thermal conductivities. Residual thermal
stresses, anisotropic degrees of cure, resin degradation, and part warpage may originate
from these large temperature gradients, diminishing the mechanical properties of the
processed part. Additionally, production of thick cross-section parts with low thermal
conductivities requires longer cycle times due to low heat transfer rates.

Virtues of microwave heating make microwave processing a viable candidate as
an alternative to many conventional polymer and composite processes. However,
despite the demonstrated advantages of microwave curing on the laboratory scale,
microwave processing has not fully developed in industry. A major factor restraining
industrial use of microwave processing is the lack of suitable tooling.

Composites have been cured using a microwave resonant cavity. Molds are
fabricated with low loss materials to the desired part geometry, charged with process

material, and placed in a cylindrical resonant cavity. This technology has demonstrated

2
potential for processing composites in an atmospheric pressure environment. However,
the low loss mold materials, such as Teﬂon, quartz, or polyethylene, typically do not
possess the mechanical properties to maintain integrity for any cyclic process using
pressure. In order to be competitive with liquid composite molding processes requiring
pressure such as resin transfer molding (RTM) and structural reaction injection molding
(SRIM), a microwave part-shaped cavity was designed. In this design, a metallic cavity
serves as the mold for the processed part and as the microwave applicator. This novel
approach allows pressure during processing and eliminates the need for fabricating a
microwave transparent mold.

In this research, the electromagnetic behavior of the part-shaped applicator
system is characterized theoretically using Maxwell’s equations. As part of this
development, key scale-up issues such as modes of propagation, penetration depths,
electric ﬁeld (E-ﬁeld) distribution, and cutoff ﬁequencies are determined. The
microwave heating patterns, using fully cured glass-reinforced polyester plates, are
experimentally characterized in the part-shaped applicator for comparison to theory and
for determining a heating scheme for maximum heating uniformity during a cure cycle.
This research also includes the development of the part-shaped applicator into a
laboratory RTM unit including a pumping system, an external heating system for hybrid
heating, and a microwave multiplexing system. Three epoxy parts are cured in the part-
shaped cavity by microwave heating, conventional heating, and hybrid heating where

both microwave and conduction heating are employed.

3

1.2. Microwave Processing of Materials

Microwaves are oscillating electromagnetic waves with a frequency roughly
between 300 MHz and 300 GHz, as shown in Figure 1.1, which correspond to
characteristic wavelengths of 1 m to 1 mm, respectively. Since the end of World War II,
microwave energy has been utilized extensively in both communication and non-
corrununication applieations‘. The primary non-communication applications are in medicine
and heating. Typically, those ﬁequencies designated by International Telecommunication
Union for industrial, scientiﬁc, and medical (ISM) applications are used for heating. The ISM
frequencies are speciﬁc to each nation”. Two commonly used ISM frequencies, regulated by
the Federal Communications Commission, for microwave heating in the United States are
915 MHz and 2.45 GHz. Non-ISM frequencies may also be used for heating purposes;
however, leakage must be controlled to avoid interference with frequencies designated
for other purposes such as communications’.

Microwave processing of polymeric materials became of interest in the late
1960s, followed by a lull and then a renewed interest in the mid-1980?. A major
research area today is in the development of microwave processing of high-performance
polymeric composite materials”. Numerous advantages over conventional polymer
processing have been demonstrated. The following section expounds on select virtues of

microwave processing of polymers and composites.

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1.2. 1. Rationale

As shown in Table 1.1, the fundamental differences in heat transfer result in
distinctions between conventional and microwave processing. Compared to the
conductive heating techniques employed by most conventional composite and polymer
processes, microwaves oﬂ‘er many advantages due to the electromagnetic interaction
with the process material. The advantages of microwave processing are typically divided
into two categories: microwave characteristics and microwave effects. The former
encompasses all virtues of microwave processing that are predictable based on
electromagnetic and/or heat transfer theory. Beneﬁts of microwave processing that can
not be explained using traditional theories are classiﬁed as microwave effects. Naturally,
many emerging theories on microwave effects create a controversial research area.
Table 1.21H3 presents a summary of microwave characteristics and eﬁ‘ects.

Attaining the desired processing temperature in the center of thick cross-section
parts is often a limiting factor in polymer and composite processing. Due to the
penetration of microwave radiation into the process material, heating occurs throughout
the material volume for relative low loss systems. In fact, microwave applicators
typically provide preferential heating in the center of the part which results in a
temperature gradient that is the inverse of that for conventional processing.

By implementing the proper microwave generator and applicator, heating for

most polymer systems is very rapid and efficient. In contrast, conduction heating, as

Table 1. I. Comparison of Conventional to Microwave Therrnoset Processing.

 

 

 

 

 

Conventional Processing Microwave ProcessinL
Temperature gradient highest temperature near inverse of conventional
mold walls - lowest temperature gradient
temperature at part center
Heat transfer to the part limited by the thermal direct energy input due
center diffusivity to penetrating nature of
electromagnetic radiation
Rate of energy input directly proportional to the directly proportional to
temperature difference the square of the E-ﬁeld
between the tooling or and the effective
surroundings and the part dielectric loss of the
surface process material
Control of energy input limited by controllability of instantaneous control
the tooling or surrounding

temperature

 

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Table 1. 2. Microwave Characteristics and Effectsw'”.

 

 

 

 

 

 

Characteristic/Effect Advantage
Penetrating radiation Volumetric heating
(Characteristic)

Rapid and efﬁcient heating

Orr/off controllability
Controllable E-ﬁelds Ability to process low-loss materials
(Characteristic)

Ability to process parts of complex geometry
Differential coupling Self-limiting heating of thermosets
(Characteristic)

Improved matrix/carbon ﬁber interfacial strength
Reaction rate enhancement Potential cycle time reduction
(Eﬂ‘ect)
Improved physical properties Potential improvement in part performance

(Effect)

 

8
employed by most conventional processes, occurs across a surface and heat transfer from
the surface to the center of the polymer system is slow due to relatively small thermal
diﬁ‘usivities. Heat transfer to process material in conventional molds is proportional to
the temperature difference between the material and the mold. The mold temperature
and, hence, the heat transfer rate are limited by polymer degradation temperature.
Figure 1.2 illustrates a qualitative processing window for thennoset processing. The
characteristic microwave penetration oﬁ‘ers potential on an industrial scale to shift the
processing time closer to t1 and, therefore, reduce cycle time and production cost.

Microwave penetration also aids in the controllability of heating. Conversion of
electromagnetic energy to thermal energy is instantaneous; therefore, microwave heating
oﬁ‘ers on—oﬂ‘ controllability of energy input into the process material. An intrinsic time
lag is often associated with the control of conventional tooling due to the mold material
heat capacity.

Electromagnetic ﬁelds can be tailored to provide very high localized ﬁeld
strengths. This is beneﬁcial for processing low-loss materials (i.e., materials with low
microwave power absorption) and irregularly shaped materials. Fellows and Hawley“ ’5
exploit this concept in the processing of V-shaped composites in a single-mode cavity
using a technique called mode-switching. Using only one mode in a single-mode cavity
during a curing cycle often results in localized heating. By switching modes, one can
generate ﬁeld patterns with varying intensities and spacial locations. To maximize

heating uniformity, Fellows identiﬁed complementary modes and switched between these

modes by mechanically adjusting the cavity length. Qiu et al."" demonstrated the same

 

 

 

 

 

 

Degradation
\ \
Short Fully Cured Part Une conomi cal

g Shot \ Operation

3

S-

{2 Partially Cured Part

t1 t2
Time

Figure 1.2. Generalized Thermoset Processing Window.

10
concept in a single-mode cavity with a ﬁxed cavity length by switching modes
electronically. This technique is termed variable ﬁequency processing, where the several
ﬁ’equencies are selected alternatively to excite modes in an intelligent switching
sequence. In general, variable frequency processing offers a more robust and efﬁcient
processing method than mechanical mode switching.

The dielectric loss is one of the physical parameters that determines the
microwave power absorption in a material. When a mixture of components with
different dielectric losses is electromagnetically heated, the various components exhibit
diﬁ'erential coupling. In some polymer processing cases, this phenomenon has been
demonstrated to improve mechanical properties. For instance, Agrawal and Drzall9
cured carbon ﬁber-reinforced epoxy specimens using microwave radiation; these samples
showed as much as a 70% increase ﬁber-matrix adhesion. This observation could
possibly be attributed to an improved epoxy network structure at the ﬁber-matrix
interface due to the relatively high energy absorption of the carbon ﬁbers.

Wei” investigated the eﬂ'ects of microwaves on the crosslinking of two
stoichiometiic epoxy mixtures: diglycidyl ether of bisphenol A (DGEBA) with a high loss
curing agent, diaminodiphenyl sulfone (DDS), and DGEBA with a low loss cruing agent,
meta phenylene diarnine (m-PDA). Wei reported an enhanced reaction rate for the
microwave cure as compared to the thermal cure at the same temperature. Additionally, the
T, analysis indicated that similar glass transition temperatures for microwave and thermally
cured samples were obtained at low conversion. However, at larger conversions, higher T,'s

were observed for the microwave cured samples. Again, improved network structure may be

l 1
respomible for the increased T, in microwave cured epoxies at high extents. This
enhancementismuchmore signiﬁcantintheDDS cured DGEBAthaninthem-PDAcured
DGEBA, possiblyduetothefactthatDDS hasamuchhigher dipolemomentthanm-PDA

For thermosets, microwaves offer self-limiting heating during processing. Cross-
linked materials are relatively less susceptible to microwave heating due to the decrease in
dipolar mobility and, hence, dielectric properties with extent of reaction. During
polymerintion, the concentration and magnitude of dipoles can decrease, ﬁirther causing
diminished dielectric properties Therefore, the microwave energy will concentrate
intrinsicallyontheunreactedresin 'I'hisisanotherillustrationofdiﬁ‘erential coupling.

As discussed above, the microwave characteristics, including radiation penetration,
controllable E-ﬁelds, and differential coupling, lead to advantages of microwave processing
over conventional processing. Some of the controversial microwave eﬁ‘ects include enhanced
polymerintion rates, greater T, and improved mechanical properties include. These
enhancementsmaybedueto oneormorephenomenaonthemolecularlevel, including
enhanced segmental vibration, preferential alignment of dipolar segments, and creation of
“energetic species“. One theory attributes the reaction rate enhancement to a non-
equilibrated temperature effect where high loss species, such as DDS, experience a higher
localtemperatm'ethanthebulk”. Furtherstudiesarerequiredtoclarifyandquantifythese

microwave eﬂ‘ects.

12

1.2.2. Applications

Microwave heating is a relatively new technology. Signiﬁcant developments in
the use of microwaves for heating purposes did not commence until after World War II.
In 1951, the ﬁrst microwave generator became commercially available for restaurant use,
followed by the infusion of the domestic microwave ovens in the 19608”. To date,
microwave processing has been well established in the food, rubber, textile and lumber
industries.

Commercial applications of microwave processing of polymers have been limited
to a few successﬁrl operations, including vulcanization of rubber‘m", production of
polyvinylchloride belts”, production of urethane foam”, and curing plywood
adhesives". Some experts speculate that possible users of microwave processing have
been turned away because of the lack of knowledge of the details and limitations of
equipment and processes”, the lack of suitable tooling’, and conservatism”. In the past
decade, there has been a concentrated effort in exploring and developing microwave
processing techru'ques for polymers and composites. The following two sections
highlight speciﬁc research ﬁndings in the use of microwave processing of polymers and

composites on the laboratory scale.

1.2.2. 1. Polymer Processing
Microwaves have been utilized in the processing of both thermosets and
thermoplastics in various capacities. With thermoplastic processing, microwaves have been

used as an energy source for such operations as drying”, polymerization", melting", and

l3

foaming”. For the processing of thermosets, microwave energy has been used primarily to

In general, the physical processes occurring during the conversion of electromagnetic
eriergy to thermal energy for thermoplastics are different than those for thermosets. Chen“
et a1. performed a thorough investigation of the factors inﬂuencing microwave absorptivity of
several amorphous and semicrystalline therm0plastics. Inﬂections in the microwave heating
rates as a ﬁinction oftemperature or ﬁequency are attributed to molecular relaxations.
Microwave calorimetiic studies revealed a critical temperature, T., above which rapid
volumetric heating is feasrble. Tc is typically between the T, and the melt temperature, T...
Since most T,s are above room temperature, microwave heating of thermoplastics ﬁom room
temperature is challenging. To overcome the low microwave absorptivity at room
temperature, thetemperanireofthethermoplasticcanberaisedto chyemploying
convartional heating or relatively lossy molds to absorb microwave radiation at lower
We

George et a1.“ presented results of heating tests conducted on polyvinyl chloride
(PVC), polycarbonate (PC), polypropylene (PP), and acrylonitrile butadiene-styrene
polymer (ABS) pellets. These polymers were dried using a tray dryer at 104°C and, for
comparison, a microwave oven with an air sweep above the oven bottom. The
penetrating nature of microwaves results in internal heating of the water bound in the
plastics, thus water is forced out from the pellet center. Conversely, conventional
heating relies on heat transfer to the center of the pellets and water diffusion from the

center to the surface of the pellets. As a result, the microwave oven dried the polymers

14
4 to 16 times more rapidly, depending on the hygroscopic nature of the polymers. The
tensile strength and impact resistance were slightly improved for the microwave drying
cases possibly due to less thermal degradation at the shorter drying times.

Several investigations have been performed on microwave curing of thermosets,
inchrding epoxies7"‘”’n'”"‘, polyestersw, polyimides”", and polyurethanesx'c'o. For
microwave curing, epoxies are the most widely studied thermosets.

During the early stages of cure, the dielectric properties of thermosets typically
increase to a maximum followed by a decrease to a constant value. To commence
crosslinldng, theresinisusuallyheatedtoacureternperature. Duringthis stage, theviscosity
of the therrnoset decreases, corresponding to an increase in molecular mobility. The dipolar
groups of the resin are less restricted to rotate with the oscillating E—ﬁeld, so the microwave
absorptivity increases. Two molecular phenomena that affect the polarizability occur during
crosslinking are changes in the types and concentration of polar groups and the formation of a
thee-dimensional network. For example, curing an epoxy with an amine curative involves a
decrease in epoxide dipoles, a change of amine dipoles ﬁ'om primary to secondary and
tertiary, and the creation of hydroxyl dipoles. As the crosslinking density increases, the
polarimbility greatly decreases causing diminished dielectric properties of the cured
thermoset. Research in modeling dielectric properties has become an increasingly intensiﬁed
area“'“. The dielectric behavior of a thermosetting system can give indications of
mechanism of microwave - molecular interaction for systems that exhibit microwave effects

(i.e., enhanced reaction rates) and provide a means of in situ monitoring of the cure process.

1 5

Reports in the literature on comparative kinetic studies between conventionally
and microwave cured samples vary dramaticallyn‘“, ranging from no statistical deviation
to a 40-fold increase in reaction kinetics for microwave cured samples. The inconsistent
results could be attributed to the lack of a standardized testing procedure. Mijovic and
Wijayam initially claimed a decrease in reaction kinetics for microwave curing of
digylcidyl ether of bisphenol A (DGEBA) and diaminodiphenyl sulfone (DDS) and later"3
argued that the difference was statistically insigniﬁcant. Mijovic also observed a broad
two-step glass transition in some of the microwave cured DSC thermographs for the
DGEBA/DDS samples. Two-step glass transitions were not observed with thermal
cures, possibly indicating a different reaction mechanism and network structure between
microwave and thermally cured epoxies. Mijovic studied DGEBA with a 4,4’-
methylenediariiline (MDA) curative and found no apparent change in reaction kinetics
with the processing method. However, DeLong et a1.“ discovered a slightly enhanced
DGEBA/MDA reaction rate under the inﬂuence of an electromagnetic ﬁeld. Further
kinetic studies are necessary to elucidate the apparent effect of reaction rate
enhancement.

Many investigators have demonstrated that controlling the microwave pulse
iicqucncy can be an important process variable. For instance, Beldjoudi and Gourdenne“
reportedﬂiataDGEBAmDMsystemreachedthegel pointin 19.0 rninutesusinga 1000Hz
pulse ﬁequency with a pulse length of0.25 ms and an average power of40W. Gelation
occurred in 23.8 minutes for the continuously radiated epoxy at 40W. A more rapid gel

time is actually a phenomenological effect due to an enhanced heating rate for the case

l 6
with pulsed energy due to what Beldjoudi explains as a double relaxation. That is, the
microwave source provides the dipolar orientation, and the lower ﬁ'equency pulses excite
molecular segmental relaxations.

Jow et al.‘2 cured DGEBA/DDS samples utilizing conventional heating and
continuous and pulsed microwave radiation. More rapid heating and higher
temperatures were realized with pulsed microwave heating. Additionally, pulse heating
provided superior temperature control.

Lewis et a1.” conducted an experimental investigation on the crosslinking of
DGEBA epoxy resin with 4,4’- or 3,3’-diaminodiphenyl sulfone using microwave
radiation. Results show that the mechanical properties of the microwave-crosslinked
resin were similar to those of the thermally crosslinked system. But microwave radiation
signiﬁcantly accelerated the crosslinking. The degree of acceleration in the rate of

crosslinking depended on both power and sample size.

1.2.2.2. Composite Processing

Polymer composites have found numerous applications as structures and
components in the automotive, aerospace, recreational, construction, and appliance
industries. Some of the beneﬁts of composites include the high strength property-to-
weight ratio, low speciﬁc gravity, and ability to integrate components. The selection of
metal versus composites, however, is often driven by the composite unit cost. Since
long processing times can contribute signiﬁcantly to the unit cost, the potentially

enhanced reaction kinetics and faster cycle times make microwave processing an

17
attractive alternative to the conventional processes such as autoclaving, RTM, and
SRIM.

Lee and Springer‘3 performed one of the earlier investigations of composite
curing experiments. A low-loss press, constructed of polypropylene and glass-reinforced
Teﬂon, provided pressure during the cure of Fiberite 82/9134B glass epoxy and
Hercules AS/3501-6 graphite epoxy specimens. The press apparatus and lay-up
composite were placed in a domestic multirnode 2.45 GHz microwave oven with
controllable power up to 700 watts. The effect of polarization angle on both
unidirectional and multidirectional laminates on the temperature distribution and
curability was investigated. Lee discovered that the glass-reinforced epoxy samples
could be cured effectively for any ply orientation and polarization angle. However,
eﬂ‘ective curing was only realized for the unidirectional graphite-reinforced epoxy
samples and a polarization angle of 90°; multidirectional laminates could not be
sufficiently cured. Poor heating of the graphite-reinforced epoxy laminates can be
attributed to electromagnetic reﬂection at the sample surface due to the large electrical
conductivity of the graphite ﬁbers.

By using a single-mode cavity, Wei et al. “ demonstrated curability of 72-ply
crossply and unidirectional graphite-reinforced epoxy samples (Hercules AS4/3 501-6) in
a single-mode cavity. The single-mode cavity offers selectivity of the resonant
electromagnetic modes and, hence, control of the heating proﬁles. In commercial

multimode ovens, the heating patterns are uncontrollable, often resulting in anisotropic

18
cure. This could explain the ineffective cure of crossply samples reported by Lee and
Springer in the multimode oven.

Power dissipation in a fully cured carbon ﬁber-reinforced epoxy disk was studied
in a cylindrical single-mode cavity by Fritz and Asmussen“. Since carbon ﬁbers are
conductive, a signiﬁcant microwave power attenuation, or a skin effect, occurs ﬁom the
disk surface. By studying temperature proﬁles, Fritz calculated a penetration depth of
approximately 1 cm for a 2.45 GHz microwave source. The penetration depth is the
distance through which the microwave power density is reduced to He of the surface
power density. Because of the skin effect, microwave heating of composites with
conductive ﬁbers is often achieved by thermal conduction from the surface layer.

Since composites are often implemented in structural applications, the eﬂ‘ect of
microwave processing on the composite mechanical properties is of interest to many
investigators. Wei et al.“ studied the failure modes of graphite ﬁber-reinforced epoxy
composites (Hercules AS4/3 501-6) using SEM analysis. The photomicrographs
revealed that microwave processed samples failed in a matrix failure mode.
Conventionally cured samples failed in a combination of interfacial and matrix failure
modes. This suggests an enhanced graphite ﬁber/epoxy bonding for the microwave
processed composites.

Agrawal and Drzal" studied the mechanical property characteristics of epoxy
composites. Single ﬁber properties were investigated by curing DGEBA with a meta-
phenylene diamine (m-PDA) curative with a single carbon, glass, or aramid (kevlar) ﬁber

in a dogbone cavity. For each ﬁber type, samples were cured using microwave and

19

conventional techniques. In accordance with the results that Wei obtained, tensile tests
and photomicrographs reveal that the ﬁber-matrix adhesion for the graphite composites
was enhanced. The interfacial shear strength for the microwave cured samples was 70%
greater than that for conventionally cured samples. Failure occurred in the matrix for
microwave cured specimens; however, conventionally cured specimens failed at the
interface. The tensile test for the aramid and glass ﬁber samples revealed that the
interfacial shear strength for the conventionally cured samples is 15% greater than that
for the microwave cured samples. This could possibly be attributed to an incomplete
network structure around the glass and aramid ﬁbers for the microwave cured samples.

By using a lap shear compression test with E-glass panels, Boey and Yue“7
demonstrated that microwave cured samples could have a greater shear strength than
conventionally cured samples. In fact, a 100% increase in shear strength was realized for
microwave cured samples that were exposed to 500 W for 15 minutes. From this work,
however, one can not conclude that microwave processing of glass-reinforced epoxy
results in improved shear strength relative to conventionally cured samples in all cases.
Several power ratings and radiation times were investigated for microwave cured
samples, while no indication was given that the time and temperature for the
conventional cure cycle was optimized to yield the best shear strength. This paper does
illustrate the importance of engineering a proper microwave power cycle. The physical

properties are highly dependent on the processing cycle.

20

1.2.3. Microwave Processing Systems

The microwave circuitry is the system through which microwaves are directed to the
applicator. Figure 1.3 illustrates a typical cavity conﬁguration. Arrangement and type of the
circuitry components may vary depending on the particular process needs. For example, high
power applications (i.e., approximately > 500W) typically use waveguides instead of coaxial
transmission lines. Waveguides have very little loss compared to coaxial lines.

Of the commercially available microwave sources, magnetrons, klystrons, traveling
wave tubes (I'WT), and gyrotrons, magnetrons are the most commonly used generators.
Magnetrons offer high power outputs, efficient operation, frequency stability, and relatively
low cost“. The desired heating rate of the process material must be considered when the
proper generator power requirements in polymer processing are engineered. Typically,
during the cure of thermosetting polymers, the resin experiences two network
transformations: gelation and vitriﬁcation. Gelation refers to the process of converting a
thermosetting resin to a crosslinked network with an inﬁnite weight-average degree of
polymerization. Vitriﬁcation refers to the transformation of the liquid or gel to a glassy
state. For maximum physical strength, complete gelation should occur before
vitriﬁcation. Using excessive microwave powers can cause vitriﬁcation before complete
gelation to produce a gelled nibber or a charred polymer. Boey and Lee“ discovered
that using an 800 W power source for curing a 10-mm thick glass-reinforced
DGEBA/DDS composite resulted in thermal degradation of the sample. This case study
illustrates the importance of implementing process control and selecting the proper

generator power.

21

 

 

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Successful commercialimtion of microwave processes requires process control and
automation to provide an attractive ‘ﬁrser-ﬁiendly” system Issues of process control and
automation of microwave systems have been investigated experimentally. Compared to a
manually operated system, an automated system can improve the overall process efficiency,
reduce sample temperature gradients, minimize temperature excursions due to reaction
exotherms, andeliminatetheneedforanexperienced operator".

The microwave applicator is the apparatus through which microwaves are applied to
the process material. Of the four basic types of applicators (i.e., multimode, traveling wave,
sinQe-mode, and specialized), the multimode applicator is the most common However, the
multimode oven is, ironically, the most empirical of the four types. A multimode oven is,
generally, a parallelopipedic box or cube in which several modes are excited over a small
ﬁ'equency band. One of the major challenges with processing with a multimode oven is
heating uniformity due to uneven distributions of electromagnetic energy in the cavity.
Several techniques have been investigated to improve heating uniformity. For example,
Kashyap and Wysloua'l"l address two methods that improved the energy distributions: multi-
slot waveguide feed with a stirrer system and ﬁ'equency sweeping. With the stirrer system,
the multi-slot waveguide feed essentially transports the microwave incident energy to select
locationsinsidethecavity. Frequencysweepingaltersthemodetypesthatareexcitedinthe
cavity, and, therefore, redistnbutes the energy.

Traveling wave applicators are chieﬂy used for continuous processing of high loss
material; low loss materials require an excessively long waveguide or slow processing speed

to absorb the necessary energy". Waveguides are hollow circular or rectangular pipes

23
through which electromagnetic energy is transmitted. For continuous processing of materials,
slotsforthefeed and exit arewtinthewaveguidesatlocations oflowE-ﬁeld tominimize
energy leakage. Theoretically, the microwave radiation in traveling wave applicators
propagatesinonedirectiononly. ’I’heenergythatisnotabsorbedbytheprocessmaterialis
directed to a matched load.

Relative to multimode and traveling wave applicators, the efficiency and ﬁeld strength
of single-mode applicators are greater. In principle, the single-mode applicator is a cavity in
which an incident wave in superposition with reﬂected waves creates a standing wave at a
resonant ﬁequency. The well-deﬁned distribution of electric and magnetic ﬁeld vectors is
termed a mode. Controllability of these modes is attained by adjusting the cavity volume or
the microwave ﬁequency. Because of the high E—ﬁeld capability, single-mode cavities are
ideal for the processing of low loss materials.

Applicators that cannot be purely classiﬁed as multimode, traveling wave, or
single-mode are categorized as specialized applicator. Specialized applicators are often
hybrids of two applicators. For instance, the part-shaped cavity is a specialized
applicator in that both resonance and traveling wave conditions exist during processing.

Details on the overall equipment and components are given in Chapter 3.

1.3. Resin Transfer Molding

Resin transfer molding (RTM) is regarded by some as being the “leading

”73

technology to provide large primary structures in the ﬁlture and “perhaps the only

composites fabrication technique that is simultaneously gaining rapid acceptance in both

24
the aerospace and automotive industries”". RTM technology has been in existence for
over halfa century and, more recently, has demonstrated the capability of producing
structural composites as well as class A surfaces for cosmetic applications. Chrysler
corporation is demonstrating this capability by manufacturing the limited-edition Dodge
Viper. Over half of the Viper exterior, including the hood, ﬁ'ont fenders, rear quarter
panels, decklid, and roof supports, is manufactured using RTM". In another case study,
the General Motors Corporation produces a glass reinforced-epoxy composite leaf
spring for the Corvette using RTM. Several other industries manufacture composite

components using the RTM technology, as outlined in Table 1.3"".

1.3.1. Process Cycle

The development of RTM began in the 19405 as a one-pot process called the
Marco process". This technique involved mixing a thermosetting resin and a catalyst
before injecting the resin system into a mold cavity ﬁlled with randomly-oriented
ﬁberglass. At room temperature, the ﬁnal composite would then cure.

Today the RTM process concept is the same; however, the processing details
have evolved to meet the speciﬁc needs of the RTM industry. Figure 1.4"“) illustrates
the various steps, in essence, involved with RTM. Initially, the mold is prepared by
removing foreign particles and preparing the seals, sprue, and vents. If necessary,
various coatings are applied to the mold walls. For instance, a mold release agent can be
utilized to assist in part removal. A gel-coat could also be applied to improve the surface

quality of the molded part by minimizing reinforcement read-through. The gel-coat may

25

Table I. 3. Examples of Components Manufactured by RTM”.

 

Industry
Aerospace

Automotive

Recreational

Construction

Appliance

Component
airplane wing ribs
cockpit hatch cover
fuselage frame
radar housing
wing ﬂaps

body panel

cross member
decklid

fender

hood

leaf spring

rear quarter panel
transmission housing
boat hull

canoe paddle
golf clubs

ice hockey stick
ski poles
snowmobile body
trailer

bathtub

safety helmet
seating

septic tank

water tank
computer housing
fan blade
television antenna

26

 

 

 

 

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Part Discharge

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Resin Cure Resin Injection

 

 

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Mold Filling and Resin Oversle

 

 

 

 

Figure 1.4. RTM Processing Cycle".

27

also protect the part surface ﬁom weathering. Next, a continuous, strand mat, woven
roving, or cloth reinforcement system is placed on the mold surface. The ﬁbers may or
may not be preshaped. The mold is closed, and a thermosetting resin is injected into the
mold cavity through a sprue that is typically centrally located. The resin displaces the air
in the ﬁber reinforcement and ﬁlls the mold to the air vents. The ﬁnal composite is either
cured at room temperature or at elevated temperatures, depending on the selected resin
system. Upon reaching the desired conversion, the composite is removed from the mold
and trimmed, if necessary.

Because of the low thermal conductivities of most thermosets and ﬁber
reinforcements, large temperature gradients can develop in parts with a thick cross-
section. Additionally, the curing stage could be excessive due to inadequate
temperatures in the composite center. Considering these issues, the penetrating
characteristic of microwave processing offers a potential solution to cycle time reduction
and uniform curing of thick composites. Electromagnetic radiation has been exploited in
curing binders used in ﬁber mats“; however, to date, microwaves have not been

utilized in the curing stage of RTM processing.

1.3.2. Polymer Matrices
Performance characteristics, processability, and cost must all be considered when
selecting a polymer system. As is evident ﬁom Table 1.3, parts manufactured by RTM

have a wide range of performance criteria and, therefore, demand a wide range of resin

28
performance characteristics. Epoxy or polyester resins are typically used; however,
RTM is adaptable to numerous thermosets, including polyurethanes, phenolics,
bismaleimides, and methacrylates“.

Epoxies typically exhibit physical properties, such as toughness, ﬂexibility, and
strength, required for higher performance parts. Additionally, epoxies oﬁ'er good surface
quality, chemical resistance, excellent adhesion, low shrinkage, and high service
temperature. Bisphenol A-based resins constitute the majority of the epoxy resins used

with RTM.

1.3.3. Fiber System

The selection of the reinforcement for a composite dictates the composite
performance properties such as speciﬁc gravity, tensile strength, and modulus.
Additionally, since the ﬁbers can constitute over half of the part volume fraction, the
reinforcement signiﬁcantly impacts the part cost.

Typical ﬁbers used in RTM include E-glass, S-glass, carbon, and aramid. E-glass
and S-glass exhibit high tensile strengths, good chemical resistance, and excellent
insulating properties. Carbon ﬁbers are very attractive for high performance applications
due to the high tensile modulus, low coefficient of thermal expansion, and high fatigue
strengths. Aramid ﬁber is a general term for an aromatic polyarnide ﬁber. This ﬁber
system oﬁ‘ers exceptionally high strength, low density, and toughness.

Selections of the ﬁber reinforcement and resin system are not independent. The
adhesion promoter sizing on the ﬁbers must be compatible with the resin system.

Achieving intimate resin-ﬁber contact is essential to maximizing mechanical properties of

29
the ﬁnal composite. Poor resin-ﬁber contact results in diminished part performance due

to weakened interlaminar shear strength and ﬁber pull-out under stress.

CHAPTER Two

THEORY
2.1. Maxwell’s Equations

The study of electrostatics reveals that a stationary volume charge density, p ,
results in an electric ﬁeld, E , about the charge volume. Analogously, the study of
magnetostatics shows that a constant volume current density, I , throughout a test
volume produces a magnetic ﬂux density, B. If I in a ﬁxed volume varies with time, p

will change as well with time according to the conservation of electric charge:
.. _. a _.
V-J(r,t)+ap(r,t) = o. [2.1]

Since I and p are coupled through Equation [2.1], E is coupled to E for time-varying
electric sources. The equations that deﬁne the relation between E and E as a function
of I and p are termed Maxwell’s equations, named after the theoretician James Clerk

Maxwell. Maxwell’s equations provide the foundation for electromagnetism.

For solving time-varying ﬁelds, a common representation of Maxwell’s equations
is in complex phasor form, valid at any point 1' in the deﬁned system. For ﬁelds
produced by a source of constant angular ﬁ'equency, co, the Maxwell ﬁeld equations are

represented as follows:
V x E(r) = -jtoi§(r) [2.2]

v.1‘3(r) = o [2.3]

30

31

V x H(r) = 7(r) + juD(r) [2.4]
V-D(r) = p(r) [2.51

where E(r) = electric ﬁeld strength [=] volts/m,
D(1‘) = electric ﬂux density [=] coulombs/mz,
H(f) = magnetic ﬁeld strength [=] amperes/m,
E(1‘) = magnetic ﬂux density [=] webers/mz,
I(1') = volume current density [=] amperes/mz,
p(1') = volume charge density =] coulombs/m3,
j = imaginary number ﬂ , and
co = angular frequency [=] radians/s.

Using a series of vector operations and Equation [2.1], it can be shown that Equations

[2.2] through [2.5] actually represent only two independent vector equations.

D and B can be eliminated by introducing the following constitutive equations
for linear “simple” matter” :

1‘5 = e‘(to)E [2.6]

E = u°(co )H [2.7]
Here, s°(co) is called the complex permittivity, and u'(m) is termed the complex
permeability. The behavior of a'(co) is dependent on the material medium. Details on
8°(ro) are given in Section 2.3. For non-magnetic material, u'(m) is simply the

permeability of free space, u...

32

2.2. Wave Propagation

Electromagnetic energy is transmitted from regions in space via electromagnetic
modes. An electromagnetic mode is a distinct ﬁeld pattern that is classiﬁed as one of four
possible categories: transverse electric (TE), transverse magnetic (TM), transverse
electromagnetic (1m, and hybrid (HE). Physically, a TE mode does not have an electric
ﬁeldcomponentinthedirectionofpropagation, andaTMmodedoesnot haveamagnetic
ﬁeld component in the direction of propagation. Both electric and magnetic ﬁeld components
in the propagation direction are zero for the TEM mode and non-zero for the HE mode.

Mathematically, the ﬁrndamental equations that deﬁne the electric and magnetic ﬁeld
behavior are Maxwell’s equations. These equations coupled with the proper assumptions
(e.g., perfect reﬂection at metallic boundaries), mode classiﬁcation (i.e., TE, TM, TEM, or
HE) and boundary conditions deﬁne a boundary-value problem with a harmonic eigenvalue
solution.

At any material interface, the tangential component of the E-ﬁeld is continuous
and the normal component of the B-ﬁeld is continuous. A special case of these boundary
conditions is a dielectric-conductor interface. Since the E-ﬁeld and the H-ﬁeld, or B-
ﬁeld, are essentially non-existent in the conductor, the tangential E-ﬁeld and normal H-
ﬁeld at the dielectric-conductor interface are zero.

The following two sections present two speciﬁc systems for transporting, storing,

or transferring electromagnetic energy: the rectangular waveguide and the rectangular

33
cavity resonator. Details of the derivations for the equations shown in the next two

sections are outlined in Appendices A and B.

2.2.1. Rectangular Waveguide

A waveguide is a structure with, typically, either a rectangular or a circular cross
section deﬁned by conductive boundaries that is used to transport electromagnetic energy.
Waveguides have also been used to heat material. Energy from a microwave generator
travels through the waveguide and is partially absorbed by the process material. The
remainder of the energy is directed to a terminating load. Waveguide applicators are
primarily used for continuous processing of high loss material; low loss materials require
an excessively long waveguide or slow processing speed to absorb the necessary energy.

For TE modes in a rectangular waveguide, the E-ﬁeld components are given in

 

phasor notation by:

E, = CIky cos(k,x) sin(kyy)exp(—jkzz), [2.8]
E, = —c,k, sin(it,x)cos(lt,y)exp(— 11,2), and [2.9]
E, = o, [2.10]
where C; = a constant,

it, = (3‘33), m = o,1,2,..., [2.11]

a
try = (%),n=0,1,2,... , and [2.12]
mi: 2 int 2
kl = Jar 28011.0(8' - ja") —(—) —(—b-) . [2.13]
a

34
These expressions deﬁne the E-ﬁeld for a positive 2 traveling wave in a waveguide with
an x-length of a and y-length of b, where a 2 b. Additionally, in or n can be zero;
however, both in and 11 may not be zero.

The traveling wave behavior is illustrated with vector E-ﬁeld plots in Figure 2.1
for the TEM mode in a rectangular waveguide. For the TEM mode, the E-ﬁeld is
independent of x and varies sinusoidally with y. As indicated by the eigenvalue indices,
there are zero variations in the x-direction and one half-wave variation in the y-direction.
The boundary condition of zero tangential E-ﬁeld at conductive boundaries is also
evident from Figure 2.1.

Conﬁning a region with conductive boundaries places a constraint on the
applicable source ﬁequency. In a waveguide, electromagnetic modes propagate above
the cutoﬁ’ frequency; below the cutoff ﬁ'equency, modes are evanescent, or
nonpropagating. Propagation of electromagnetic energy for a given mode through a
waveguide is only permissible using source frequencies above the cutoff frequency. The
mode becomes nonpropagating, or evanescent, below the cutoff frequency. For a
rectangular waveguide ﬁlled with a dielectric material, the cutoff frequency is a function
of the waveguide dimensions, the electromagnetic mode, and the dielectric constant of

the material as given by

 

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35

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37

2.2.2. Rectangular Resonant Cavity

The resonant cavity is a system, enclosed by conductive boundaries, that can store or
transfer electromagnetic energy. Typically, resonant cavities are excited by a coupling probe
or an aperture. The superposition of incident and reﬂected waves creates a standing wave
phenomenon, or resonance, in the cavity.

For processing applications in a resonant cavity, the heating uniformity inside
dielectric materials is a strong function of the heating mode. The E-ﬁeld for the TB modes

in a rectangular resonant cavity is given in phasor notation by:

E, = c,lt, cos(k,x)sin(lt,y)sin(lt,z), [2.15]
E, = -c,it, sin(lt,x)cos(k,y)sin(it,z), and [2.16]
E, = o, [2.17]
where C; = a constant,
it, =(ﬂ),m=o,i,2,..., [2.18]
a
it, =83?) , n=o,1,2,... , and [2.19]
it, = (E), p = 0,1,2... [2.20]
C

By convention, a < b < c, where a, b, and c refer to the x-length, y-length, and z-length
of the cavity, respectively. Additionally, in or n can be zero; however, both in and it may
not be zero.

A plot of the TEoir mode in a rectangular cavity is illustrated by vector plots for

various x-y planes in Figure 2.3. Only the x component of the E-ﬁeld is present for the

38

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2.... .m 2.... m...

Ailiili

lLJiiililililiLlJiJ

 

 

 

1 1 i . 8.. - 8.. 8..
.8... u a 2...... u a .92... a a 9...... u a . 9...... u u

 

 

 

 

 

 

oo— .o
on...
03....
02.6
coed

 

 

 

 

 

39

TEou mode. In accordance with the eigenvalue indices, there are no E—ﬁeld variations in
the x-direction, one half-wave variation in the y-direction, and one half-wave variation in
the z—direction. The strongest E-ﬁeld is located at the intersection of the planes at y =
0.5 b and z = 0.5 c (i.e., the cavity center). The E-ﬁeld diminishes to zero at y = O, y =
b, 2 = 0, and z = c due to the zero tangential E-ﬁeld boundary condition at conductive
surfaces.

Resonant conditions for a speciﬁc mode in a cavity can be attained by adjusting

the source frequency or cavity dimensions to satisfy the following relationship.

., = émJGz—J’ «13’ «151’ mu

Figure 2.4 illustrates this relationship for a 6.4 cm by 7.6 cm cavity.

 

2.3. Microwave Energy Absorption Mechanisms

In the microwave frequency range, there are primarily two types of mechanisms
for dissipating microwave radiation: dielectric loss heating in electrically insulating
material and electrical resistive heating in conductive material. Microwave power
absorption in polymeric materials is typically either purely dielectric heating or a summed
contribution of dielectric and resistive heating if the composite is partially composed of
conductive materials such as carbon ﬁbers or metal ﬁlings.

With conductive materials, the ﬁ'ee electrons move in response to the E-ﬁeld
causing power absorption due to electrical resistive heating; whereas, dielectric materials

principally absorb microwave radiation via reorientation and distortion of induced or

40

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..5. 593 £26

mN ON 3 o. m

 

582 x A 582 > A 582 N

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

ﬂour—x 1 -

oZE M
.8... i /
b
5E - v m
.8 m

m...
ORE // - m
1 o

 

 

 

41

permanent dipoles. This power absorption mechanism is called loss heating, which
originates ﬁ'om the inability of the dipoles to maintain phase with the electric ﬁeld. The
coupling eﬁciency of the E-ﬁeld with the dipoles is dependent on the dipole moment
magnitude, concentration of dipoles, and mobility of the dipoles. Dielectric loss due to dipole
relaxation is typically in the ﬁ'equency range of 106 to 1010 Hz ‘32).

The interaction between microwaves and a dielectric or a conductor is dictated by the
complexpennittivityofthematerial, e',where 8' = 8’ -je". e' istherealpermittivity
(i.e., electrical polarizability) of the material, the magnitude of which reﬂects its ability to
storeelectricalenergyasheat; jistheimaginarynumber E; e" isthelossfactorofthe
material, and the magnitude of this determines its ability to dissipate electrical energy into
thermal energy. The loss factor in the microwave frequency range captures the contributions

ﬁom both dielectric loss heating and resistive heating:

a" = 8;; + :0:— , i [2.22]

O

O
(08

where as is the loss ﬁom dipolar reorientation polarization and is the loss due to

 

resistive (i.e., ohmic) heating. a is the electrical conductivity of the material, a) is the angular
frequency of the microwaves (co=21r£ where f is the microwave ﬁequency), and e. is the

permittivity of he space (8.851(10'12 Fm" ). Separation of the dipolar and resistive heating
contributions experimentally is not feasible. Therefore, only a " is I‘CPOrted for materials in

the literature.

42
Microwave power dissipation per unit volume of homogeneous material, derived

ﬁ'om the ﬁrst law of thermodynamics and Poynting’s theorem, is given by

 

Pm = ~1- eom[eg + ° )Eﬁ, [2.23]
2 6,0)
where PW = power absorption [=] watts/m3,
a. = permittivity of ﬁ'ee space [=] farads/m,
0) = angular frequency =] radians/s,
a}; = dipolar loss,
a = electrical conductivity [=] Siemens/m, and
E0 = electric ﬁeld strength [=] volts/m.

From this expression, one can deduce that microwave power absorption is directly
proportional to the microwave frequency, dielectric loss, and the square of the electric

ﬁeld strength.

2.4. Penetration Depth

In composite and polymer processing, microwave power penetration is oﬂen of great
practical interest. Electromagnetic penetration into the material is crucial for energy to
couple with the process material.

The power penetration depth, D,, is deﬁned as the distance from the surface of an
inﬁnite plane at which the power diminishes to He (:43 7%) of the surface power. As
electromagnetic energy passes through lossy material, the E-ﬁeld magnitude decreases as a
result of the transfer of energy to thermal energy. Assuming zero reﬂected energy, the power

penetration depth in a lossy, homogeneous, and inﬁnite medium is calculated by

43

D = l [2.24]

P 2 ’
mJZuou'eoe'[ 1+(8—J -]
e

where u. is the permeability ofthe free space, 4rtx10'7 Hm"; and p.’ is the relative permeability
of the medium which equals one for non-magnetic materials. As is evident from Equation
[2.24], penetration depth in an unconﬁned region is inversely proportional to the
ﬁequency.

Placing a constraint on the boundaries of propagation alters the penetration depth
calculation. The penetration depth in a lossy waveguide is given by

1
D1: = , [2.25]

J1me

 

2 2
where R = mzeouoe' — (M) — (“I“) [=] l/mz, [2.26]
a
1: —co 2801108" , [2.27]
m = mode index for x-direction,
n = mode index for y-direction,
a = x-dimension of waveguide [=] m, and
b = y-dimension of waveguide [=] m.
For an unconﬁned system (i.e., a = b = co), Equation [2.25] reduces to Equation [2.24].
The penetration depth in a lossy waveguide decreases with increasing frequency above
an optimum ﬁequency where the penetration depth is a maximum, fa". Below ﬂy, the
penetration depth decreases to zero at the cutoff frequency. This behavior is illustrated

in Figure 2.5 for a 4.2 cm x 8.9 cm waveguide ﬁlled with a material of various complex

perrnittivities. From this ﬁgure, one can see that for a given dielectric constant, the

8.82.3 38.05%... a. a... .3 .5 N... a a. 08:. am. 2.. 8. 5...... 8888.. a}... .3 9...»...—

Anmmvv 5:259...—

 

44

 

o.» +
Quill
o. . IOI
.« €59.00 c.3205

 

 

 

 

 

3.0 I .w

A. o.o

 

.d
. a
u
a
l
m
. o... m.
n
a
. a
d
1
u.
)
. m
(
. 0N

 

 

45
penetration depth increases with a decreasing dielectric loss at a constant frequency.

Additionally, the cutoff frequency decreases for an increasing dielectric constant,

consistent with Figure 2.2. Details of the derivation of equations [2.24] and [2.25] are

shown in Appendix C.

CHAPTERTHREE

EXPERIMENTAL
3.1. Experimental System

The development of the part-shaped applicator consists of two phases: cavity
characterization and curing demonstration. The purpose of the former is to acquire an
understanding of the electromagnetic behavior in the process material. A portion of the
cavity characterization is theoretical, using concepts presented in Chapter 2. These
theoretical issues are addressed in Chapter 4. This chapter focuses on the details of the
laboratory experimental system for both the cavity characterization and curing
demonstration.

The microwave processing system is comprised of microwave applicators (e.g.,
part-shaped cavity and external applicators), microwave circuitry, thermometry system,
and process material. These components and systems are discussed in detail in the

following ﬁve sections.

3.1.1. Part-Shaped Cavity

The brass part-shaped cavity, as illustrated in Figure 3.1, is capable of processing
a parallelopipedic 31.8 cm x 8.9 cm x 4.2 cm part. The numbers in brackets in this
section refer to the various features in Figure 3.1.

Electromagnetic radiation is admitted to the process material in the cavity

through external applicators and aperture plates that can be secured to a centrally located

46

47

 

 

 

 

 

 

’/\\
00°
0

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

8
//\\
O O

O
2
o 0
ok__/
0 O o

 

 

 

 

 

 

 

 

 

 

    
      
        
  
     

   

  
  
 

    

Ilt'nh g
E€a%.€g
ll vgio
Eﬁﬁgggogg
'iaLgé9seg
E g—o‘m‘ieaa
(Cir—"IIIt‘:'l:i'-5\.:.a
gagggge
..“\:" H8
gin“ as.
Mat‘ﬂa gnu
l‘
///
U—‘/ 1
§
‘ z.
oo . S
I 3%

 

T

l‘

Leﬂ View

     
 

  
      

0‘

Figure 3.]. Drawing of the Part-Shaped Cavity.

48
port (i.e., port A [1]), an end port (i.e., ports B [2] and/or C [3]), and an oﬁ‘-center port
(i.e., ports D [4] and/or E [5]). The aperture acts as a shunt admittance device to the
mold interior. In this thesis, ports with an external applicator will be referred to as
“active” ports.

During cavity characterization experiments and curing runs, all non-active ports
were sealed with a brass plate. To provide a resin barrier, all liquid molding experiments
used Teﬂon spacers between the cavity and external applicators at the active ports. An
insigniﬁcant amount of electromagnetic energy is absorbed by Teﬂon as its dielectric loss
is 0.0003 ('6’. In an industrial liquid molding process, the aperture should be ﬁlled with a
more rigid low-loss material such as reinforced polyimide.

The cavity has 74 diagnostic holes [9] for measuring temperature inside the
process material or E-ﬁeld strength at the wall. For the resin curing experiments, these
holes are either ﬁlled with silicone sealant or with glass capillary tubes with one sealed
end. The capillary tubes provide a barrier between the resin and the temperature probes.
The wall thickness of the tubes is 0.75 mm.

Conventional heating is achieved ﬁom ﬁve ﬂexible silicone heaters [10] afﬁxed to
the external surfaces of the cavity. The heaters are wired to deliver a total power of
1280W. When conventional heating is employed, the heater duty is controlled via a
Digi-sense proportional controller with a type K thermocouple attached to the cavity

exterior near port A.

49
A centrally located sprue [6] is placed on the top of the cavity for resin transfer.
Four vents [7] are located at the each of the bottom corners of the cavity to ensure

proper wetting of the ﬁber mats.

3.1.2. External Applicators

The external applicator, as shown in Figure 3 .2, is a single-mode cavity with one
open surface through which electromagnetic energy is fed to the part-shaped cavity.
Resonant conditions are attained in the external applicator by mechanically adjusting the
depth of the microwave coupling probe [2] and the height of the traveling short [1] to
match the impedance of the input radiation. For all experiments, the external applicators
were tuned by minimizing reﬂected power near a cavity length of 10.3 cm, the theoretical
TEou cavity length. Resonance in the external applicators is affected by the dielectric
properties of the material in the part-shaped cavity. Since the dielectric properties of the
process material are functions of temperature, the external applicators required slight
tuning during each run to maintain heating efﬁciency. A microwave excitation probe,
consisting of an aluminum inner conductor with a diameter of 0.95 cm and an outer brass
conductor with a diameter of 2.26 cm, can be adjusted ﬁom a depth of 0 to 6.37 cm. A
grounding contact ring (Instrument Specialties 97-154) is soldered to the outer
conductor to provide electrical contact with the external applicator walls. The external
applicator length is ﬁxed by a 7.6 cm x 6.4 cm shorting plate that is mechanically

adjusted. Electrical contact between the external applicator walls and the traveling short

 

50

(4)

ll

(1)/
il\( (5)/

 

 

 

(6) *

 

 

 

- - .\x;..\~\ :\.~. <\~\..\\\ 9... .\~.‘\§\\.‘\\¢\\
-

(3)

wmww 1

 

 

 

 

 

 

shafts, (5) grounding contacts, (6) type N coaxial connector, and (7) choke

lkgend: (1) traveling short, (2) inner conductor, (3) outer conductor, (4) threaded

 

Figure 3.2. Cross-section of the external applicator.

5 1
is attained by grounding contact strips (Instrument Specialties 97-139) soldered to the

perimeter of the short.

3.1.3. Experimental Setup and Microwave Circuitry

The external applicator conﬁguration dictates the type of wave propagation in the
process material. For instance, activating one port and sealing all other ports results in a
potential standing wave inside the cavity. Electromagnetic wave reﬂection at all sealed
ports in superposition with incident waves from the external applicator could create the
standing wave phenomenon. The circuitry and three standing wave conﬁgurations,
which were investigated, are illustrated in Figure 3.3.

During runs with two or more active ports, incident energy is directed to one
external applicator through an SPST electromechanical switch. All other external
applicators act as a matched termination. Therefore, the waves inside the cavity are
primarily incident waves. The cavity in this conﬁguration becomes a traveling wave
applicator. The curing demonstration utilized the traveling wave conﬁguration, as
shown in Figure 3.4. Cavity characterization experiments were performed for both
traveling wave and standing wave conﬁgurations.

A magnetron microwave generator (Opthos MPG 4) with a ﬁxed frequency of
2.45 GHz and a power of 120 W served as a high power microwave source for all
experiments. 50-ohm impedance coaxial transmission lines are used to transmit power
from the power source to the external applicator. The power source and the microwave

switch are protected from reﬂected energy by a circulator. Incident and reﬂected signals

52

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(1)
-"°
, L ............................... I.
(6) = ......... .o.-_-'.
[g] e
(2 (3) '
(4) J“)
E '1»
T© I
o (A)
(11)
U
(11): O@[ 302) (C)
n
(11)
0
@

 

 

 

 

 

(D)
I: 3..

 

Legend: (1) 120 W magnetron microwave generator, (2) directional coupler, (3) power sensor, (4)
circulator, (5) reﬂected power meter, (6) terminal resistor, (7) input power meter, (8) computer, (9)
Luxtron thermometry system, (10) part-shaped cavity with external applicator, (1 1) external applicator,
(12) part-sluped cavity, [—-] coaxial transmission lines, [ ----- ] silica temperature probes,

[- ----- ] computer data transmission lines, (A) port A heating conﬁguration, (C) port C heating
conﬁguration, and (D) port D heating conﬁguration.

 

 

 

Figure 3.3. Microwave Circuitry and Cavity Conﬁgurations for Standing Wave
Operation.

53

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54
ﬁ'om the Opthos generator and the cavity, respectively, are sampled by Narda coaxial
direction couplers with a 20 dB attenuation and measured using 3.5 W power meters
(HP 43 SB). Power meter data and temperature data are stored in a computer.

Liquid resin is transferred to the cavity via a pumping system comprised of a
Sherwood bronze gear pump, driven by a l-hp U.S. Unimount 125 electric motor. The
motor is capable of a rotation speed of 3475 RPM and is controlled with a Magnetek
GPD 403 variable frequency drive, which allows the user to control the pump speed at a

constant torque.

3.1.4. Thennometry System

Temperature measurements are sampled inside the part-shaped cavity using a
Fluoroptic thermometry system (Luxtron 750 and 75 5) with temperature probes
constructed of silica ﬁbers coated with Teﬂon. The E-ﬁeld is essentially unaﬁ‘ected by
the temperature probes, as both silica and Teﬂon are very low loss materials.

The thermometry unit contains a xenon ﬂashlamp that emits a blue-violet light,
which travels through the ﬁber optic probe body, at speciﬁed intervals. A manganese-
activated magnesium ﬂuorogerrnanate material, located in the probe tip, absorbs the
blue-violet light and emits a deep red ﬂuorescence for a given time period which is a
ﬁmction of temperature. The decay time of the red ﬂuorescence is measured and
correlated with the temperature at the probe tip. Data acquisition of the temperature is

achieved via an RS-232 interface to a personal computer.

55

3.1.5. Process Material

For the cavity characterization experiments, fully cured glass-reinforced polyester
plates are used as a dielectric media in the cavity. Eleven 31.8 cm x 8.9 cm x 0.32 cm
plates are stacked and heated in the cavity. The complex permittivity of glass-reinforced
polyester at 2.45 GHz and 25°C is 4.30 - j 0.049. The permittivity was measured by the
cavity perturbation technique using the TM, 12 mode in a cylindrical cavity with a
diameter of 15.24 cm.

For RTM applications, a common composite is a glass-reinforced epoxy system,
speciﬁcally diglycidyl ether of bisphenol A (DGEBA). A DGEBA resin, Epon 828 (Shell
Chemical Co.), cured with diaminodiphenyl sulfone (DDS) and reinforced with
continuous random E-glass ﬁbers (U niﬁlo U101), served as the composite system for the
curing demonstration. The chemical structures of difunctional DGEBA and
tetrafunctional DDS are given in Figure 3.5.

The crosslinked structure of an epoxy is formed by step-growth polymerization.
An epoxy resin, which typically has two or more epoxide functionalities, reacts with a
curative which is capable of opening the epoxide ring via nucleophilic substitution.
Typical curatives include amines and anhydrides. During polymerization of an epoxy
with an amine, for example, the predominant reaction consists of the amine opening the
epoxide ring and transferring a hydrogen to the oxygen of the epoxide ring.

Respectively, a primary or a secondary amine after this reaction becomes a secondary or

tertiary amine.

56

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57
3 .2. Cavity Characterization Methods

Achieving uniform and rapid heating rates for a given process material is one of
the main objectives in the development of the part-shaped cavity. Some of the dominant
process variables that inﬂuence heating patterns and rates include the following:

microwave power,

microwave frequency,

aperture width,

aperture orientation relative to the coupling probe,
aperture conﬁguration (e. g., single vs. double aperture),
number of external applicators,

location of external applicators,

resonant mode in the external applicator,

process material dielectric properties,

geometry of ﬁll material in the aperture, and
dielectric properties of ﬁll material.

Experimental efforts for the cavity characterization are focused on determining the effect
of aperture width and orientation on the E-ﬁeld pattern behavior and, hence, heating
patterns and rates. Additionally, the number and location of external applicators are
investigated, as explained in Section 3.1.3.

Qualitative heating pattems are determined by securing thermal paper to each
polyester plate. The thermal paper begins to indicate at 60°C and progressively darkens
at higher temperatures. This allows a ﬁngerprint of the heating zones to be taken during
a run.

The unsteady-state energy balance for a reactive process material in the mold is

given by:

58

mpg—=V-(kvr)+1>w+sm, [3.1]
where p is the material density [=] kg/m’,

C, is the material heat capacity =] J/kg-K,

T is the material temperature [=] K,

t is time [=] s,

k is the material thermal conductivity =] W/m-K,

PW is the rate of microwave power absorption [=] W/m3, and

$3,... is the reaction source term [=] W/m’.

At the beginning of a run, the thermal conduction temt, V-(kVT), can be neglected due
to the insigniﬁcant temperature gradients throughout the cavity volume. Additionally,
the reaction source term is zero since the polymer plates are non-reactive; therefore, the
initial heating rate, dT/dt, is directly proportional to the microwave power absorption:

ﬂ_1

dt .. pC, W. [3.2]

 

By recording temperature as a function of time at a ﬁxed location in the cavity, one can
determine an indication of microwave power absorption for the polyester plates as a

ﬁmction of a processing parameter, such as aperture width.

3.3. Glass Fiber-Reinforced DGEBA/DDS Processing

Three methods of curing composites are demonstrated in the part-shaped
applicator: conventional, microwave, and hybrid. With conventional curing, resistive
heaters afﬁxed to the cavity surface provide the thermal energy for processing. During
the microwave curing run, microwave radiation provides the energy for promoting

crosslinking. Hybrid curing combines conventional and microwave curing techniques.

59

The resin is prepared according to the manufacturer recommendation of 4.0 g of
DGEBA per 1.0 g of DDS, corresponding to a mixture with excess DGEBA. Epon 828
has an epoxide equivalent weight of 185 to 192, and the molecular weight of DDS is
248.3. Each molecule of DDS has four reactive sites that can react with one of the two
epoxide groups per molecule of DGEBA

Approximately 2000g of DGEBA is placed in an insulated 4-liter beaker,
agitated, and heated to 130°C on a hot plate. When the resin reaches 130°C, 500 g of
DDS is slowly added to the DGEBA, while maintaining the mixture temperature near
130°C. After the DDS is ﬁilly dissolved in DGEBA, the resulting mixture is degassed in
a vacuum oven at approximately 100°C for 15 minutes to minimize void formation
during processing.

As shown in Figure 3.4, the‘resin reservoir and the tubing is heated. The resin
temperature is maintained above 60°C to facilitate resin transfer; at lower temperatures,
the viscosity is too high for pumping.

To aid in part removal, the cavity is treated with Mono-Coat RPM 711 sealer and
Mono-Coat RPM 115 mold release agent (Chem-Trends, Inc). The cavity is charged
with 30 E-glass ﬁber mats and heated to 100°C. Five capillary tubes are placed inside
the cavity through the diagnostic holes. Four tubes are located in the center of the cavity
near port A (2 tubes), port B (1 tube), and port C (1 tube). The ﬁfth tube is located near
the mold wall. The resin is transferred to the cavity via the gear pump, powered by a 1-
hp electric motor controlled with a variable frequency drive. Upon ﬁlling the cavity with

resin, the vent valves are closed and pumping is terminated.

60
For each curing method, the process cycle consisted of cure and post-cure stages.
The manufacturer recommendation of a ﬁve hour cure cycle at 125°C and a one hour

post-cure cycle at 200°C is observed.

3.4. Mechanical Strength Tests

For comparison of part strength for each of the three processing methods, the
modulus of elasticity of samples was determined by the three-point bending test using
procedure A in accordance with ASTM 790. Figure 3.6 illustrates the approximate
locations of the samples that were removed from the cured composite. Ten samples with
a length of 76.2 mm and a width of 25.4 mm were sized with a D.R. Bennett, model PX,
table saw using a water-soluble oil coolant system to prevent any thermal effect during
sample preparation. A depth of approximately 3.18 mm was achieved using a Struers
Abramin polisher and a Leco BG-20 belt grinder equipped with a water cooling system.

The three-point bending tests were performed on an SFM 20 United Testing
System unit using a 1000-lb loadcell. The diameter of both the loading nose and the
supports is 6.35 mm. With a support span of 50.8 mm, samples were deﬂected at a rate

of 1.27 mm/min.

 

30-ply E-glass ﬁber mat stack

Random-orientation E-glass ﬁber mat

61

 

 

 

|<— 76.2mm —>|

 

38.0 m

T
i

 

|<———318mm

Depth = 3.18 mm

18 mm(x10)

Sample size

Approximate locations of samples extractions

Figure 3.6. Fiber Mat Orientation and Sample Locations for Curing Experiments in Part-Shaped Cavity.

62
Assuming that the glass-reinforced epoxy specimens behave as a linear elastic

material at small deﬂections, the modulus of elasticity is given by:

L’m
" 4bd3 ’ [3'3]
where E; = modulus of elasticity [=] N/mz,
L = support Span [=] m,
m = tangent of the initial linear portion of the load-deﬂection curve [=] N/m,
b = width of specimen [=] m, and
d = depth of specimen [=] m.

Equation [3.3] is only valid at small deﬂections. At large deﬂections, composites usually
exhibit nonlinear stress-strain behavior". Additionally, ﬁiction forces at the supports

become signiﬁcant.

CHAPTER FOUR

RESULTS AND DISCUSSION

4.1. Cavity Characterization Results

4.1.1. Temperature Proﬁles and Heating Pattems

As described in Chapter 3, the E-ﬁeld is characterized inside the part-shaped
cavity by studying heating rates and heating patterns using the E-glass reinforced
polyester plates. Figures 4.1 through 4.3 illustrate the temperature proﬁles for
microwave heating of the eleven plates using the standing wave cavity conﬁgurations
shown in Figure 3.3. The approximate location of each temperature probe is shown in
the ﬁgures; each probe is located in the center of the middle plate.

As is evident in Figures 4.1 through 4.3, heating rates diminish at distances
farther away from the microwave source (i.e., the active port). This observation in
Figures 4.1 and 4.2 can be attributed to energy absorption. Through lossy material, the
E-ﬁeld intensity decreases in the direction of propagation due to the conversion of
electromagnetic energy to thermal energy.

Temperature probe 1 in Figure 4.3 detects no rise in temperature over the 8.5
minute experiment. This is not due to microwave energy attenuation, but rather the
electromagnetic ﬁelds are not coupling with the entire length of the cavity. However,
temperature probes 2 and 3 indicate that the E-ﬁeld couples very well with the polyester

plates near port D.

63

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A map of the heating patterns throughout the cavity can be determined by placing
temperature-sensitive thermal paper on each of the eleven plates. Figure 4.4 shows
images from the center plate for heating runs using one external applicator at ports A, C,
or D. The dielectric properties of the polyester plates allow propagation of the observed
heating mode. The distinct lobes on the images from heating the polyester plates
demonstrate the standing wave phenomenon.

Diminishing E-ﬁeld magnitude and heating intensity are particularly evident for
the runs with the external applicator at port C. The heating intensity is the greatest at
the center of the cavity near the source and the least near the top and bottom of the
cavity. Thermal paper on the top and bottom plates for most runs provided no indication
of heating above 60°C.

For the experimental system, greater heating uniformity can be realized by
admitting electromagnetic energy from more than one port: namely ports A, B, and C.
As explained in Chapter 3, heating with this conﬁguration results in a traveling wave. A
traveling wave is an incident wave, and a standing wave is the result of the superposition
of an incident wave and a reﬂected wave. One would expect, for a given power source,
the heating rates to be lower for the traveling wave operation. This is evident in
comparing Figure 4.5 with Figure 4.1 and Figure 4.6 with Figure 4.2.

Thermal paper images from heating the polyester plates in the traveling wave
operation, as shown in Figure 4.7, exhibit a more uniform heating pattern relative to the
standing wave operation; the lobes are not as sharply deﬁned for the former. Slight

reﬂection inside the cavity system ﬁom the aperture plates and from the non-active

68

 

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72
external applicators creates the lobed heating patterns when using more than one
external applicator.
The image from heating with the external applicator at port C illustrates the
terminating nature of the non-active external applicator. The high temperature indication
ceases where port A is located, mid-way across the plate. The incident energy from port

C couples with the external applicator at port A and is directed to a dummy load.

4.1.2. Aperture Effects

As explained in Section 3.2, the initial microwave heating rate of a non-reactive,
dielectric material is directly proportional to the microwave power absorption. A
straight-line behavior near the initiation of a heating experiment is evident in Figures 4.1
through 4.3. As the heating progresses, however, the dielectric properties change with
temperature, heat losses at the mold surfaces increase, and thermal conduction becomes
signiﬁcant due to temperature gradients. These phenomena create the non-linear
temperature behavior at longer times.

Part of the cavity characterization study focused on the effect of aperture
orientation, conﬁguration, and width on the heating rates and patterns. Aperture
orientation refers to the orientation of the opening in the aperture plate with respect to
the excitation probe. The three conﬁgurations investigated were single symmetric, single
asymmetric, and double symmetric apertures. The investigated aperture characteristics

are illustrated in Figure 4.8.

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The desired resonant mode in the external applicator using a double symmetric
aperture or an aperture perpendicular to the probe is difﬁcult to excite. The resonance
curve for these cases had a very narrow bandwidth. As a result, very frequency tuning
during heating was required. Single apertures parallel to the excitation probe provide a
more stable conﬁguration for the desired resonance mode.

A single asymmetric aperture at port D proved to diminish the intense heating
pattern characteristic of the off-center port. By locating the aperture closer to the center
of the cavity, the E-ﬁeld could be dissipated over more material.

For single symmetric apertures oriented parallel with the excitation probe, the
initial heating rates are plotted versus aperture width in Figure 4.9 through 4.11 to
identify the effect of aperture width on the microwave power absorption. At each type
of port (i.e., center, off-center, or end), the initial heating rate increases with aperture
width at small apertures. Above an aperture width of 5.0 cm at port A, the near-ﬁeld
heating appears to become more intense.

The thermal paper images for three experiments utilizing different aperture
widths demonstrate that microwave heating patterns are essentially independent of
aperture width. As shown in Figure 4.4, characteristic of each run using the external
applicator at port A is a near-ﬁeld heating effect directly below the port. On either side
of the central heating lobe, three lobes appear, diminishing in intensity away from the
central lobe. The width of the aperture controls the admitted E-ﬁeld and, hence, the

heating rates. As outlined in Table 4.1, the locations of the E-ﬁeld lobes are

75

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Table 4.1. Thermal Paper Images for Various Aperture Widths.

 

Thermal Paper Image

 

 

 

 

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79
approximately the same, within measurement uncertainty. This observation indicates

that the electromagnetic mode in the cavity is not affected by the aperture width.

4.2. Theoretical Calculations

4.2.1. Electromagnetic Modes in the External Applicators

A resonant cavity is described in Chapter 2 as being a system, enclosed by conductive
boundaries, that can store or transfer electromagnetic energy. Because the external applicator
is not completely enclosed by conductive boundaries, strictly speaking, it is not a resonant
cavity. The presence of process material directly below the aperture creates a partially
reﬂective surface. However, analysis of the theoretical cavity modes provides instructive
information on the electromagnetic behavior inside the cavity.

Experimentally, two modes can be excited in the external applicators: one at lengths
ranging from 7.0 to 9.6 cm and one ranging ﬁ'om 18.7 to 19.5 cm. As is evident (tom the
mode chart shown in Figure 4.12 for a resonant cavity with the cross-sectional dimensions of
the external applicator, 6.4 cm x 7.6 cm, three modes should be excited over the range of the
extemal applicator length at 2.45 GHz. Excitation of the TEou,'I'Em, and TEorz modes is
possible at 10.3 cm, 20.8 cm, and 20.6 cm, respectively. The heating characterization runs
and curing experiments all used the observed mode near the theoretical TEou mode.

The theoretical TEou mode resonates with an E-ﬁeld component in the x-
direction only and zero E-ﬁeld at z=0 and z=c, where c is the length of the cavity in the
z-direction. In the external applicator, the excitation probe and the aperture length are

oriented parallel with the x-direction, as shown in Figure 4.13. At z=c, the E-ﬁeld in the

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82
external applicator is zero due to the conductive boundary; however, at z=0, admittance
of the E-ﬁeld to the cavity would not be possible if the E-ﬁeld were zero. Therefore, the
E-ﬁeld is zero on the conductive aperture plate and non-zero at the aperture. The
theoretical E-ﬁeld vectors shown in Figure 4.13 do not strictly represent the actual B-
ﬁeld magnitudes. Due to the aperture, deviation from the theoretical calculations would

be most evident near 2 = 0.

4.2.2. Electromagnetic MOdes in the Cavity

Propagation of electromagnetic energy in the cavity is deﬁned by waveguide
theory, as the cavity is a dielectric-ﬁlled waveguide. In a waveguide, electromagnetic
modes propagate above the‘cutoﬂ‘ frequency; below the cutoff frequency, modes are
evanescent, or nonpropagating.

Cutoﬂ‘ frequency calculations, theoretical E-ﬁeld calculations, thermal paper
images, and heat transfer analysis are used in determining the propagating mode in the
cavity. Using the cross-sectional dimensions of the cavity, 4.2 cm x 8.9 cm, Figure 4.14
illustrates the cutoff frequencies for a range of dielectric constants typical of polymeric
systems. Over this range, ﬁve modes could potentially propagate through the cavity at
2.45 GHz: TEm, T520, TEM, TE“ or TM", and TE” or TM21. The glass-reinforced
polyester used for heating characterization has a complex permittivity of 4.3 - j0.049.
With a dielectric constant of 4.3, any of the aforementioned modes could propagate

through the cavity at 2.45 GHz. ’

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The E-ﬁeld intensity patterns for the seven lowest order modes for the cavity are
calculated and plotted to theoretically determine the areas of heating intensity. The B-
ﬁeld strength plots, shown in Figure 4.15, are x-z plots with attenuation in the negative x
direction. The dark regions indicate areas of higher E-ﬁeld intensity. For comparison, a
thermal paper image is included in Figure 4.15. As is evident from comparison of the
heating pattern to the E-ﬁeld strengths, TEzo, TE", TE21, and TM21 are not the modes of
propagation observed experimentally.

The plots in Figure 4.15 show the Euclidean norm of the E-ﬁeld as a ﬂinction of
space in an x-z plane. Vector plots of the TB“), TE, 1, and TM" modes, illustrated in
Figure 4.16 for an x-y plane, provide more detail about the direction of the E-ﬁeld.

Assuming that a mode similar to the T1301: mode is resonating in the external
applicator, only an x-component of the E-ﬁeld exists in the external applicator. This
suggests that the TEM mode is propagating in the cavity since the TEM E-ﬁeld has an x-
component only. As shown in Figure 4.16, a y-component only exists for the TB“.
mode. The TM" mode has an x—, y-, and z-component. Additionally, the z-location of
the E-ﬁeld nodes for the TM“ mode is a function of y. Experimentally, the thermal
paper from each plate indicated identical positions of the intense heating regions.

The theoretical E-ﬁeld magnitude for the T1301 mode is constant in the x-
direction. However, from the thermal paper images, the heating patterns indicate high
temperatures at the center of the cavity and diminishing temperatures near the cavity wall

in both the x- and y—directions. To elucidate this observed phenomenon with the heating

85

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Figure 4.15. Theoretical E—ﬁeld Strength Plots and Thermal Paper Image For the Mid-
Plane (i.e., y = 0.5 b) of a 4.2 cm x 8.9 cm Waveguide ﬁlled With Dielectric Material (8‘
= 4.30-j0.049).

86

 

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87
patterns and demonstrate the virtues of microwave heating, a one-dimensional ﬁnite

diﬂ‘erence model was developed based on the energy balance:

pop-:3: wing”, [4.1]
where p is the material density ‘ [=] kg/m”,

C, is the material heat capacity ' =] J/kg-K,

T is the material temperature . =] K,

t is time [=] s,

k is the material thermal conductivity =] W/m-K, and

PW is the rate of microwave power absorption [=] W/m3.

This model assumes an isotropic thermal conductivity and no heat of reaction. The
physical system is an 8.9 cm slab of polymer between two 1.25 cm slabs of brass, which
are the dimensions of the cavity and brass walls in the x-direction. The details of the
boundary conditions and the FORTRAN code are outlined in Appendices F and G,
respectively.

The half-wave E-ﬁeld variation in the y-direction for the TEM mode is
responsible for the strong center heating and absence of heating near the cavity wall at
y=0 cm and y=4.2 cm. The E-ﬁeld in the x-direction, however, is invariant with position.
The relative differences in heating intensity in the x-direction can be explained using the
heat transfer model. A constant microwave power input of O. l 15 W/cm3 was selected
based on matching the simulated rate of temperature rise with that of experiments with
an external applicator at port C. Using a free-convective boundary condition of 5.0
W/mz-K, the model predicts the temperature proﬁles shown in Figure 4.17. At short
times, the temperature proﬁle is relatively ﬂat, due to the insigniﬁcant temperature

gradients. During this regime, the heating rate is equal to the constant? microwave power

88

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89
absorption. When the temperature of the polymer increases, energy is transferred
outward toward the brass walls and into the surroundings. When conduction becomes
signiﬁcant, the low thermal conductivity of the polymer creates a resistance to heat
transfer and, eventually, a temperature gradient. A constant E-ﬁeld does not necessarily

translate into constant temperature proﬁles.

4.3. Processing of Glass-Reinforced DGEBA/DDS

4.3.1. Complex Permittivity

The complex permittivity of DGEBA/DDS as a function of E-glass ﬁber content
at 25°C was measured by the cavity perturbation technique using the Mn mode at
2.45 GHz in a cylindrical cavity with a diameter of 15.24 cm‘3”. The Teﬂon cylinder
used to introduce the epoxy-ﬁber sample into the cavity has an inner diameter of 1.12 cm
and a length of 2.60 cm. Resin samples were prepared with the 3:4 amine to epoxy ratio.
Epoxy-ﬁber sample volumes were calculated based on the densities of each component

in the mixture. Assuming volume additivity in a two-component system, the sample

 

volume is given by
M ' 1 Xrbe)
Vmix = mm = Mmix[ 1‘X. r + i f » [4-2]
pmix presin ( {be ) Plies:

where M.;,. is the mass of the ﬁbers and resin, 15..., is the ﬁber mass fraction, p...“ is 1.09
g/cm3, pm. is 2.54 g/cm3. Air pockets in the epoxy-fiber mixture at ﬁber mass fractions

greater than 0.5 introduce uncertainty in the measurement of sample volume.

90 .

The complex permittivity of the E-glass ﬁbers in resin appears to behave linearly
with ﬁber mass fraction for mass ﬁ’actions below 0.5 as is evident in Figure 4.18. At
room temperature and zero extent of cure, DGEBA/DDS in a 3 :4 amine to epoxy ratio
has a larger dielectric loss and a smaller dielectric constant relative to the ﬁbers. For
instance, a ﬁber mass fraction of 0.25 increases the dielectric constant of the unloaded
DGEBA/DDS sample by 15% and reduces the loss by 8%. The uncured DGEBA/DDS

samples have a complex permittivity of 3.50 - j0.23.

4.3.2. Curing Results

By conventional, microwave, and hybrid heating, three E-glass-reinforced
DGEBA/DDS composites are cured. Temperatures during cure are measured at
locations near ports A, B, and C in the center of the part and at a wall location near the
cavity center. Temperatures at the select locations inside the cavity were used to provide
a basis for manual control of the microwave power input for microwave and hybrid
heating. Figures 4.19 through 4.21 illustrate the various temperature proﬁles for the
three types of heating. A comparison of the average axial-to-wall temperature gradients,
ﬁom using three curing methods in the part-shaped cavity, is shown in Figure 4.22.

Over one hour is required for the composite to achieve the cure temperature,
125°C, using conventional heating. Microwave and hybrid heating techniques heated the
composite to 125°C in 10 minutes or less, corresponding to more than an 85% reduction

in the time required to attain the cure temperature.

91

 

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Figure 4.18. Complex Permittivity of Uncured DGEBA/DDS at 25°C as a Function of
Fiber Content.

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96

At the beginning of the cure cycle, the relatively large dielectric loss of the
composite provided excellent coupling of the electromagnetic energy with the process
material molecules. As the reaction progressed, the loss decreased due to a decrease in
the molecular mobility (i.e., crosslinking). A thermal boost from the heaters on the
cavity was required to attain the post-cure temperature during the microwave heating
run. A microwave source larger than 120 W is required to achieve the 200°C post-cure
temperature due to the decrease in the microwave power absorption combined with heat
losses to the external applicators and the surroundings.

Microwave heating resulted in the largest average axial-to-wall temperature
gradient throughout the cure and post-cure cycles. As explained in Section 4.2.2., this
gradient develops due to the heat loss at the cavity boundaries coupled with the low
thermal conductivity of the composite. Gradients can also develop as a result of the type
of electromagnetic mode propagation inside the cavity.

Hybrid heating provided the most uniform temperature proﬁles for the three
processing methods. Initially, the conventional heating technique results in a
temperature gradient between the center and the mold wall is formed. The heat transfer
model can be applied as a semi-quantitative tool in the temperature proﬁle analysis.
Table 4.2 outlines the parameters used in the model for each process method.

The simulation of conventional heating, as shown in Figure 4.23, clariﬁes the
observed initial temperature gradient. Because of the low thermal conductivity of the

polymer, the temperature of the midplane lags the rising temperature of the brass wall.

97

Table 4. 2. Heat Transfer Model Parameters for Conventional, Microwave, and Hybrid
Heating.

 

 

 

 

 

 

 

Parameter Conventional Microwave Hybrid
Heating Heating Heating

Physical Properties

Polymer

Thermal conductivity 2.0 2.0 2.0

(W/mz-K)

DeiLsity(kg/m3) 1100 1100 1100

Heat capacity (J/kg-K) 2000 2000 2000

Brass

Thermalconductivity 110 110 110

(W/mz-K)

Density (kg/m3) 3500 8500 8500

Heat capacity (J/kg-K) 380 380 380

Physical System

Convective heat transfer 1000 5 .0 1000

coeﬂicient

(W/mz-K)

Polymer thickness (cm) 8.9 8.9 8.9

Brass thickness (cm) 1.25 1.25 1.25

Initial system temperature 25.0 25.0 25.0

(°C)

Ambient temperature 125.0 25.0 125.0

(°C)

Microwave power input 0.0 2.0 2.0
Mlm’)

Finite Diﬁ'erence

X-interval (cm) 0.178 0.178 0.178

Tithe-step (s) 0.10 0.10 0.10

 

98

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99
Conductive heat transfer in the material near the wall is greater than that near the
midplane.

A simulation of microwave heating with a constant microwave power absorption
of 2.0 W/cm3 is performed. This microwave heating simulation employs a greater power
absorption than the previous one in Section 4.2.2. Comparing Figures 4.24 and 4.17
shows the same type of temperature proﬁle: maximum in the center and minimum at the
walls. For the TEM mode, greater microwave power results in faster heating times and
more uniform temperatures. Energy input due to microwave power becomes the
dominant energy transfer mechanism, and the effects of conductive heat transfer are only
seen near the mold walls. A

As observed experimentally, minimizing temperature gradients and reducing the
heating time are feasible by combining microwave and conventional heating. The same
behavior is demonstrated theoretically, as shown in Figure 4.25. With hybrid heating,
the large outward ﬂux of thermal energy near the mold wall associated with microwave
heating is offset by the large inward ﬂux of energy near the mold walls associated with

conventional heating.
4.4. Mechanical Strength Tests

Ten test samples ﬁ'om each cured composite were prepared according to the
description given in Section 3.4. During each test, the normal load on the sample surface
and the deﬂection of the sample midpoint are recorded. An example of the typical load-

deﬂection relationship is shown in Figure 4.26. As is evident from this plot, the sample

100

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103
deﬂection behaves linearly with the applied normal load, for this particular sample, to
approximately 12 mm. The deﬂection range from 0 to 12 mm is deﬁned as the region of
linear elastic behavior. This slope is used in the calculation of the modulus of elasticity
as given in Equation 3.3.

The moduli of elasticity for samples cured by conventional, microwave, and
hybrid heating are 6.56 :l: 0.30 GPa, 6.68 i 0.26 GPa, and 6.47 :l: 0.22GPa, respectively.
Only the samples in Table 4.3 that are designated as ‘valid data’ are used in the
calculation of these averages. The invalid samples contained three or more visible
microvoids; voids can affect data accuracy. By observing the manufacturer
recommendation for the cure cycle, each of the three processing methods have the

capability of producing composites with comparable moduli of elasticity.

104

Table 4. 3. Modulus of Elasticity Data for E-glass-Reinforced DGEBA/DDS.

 

Sample Width Length Depth Number of Valid Modulus of

 

(mm) (mm) (mm) Visible Data Elasticity
Microvoids (GPa)
Conventional Cure
1C 2.53 7.62 0.324 0 Yes 6.22
2C 2.53 7.60 0.303 1 Yes 6.84
3C 2.52 7.62 0.354 1 Yes 6.23
4C 2.53 7.62 0.309 11 No 6.64
5C 2.53 7.61 0.326 50 No 6.45
6C 2.52 7.61 0.283 50 No 6.54
7C 2.53 7.62 0.336 20 No 6.30
8C 2.53 7.62 0.358 0 Yes 6.50
9C 2.53 7.62 0.329 0 Yes 6.89
10C 2.53 7.63 0.332 0 Yes 6.69

Average: 6.56 :l: 0.30

 

Microwave C are

 

1M 2.54 7.62 0.336 1 Yes 6.84
2M 2.54 7.61 0.277 0 Yes 6.69
3M 2.53 7.62 0.324 0 Yes 6.84
4M 2.54 7.51 0.324 0 Yes 6.86
5M 2.54 7.62 0.330 0 Yes 6.9]
6M 2.54 7.62 0.381 50 No 5.96
7M 2.53 7.59 0.326 15 No 6.38
8M 2.50 7.62 0.310 2 Yes 6.26
9M 2.53 7.62 0.290 0 Yes 6.39
10M 2.54 7.62 0.329 3 No 6.49
Avergge: 6.68 :l: 0.26
Hybrid Cure
1H 2.46 7.60 0.286 0 Yes 6.75
2H 2.54 7.64 0.329 0 Yes 6.64
3H 2.53 7.63 0.330 0 Yes 6.26
4H 2.52 7.59 0.343 0 Yes 6.62
5H 2.53 7.62 0.336 25 No 6.36
6H 2.53 7.63 0.307 25 No 6.16
7H 1.88 7.61 0.338 0 Yes 6.54
8H 2.19 7.61 0.333 0 Yes 6.31
9H 1.71 7 60 0.362 0 Yes 6.16

10H 2.55 7.62 0.331 1 (large void) No 6.09
Average: 6.47 :l: 0.22

CHAPTER FIVE

CONCLUSIONS
5.1. Part-Shaped Cavity Processing Capabilities and Limitations

The experimental part-shaped cavity has demonstrated:

0 more than an 85% reduction in the time required to heat the center of a thick
cross-section part to the cure temperature, which translates to a potential
reduction in the cycle time, and

0 the ability to produce composite parts by conventional, microwave, and
hybrid heating with comparable moduli of elasticity.

The geometry of the experimental cavity is a simple, parallelopipedic shape,
permitting the propagating electromagnetic mode. In order for microwave energy to
couple with the process material, the geometry of the mold must be able to support an
electromagnetic mode. Additionally, the process material must have dielectric properties
that allow mode propagation through the mold cavity. Knowledge of 8' over the cure

cycle is crucial, as the dielectric properties of polymers signiﬁcantly during cure.
5.2. Project Status

To date, this project has investigated the ability of the part-shaped applicator to
heat dielectric materials (e. g., fully cured glass-reinforced polyester) and to cure E-glass-
reinforced DGEBA/DDS. The developments for this project have demonstrated the
following:

1. a fundamental comprehension of the electromagnetic behavior in the external
applicator and the cavity:

105

 

106

A mode similar to the TEM; mode resonates in the external applicator, as
determined by the theoretical cavity length.

The pseudo-TEM; mode is a stable mode when using a single aperture oriented
parallel with the coupling probe, and hence the E-ﬁeld, in the external applicator.

The magnitude of the admitted E-ﬁeld to the process material can be controlled
by the aperture width.

The TEM mode propagates in the cavity, as determined by thermal paper images,
heat transfer analysis, and electromagnetic analysis of the heating intensity
locations.

. the ability to cure a glass-reinforced DGEBA/DDS RTM composite by three
processing methods, conventional, microwave, and hybrid heating, with the
following characteristics:

thick cross-sections: 8.9 cm x 4.2 cm (3.5 in. x 1.6 in.)

comparable moduli of elasticity: 6.56 :1: 0.30 GPa, 6.68 :l: 0.26 GPa, and 6.47 i:
0.22 GPa for conventional, microwave, and hybrid heating, respectively

. a theoretical explanation, based on a heat transfer model, of the observed distinctions
between the three heating techniques.

Conventional heating: Because of the low thermal conductivity of the
polymeric process material, the temperature of the midplane lags the rising
temperature of the mold wall. Conductive heat transfer in the material near the
wall is greater than that near the nridplane.

Microwave heating: When the temperature of the polymer increases, a ﬂux of
energy is directed to the mold walls and into the surroundings. When conduction
becomes signiﬁcant, the low thermal conductivity of the polymer creates a
resistance to heat transfer and, eventually, a temperature gradient. The center
temperature is greater than the wall temperature due to the resistance of heat
transfer through the process material.

107

0 Hybrid heating: The outward ﬂux of thermal energy near the mold wall
associated with microwave heating alone is offset by the large inward ﬂux of
energy near the mold walls associated with conventional heating. Hybrid heating
provides the most uniform heating.

4. the dielectric characterization of the composite system as a ﬁmction of glass content
is determined.

Chapter six outlines suggestions for further development of the part-shaped applicator

technology.

CHAPTER SIX

RECOMMENDATIONS

The advantages of hybrid heating in the part-shaped applicator have been
demonstrated experimentally and theoretically. To ﬁJlly realiZe the capabilities of this
processing technique, however, methods of cycle time reduction should be investigated.
Additionally, hardware modiﬁcations are required to improve coupling eﬁiciency from

the external applicator to the process material. Some hardware modiﬁcations have the

 

potential to improve the heating uniformity as well. The remainder of this chapter
presents concrete suggestions on achieving lower cycle times and attaining more efﬁcient
coupling and greater heating uniformity.

Because of material costs and mold and material preparation time, optimization
should ﬁrst be explored theoretically using the unsteady-state energy balance. Modeling
may also provide additional insights with respect to the processing methods:
conventional, microwave, and hybrid heating. Processing schedules should be
determined to minimize cycle time while avoiding temperature excursions from the
exothermic heat of reaction. Heat transfer, considering the heat of reaction and

microwave source term, is deﬁned as:

6T
pep-5t— = V—(kVT) +(-r,)(-AH,,,,) + PW, [6.1]
where p is the material density [=] kg/m’,
C, is the material heat capacity [=] J/kg-K,
T is the material temperature [=] K,
t is time =] s,

k is the material thermal conductivity [=] W/m-K,
108

l 09

«A is the rate of reaction [=] kg/m3-s,

-AHR,. is the heat of reaction [=] J/kg, and

PW is the rate of microwave power absOrption [=] W/m3.

For a non-isothermal, reacting system, equation [6.1] is a non-linear equation. The
microwave power absorption is a function of the dielectric loss, which is strongly
dependent on the polymer fractional conversion and temperature. The power absorption
also depends on the E-ﬁeld strength. As the dielectric properties change, the E-ﬁeld is
aﬂ‘ected. The physical properties and reaction rate change with conversion and
temperature. Solving equation [6.1] requires the knowledge of the E-ﬁeld magnitude,
the behavior of the dielectric loss and physical properties as a function of temperature
and conversion, and a model of the reaction kinetics and microwave power absorption.
These constitutive relationships should be developed, and the optimum cycle time for
conventional, microwave, and hybrid heating should be determined.

Aside from thermal runaway due to the heat of reaction, another limitation in the
reduction of the process cycle time is premature Vitriﬁcation. As discussed in Section
1.2.2.2, Boey and Yue" demonstrate the importance of engineering a proper microwave
power cycle. Curing DGEBA/DDS on glass substrates for several microwave power
levels over a range of radiation times, they demonstrated that the shear strength is highly
dependent on the processing cycle. For the microwave powers investigated, the
maximum shear strength increased with power. At higher power levels, longer
irradiation times resulted in a sharp drop of the shear strength due to premature

vitriﬁcation. Thermosetting polymers must undergo gelation before vitriﬁcation to

realize optimum physical properties. Thermal runaway is more likely than premature

l 10
vitriﬁcation when processing thermosets with thick cross-sections. However, physical
properties should be tested for each run with the part-shaped cavity to verify this
assumption.

Implementing a larger microwave power source will be required for the optimum
cure cycle. The post-cure temperature of 200°C for the glass-reinforced DGEBA/DDS
system could not be attained for the run with microwave heating with the 120W power
source. This could partially be attributed to the decrease in the dielectric loss with
conversion, as excellent coupling was realized between microwave energy and resin with
low conversion. At greater ﬁ'actional conversions, the composite temperatures became
less sensitive to microwave power input. A larger microwave power source would
provide faster control of energy input for the low loss material.

With the current laboratory setup, the external applicators are oriented to excite
the TEol mode in the cavity. The applicators should be rotated 90° to excite the
dominant mode, TE"). For the rectangular waveguide, greater penetration depths and
smaller wavelengths are feasible with lower order modes. Figure 4.15 illustrates the
smaller wavelength of the TE") mode relative to the TEM mode. By reducing the
distance between regions of high E—ﬁeld intensity, the axial temperature gradients can be
reduced. Using a waveguide applicator, Outifa et al.88 discovered that exciting higher
order modes resulted in an uncontrollable energy distribution of hybrid modes.

To date, the development of the part-shaped cavity boasts great potential for
cycle time reduction and improved heating uniformity in industrial applications.

However, the experiments have been performed in a parallelopipedic mold. The next

l l 1
phase of the development should be focused on applying the part-shaped applicator
concept to a commercial part with a degree of complexity (e. g., curvature, varying
thickness, etc.). Issues of E-ﬁeld propagation, such as internal reﬂection and hot spots,
in unconventional waveguides should be investigated.

By performing a preliminary economic estimate, Akyel and Bilgen” determined
that the product costs of microwave curing of polymers and vulcanization of rubber
could be reduced by 6 to 23%, depending on the production rate. A more detailed
economic evaluation of the part-shaped applicator process should be performed for
comparison to the conventional process. [Because of the external applicators and the
microwave circuitry, microwave tooling will require a greater initial capital investment.
However, a smaller total product cost per unit for the part-shaped applicatortechnology

may be attainable due to the feasible increased production rate.

 

APPENDICES

APPENDIX A

VECTOR WAVE EQUATION DEVELOPMENT

Maxwell’s equations in complex phasor form are given by
V x E(r) = —jcoB(‘r‘) [A. 1]

V x 11(1) = 1(1) + jcoD(r) [A2]
V-D(1’) = p('f) [A3]
V . 1‘36) = 0 [A4]
where E = electric ﬁeld strength [=] volt/m,

B = magnetic ﬂux density [=] weber/mz,

H = magnetic ﬁeld intensity [=] ampere/m,

I = volume current density [=]ampere/m2,

D = electric ﬂux density [=] coulomb/m2,

p = volume charge density [=] coulomb/m3,

j = imaginary number, Jr] , and
a) = angular frequency (=21rf) [=] radians/s.

In complex phasor form, a vector is represented as
A(f,t) -_- Re[A(r)e’“]. [A.5]
In this development, we will assume that the system is source-free.

p= 1 =0 [A6]

We will also assume that the system is composed of simple, homogeneous material.

Therefore, the electric and magnetic ﬂux density vectors can be expressed as
15(r) = e'E('r‘) [A7]

112

' , pprldﬁh'm‘m

 

1 13
136) = uoﬁfr‘) [A3]

where e' = complex permittivity (8,,[8' -— je"]) [=] farad/m,
e' = dielectric constant,
a" = dielectric loss,
a. = permittivity of free space, 8.85 x 10'12 farad/m, and
u. = permeability of free space, 1.26 x 10“ henry/m.

Combining Equations [A. 1] through [A.8] yields the following:

V x F: = -jmuoH [A.9]

V x H = jroe'E [A. 10]
V-(a’E) = o [All]
V-(uoH) = o [A.12]

For isotropic media,

V-E=0and [A.l3]
V-H = 0. [AM]
Curling equation [A.9] gives

VxVsz-jmroxH [A.lS]

which when combined with equation [A. 10] gives
V x V x E = -j03uo(j(oe'E) = m’e'uoE. [A.16]
Using a vector identity, [A.16] can be rewritten as
VxVxE=V(V-E)—V2E=o2e‘p,l‘3. [A.17]

Combining [A. 13] with [A.17] yields

 

 

 

V’E + o’e'oofa = o [A. 18]

 

1 14
Similar step can be taken to express the magnetic ﬁeld intensity in the vector wave

equation form. For example, curling equation [A.10] gives
VxVxH=jme'VxE. [A.19]

Substituting equation [A 9] results in

V x V x 1‘1 = jaie‘(—jcouoH) [A20]
which can be reexpressed as
vaxﬁ -.- V(V-H)—V2H =o2e‘ooi‘i. [A.21]

Using equation [AM] in equation [A.21] gives

 

 

 

V213 +o’e‘nor‘i = 0. [A22]

 

Equations [A.18] and [A22] are the vector wave equations for the electric ﬁeld intensity
and magnetic ﬁeld intensity, respectively. These equations are second order,
homogeneous, partial differential equations. The solution is outlined in the following

section.

 

APPENDIX B

SOLUTION TO VECTOR WAVE EQUATIONS

One can reduce equations [A18] and [A22] to
V2E+k2E = o [13.1]
V’H + k’H = o [3.2]

by letting k, the complex wavenumber, be represented as follows

 

k = co‘leo(e' — je")[l0 . [B3]
A scalar counterpart to equations [B. l] and [B2] can be written as
V"? + k2‘1’ = 0. [B4]

For the TE modes in Cartesian coordinates”, the E-ﬁeld and H-ﬁeld components are:

 

 

 

3. .22: H, = . 1 6;?
6" JmlJ'o 6x62
Eyﬁ. Hyz} 932: [35]
6x Jmllo ayal
Ez=0 Hz: ,1 (Ly—2+“)?
J‘Wo 02

Therefore, a solution to [3.4] can indirectly give a solution to the E-ﬁeld and H-ﬁeld
components.

Because of linearity,
‘I’(x,y,z) = X(x)Y(y)Z(z) [B6]

115

116

Expanding [B.4] gives

2
V"I’+k"1’=:‘:l+:;f+:\f+k2‘1’=0. [B.7]

 

Substituting [B.6] into [3.7] and dividing by XYZ yields

1d’x ld’Y 1d’Z

de2+Ydy2+Zdzz+k2:O' [B.8]

 

The wavenumber can be represented as a linear combination of contributions ﬁ'om each
Cartesian direction as
it2 = k,’ + it,2 + it}. [3.9]

Therefore, equation [B.8], aﬁer expanding and collecting terms, is

2 2 2
arses)«aides—fez)=0.
y 2

 

 

Setting each of these three terms equal to zero would be a solution.

 

 

2

d),(.~lt,,’x=o [3.11]
d’Y 2

dy’ +ky Y=0 [B.12]
d’Z 2

y+kzz=0 [3.13]

The solutions to equations [B.11] through [B. 13] depends on the physical system. For a

positive 2 traveling wave in a waveguide with conductive boundaries, Equation [B.6]

becomes

‘1’ = c, eos(k,x)eos(lt,y)exp(-jrt,z) [3.14]

 

117

where CI is a constant.

According to equation [3.5],

3, = c,k, cos(k,x)sin(k,y)exp(-jk,z) [3.15]
E, = —c,k, sin(k,x)cos(k,y)exp(-jk,z) [3.16]
E, = o [3.17]

For a conductive waveguide with an x-length of a and a y-length of b, the boundary

condition of zero tangential E-ﬁeld results in

k, = (1'?) and [3.18]
k, = (%j [3.19]

Therefore, according to equation [3.3] and [3.9],

 

m’souo(e' - je") = (L11): +0sz + k: or [3.20]
k, = J03280|10(8' — je") —(E;—n)2 -613): . [3.21]

Equation [3.21] is of the form JR + jI , which can be separated into real and imaginary

components as shown below

 

 

,/R+j1= %J(R2 +12)” +3 :1: jﬁﬂR’ +Iz)y2 - R = A— jB [3.22]

2 2
where R = (ozaouoe' - (£815) - ("—3) [3.23]

118

I = -0) 2801108" [3.24]

 

 

i1 2 2 l2
A:f (R +1 )/+R [3.25]
3 = 5%ﬁ12 +12)% - R and [3.26]
k, = A— [3. [3.27]

The exponential contributions of equations [3.15] and [3.16] now become
exp(—jk,z) = exp(—j[A — j3]z) = exp(—Bz — jAz). [3.28]
According to Euler’s theorem,

exp(jx) = cos(x) — j sin(x) , [3.29]
Equation [3.28] can be rewritten as

exp(—3z — jAz) = exp(—3z)[cos(— A2) + j sin(— Az)]. [330]

Therefore, the E-ﬁeld components can be written as
E, = C16?) cos(—";—1c x) 34% y) cos(- Az) exp(—32) [3.31]

Ey = C,(¥) si £81: )coscgiy) cos(—Az)exp(—32) and [3.32]
31 = 0. [3.33]
To complete the representation, the E-ﬁeld in vector notation in Cartesian coordinates is

represented with the above components as

1'36) = Exsx + 15,9, + 13.9. - [3.34]

APPENDIX C

POWER PENETRATION D3313 CALCULATION

The power penetration depth is deﬁned as the distance through a dielectric material
through which the microwave power is reduced by l/e (~3 6.8%). According to
Poynting’s theorem, the microwave power absorption is directly proportional to the
square of E-ﬁeld intensity or

(P) “1‘312 - [C. 1]

As shown in Appendix 3, the dependence of the E-field on the direction of propagation,
the z-direction, is given by

|1‘=.|2 ~ exp(-—232). [02]

Therefore, the distance through which the square of the E-ﬁeld is attenuated to l/e is

given by

1
Z = DP = '2—B' , [(3.3]
01'

D = 1 = 1
2%«1? +I’)y’ -R JAG/”(nay -1)

where R and I are given by [3.23] and [3.24].

[04]

119

 

APPENDIX D
CUTOFF FREQUENCY

The cutoff frequency, f., in a lossy waveguide is the frequency below which
electromagnetic energy does not propagate. Setting the power penetration depth equal
to zero is one method of calculating fc.

1

Jamie/Tr — 1)

Equation [DJ] is satisﬁed if R approaches zero. According to this condition and

 

= o [3.1]

equation [3.23], the cutoff frequency in the lossy waveguide is given by

3.351311%)? 32]

 

 

120

APPENDIX E
GUIDED WAVELENGTH

Examination of Equations [3.31] and [3.32] can provide an expression of the
wavelength in the part-shaped applicator. The cosine functions dictate the wavelength in

the Z-direction, so the wavelength can be written as:

_ 21: 2‘51:

i, __=
. A J(R’+Iz)%+R

[El]

where R and I are given by [3.23] and [3.24].

For a low-loss system where the contribution from a" is insigniﬁcant, I can be neglected

 

to give:
21: 21:
A, =—= .2
' Ji J 2 , (mu)2 (My [E ]
no 80110 - — — —
a b

Several algebraic manipulations yield the following:

'=F.z.:...[(’:")2+(%)’]'

Substituting for the cutoff frequency, source frequency, and the speed of light gives:

[53]

 

121

 

122

C

M: "FTP i 2, [54]
[12:32 [1%)

which is a common expression given in the literature, where 71. is the wavelength in the

 

inﬁnite dielectric medium, f¢ is the cutoff frequency of the waveguide, and f is the source

frequency.

APPENDIX F

UNSTEADY-STATE ENERGY BALANCE DEVELOPMENT

A heat transfer model is developed to determine the temperature proﬁles
characteristic of the three types of heating investigated for this project: conventional,
microwave, and hybrid. The heat transfer model also assists in the interpretation of the
heating patterns of the polymer plates, as recorded by thermal paper.

A general energy balance around an arbitrary homogeneous system is given by:

6T

pCPE=V-(kVT)+PMw-I~Sm, [F.l]
where p = the material density [=] kg/m3,

C, = the material heat capacity [=] J/kg-K,

T = the material temperature [=] K,

t = time [=] s,

k = the material thermal conductivity =] W/m-K,

PW = the rate of microwave power absorption [=] W/m’, and

SR. = the reaction source term [=] W/m3.

Assuming an isotropic thermal conductivity and non-reactive material, Equation [F . 1]
can be reduced to

1
9C.

2T— : aV2T+

 

PW. 1321

where or is the material thermal diffusivity.
Employing the Crank-Nicolson method of ﬁnite differences to Equation [F.2]

yields the following expression:

123

124

At
—BTi-l.n+1 + (l + 2B)Ti.n+l _ BTiHmH = BTi-lm +(1— 23)Ti.n + BTiHmH + EPMW ’

P
[33]
where i is the spacial index,

n is the time index,
13 _ orAt

2(AX)’ ’
At = time step, and
Ax = spacial increment.

This expression approximates the heat transfer for the interior nodes in the polymer. The
microwave power absorption in the brass is negligible; therefore, heat transfer in the
brass for the interior nodes can be approximated by Equation [F.3] minus the term
deﬁning microwave power absorption.

The nodes at or neighboring the boundaries require special treatment according
to the speciﬁed boundary conditions, as outlined in Figure F. 1. Implementing the
convective heat transfer boundary condition at the ﬁrst node gives the following ﬁnite

difference expression:

hAxTa,
T , [R41
3

 

[hAx +1)Tl..,l — T2.n+1 =
3
where h is the convective heat transfer coefﬁcient and k3 is the brass thermal

conductivity.

 

125

 

-o—t-c—o...

 

Bound C nditions

1.) Free convection at brass boundaries -

43%.,” = h[T,, — T(o,t)]

2.) Symmetry about the midplane -
a] = .
2 C

3.) Equal heat ﬂuxes at brass-polymer interface -

3T 6T
-kBE1x=xi = —kpa1x=x§

Figure El. Boundary Conditions of the One-Dimensional Heat Transfer Model.

126
Based on the boundary condition of equal heat ﬂuxes between two distinct
media, Carnahan et al.91 developed the expression for the interfacial temperature. The
explicit form of this temperature is

At

2(Ax)2 0‘” k k

TX.,n+1 = Tx'.n + TX.+1,II — 1+-_B TXB.D +—BTX."1JI [F.S]
212 4. 52 k? k?
“P k?

where TX. = the interfacial temperature,

sub P = polymer physical property, and
sub 3 = brass physical property.

The ﬁnal boundary condition to apply is the adiabatic condition, which can be
applied due to symmetry. Exploiting symmetry increases the program efﬁciency by
cutting the number of calculations in half.

The ﬁnite difference equations of the entire system form a tridiagonal matrix.
Solution to this matrix is attained by the Thomas algorithm”. The FORTRAN code for

the heat transfer model is shown in the following section.

 

APPENDIX G

FORTRAN CODE FOR HEAT TRANSFER MODEL

CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC
C Program Description: Unsteady-state heat conduction in a one-dimensional C
C polymer slab with brass boundaries and constant microwave power input to C
C polymer slab. C
CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC
C Variable Explanations

A sub-diagonal coefﬁcient to the tridiagonal matrix
ALPHAB brass thermal diffusivity [=] mz/s

ALPHAP polymer thermal diffusivity [=] m2/s

3 diagonal coefﬁcient to the tridiagonal matrix

C super-diagonal coefﬁcient to the tridiagonal matrix
CPB brass heat capacity [=] J/kg-K

CPP polymer heat capacity [=] J/kg-K

D constant vector

DT time step [=] s

DX interval length [=] m

H convective heat transfer coefficient [=] W/mz-K
ICOUNT counting variable

[PRINT printing ﬁequency

K3 brass thermal conductivity [=] W/m-K

KP polymer thermal conductivity [=] W/m-K
number of intervals in the brass

NIP number of intervals in the polymer

NN total number of nodes

PMW microwave power absorption [=] W/m3

RATIOP polymer constant

RATIOB brass constant

RHOB brass density [=] kg/m3

RHOP polymer density [=] kg/m3

STORE dummy storage variable

T temperature array [=] K

TIME simulation time [=] s

TINF ambient temperature [=]°C

TMAX maximum center temperature to be computed [=]°C
T0 initial system temperature [=] °C, K
TOLD temperature of previous time step [=] K
x-position [=] m

000000000000000OOOOOOOOOOOOOOOOO
000000000000000OOOOOOOOOOOOOOOOOO

X

127

 

128

C XP thickness of the polymer [=] cm, In C
CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC
PROGRAM 1DHX .

t

* VARIABLE DEFINITIONS

.

DIMENSION X(lOO),A(lOO),B(100),C(lOO),D(lOO),T(100)
REAL‘B DT,TMAX,RATIOP,RATIOB,DXPLTINETIMEJMW,
KP,RHOP, CPP,KB,RH03,CP3,T0,ALPHAB,ALPHAP,TOLD,A,3,
C,D,T,XP

INTEGER NIP,NIB,NN,IPRINT,ICOUNT

CHARACTER“ so STORE

292»

t

" INPUT DATA/COMPUTE CONSTANT S

a
OPENCUNIT =2 1, STATUS='OLD',FILE=' 1DHX.DAT')
OPENCUNIT=22, STATUS='UNKNOWN',FILE=' lDTDATADAT')

READ(21,101) RHOP,CPP,KP,RHOB,CPB,KB,XP,TO,TINF,H,PMW,
& DT,TMAX,NIP,NIB,IPR1NT

ALPHAP = KP/(RHOP‘CPP)
ALPHAB = KB/(RHOB'CPB)

XP = (KP/200.0)

DX = XP/NIP

RATIOP = DT*ALPHAP/(2.0*DX"2)
RATIOB = DT‘ALPHAB/(2.0"DX"2)
TINF = TINF + 273.15

T0 = To + 273.15

TMAX = TMAX + 273.15

NN = N13 + NIP + 1

WRITE(22,102)
REWIND(21)
READ(21,7) STORE
READ(21,7) STORE

Do 201 = 1,100
READ(21,7,END=25) STORE
WRITE(22,7) STORE
20 CONTINUE

 

 

 

129

25 CONTINUE

*

COMPUTE COEFFICIENTS

3(1) = H*DX/K3 + 1.0
C(1)= .10
D0) = H‘DX‘TINF/KB

Do 301 = 2,NIB-1
A0) = -1.0"'RATIOB
3(1) = 1.0 + 2.0*RAT103
CO) = -l.0*RATIOB
3o CONTINUE

A(NIB) = -1.0*RATIOB
3(NIB) = (1.0+2.0*RATIOB)

3(N13+2) = 1.0 + 2.0*RATIOP
C(N13+2) = -1.0*RATIOP

DO 401 = (N13+2),(NIE+NIP)
A(I) = -l.O*RATIOP
3(1) = 1.0 + 2.o*RATIOP
C(I) = -1.0*RATIOP
4o CONTINUE

A(NN) = -2.0*RATIOP
3(NN) = 1.0 + 2.0*RATIOP

t

' SET BOUNDARY CONDITIONS & INITIALIZE TEMPERATURE ARRAY
up
DO so 1= 1,NN
xa) = (1-1 .0)/(NIB+NIP)
T(I) = To
so CONTINUE

TIME = 0.0
WRITE(22,103) TIME/60.0

Do 601= 1,NN
WRITE(22,104) X(I),T(I)-273. 15
60 CONTINUE

 

t

130

" COMPUTE TEMPERATURES - MAIN LOOP

*

70

t

998°

:11

aorta

82°

ICOUNT = 0

CONTINUE

TIME=TIME+DT
ICOUNT=ICOUNT+1

BRASS

TOLD = T(NIB+1)

T(NIB+1) = TOLD + ((2.0‘ALPHAB*DT/(DX*DX))/
(ALPHAB/ALPHAP + KB/KP))*(T(NIB+2H1.0+KB/KP)*
T(NIB+1) + (KB/KP)*T(NIB))

Do so 1= 2,(N13-1 )
D(I) = RATIOB‘TO-l) + (l.0-2.0*RATIOB)*T(I) +
RATIOB*T(I+1)

CONTINUE

D(N13) = RATIOB*T(N13-I) + (1.0-2.0*RAT103)*T(N13) +
RAT103*TOLD + RA'I'IOB*T(NIB+1)

CALL TRIDAG(1,NIB,A,3,C,D,T)

POLYMER

D(NIB+2) = RATIOP*T(NIB+1) + RATIOP*TOLD +
(l.0-2.0*RATIOP)*T(NIB+2) + RATIOP*T(NIB+3) +
DT‘PMW/(RHOP‘CPP)
DO 90 I = (NIB+3),(NIB+NIP)
D0) = RATIOP*T(I-I) + (l.0-2.0*RATIOP)*T(I) +
RATIOP*T(I+1) + DT*PMW/(RHOP*CPP)
CONTINUE

D(NN) = 2.0*MTIOP*T(NN.1) +
(l.0-2.0*RATIOP)*T(NN) + DT*PMW/(RHOP*CPP)

CALL TRIDAG(NIB+2,NN,A,B,C,D,T)

IFOCOUNT.GE.IPRINT)THEN

 

131

WRITE(22,103) TIME/60.0

 

Do 100 1= 1,NN
WRITE(22,104) X(I),T(I)-273.15
100 CONTINUE
ICOUNT = o
ENDIF
IF(T(NN).LE.TMAX) GOTO 70
t
'1 FORMAT STATEMENTS
‘
7 FORMAT(A)
101 FORMAT(7(/),F8.2,//,F8.2,//,F8.S,4(/),F8.2,//,F8.2,//,F8.5,6(/),
& F8.2,//,F8.2,//,F8.2,//,F8.3,//,F12.2,6(/),F9.5,//,F8.2,//,I3,//,
& 13,//,110)
102 FORMAT('Output Data for lDMW.FOR',/,' ')
103 FORMATC ','TIME (MIN) = 31:9. 1)
104 FORMAT(F5.3,2X,F6.1)
STOP
END

SUBROUTINE TRIDAG(IF,L,A,B,C,D,V)

&

200

210

DIMENSION A(100),3(100),C(100),D(100),V(100),BETA(100),
GAMMAUOO)
REAL‘S A,3,C,D,V,3ETA,GAMMA

3ETA(IF) = 3(13)

GAMMA(IF) = D(IF)/BETA(IF)

11731 = IF + I

DO 2001 = IFP1,L
BETA(I) = B(I)-A(I)*C(I-l)/BETA(I-1)
GAMMA(I) = (D(I)-A(I)*GAMMA(I-l))/BETA(I)

CONTINUE

V(L) = GAMMA(L)

LAST = L-IF

Do 210 K =1,LAST
1 = L-K
V(I) = GAMMA(I) - C(I)"‘V(I+1)/3ETA(I)
CONTINUE

RETURN

END

 

132

Input Data for 1DHX.FOR

 

Physical Constants

Polymer

Density [=] Its/m3

1 100.0

Heat Capacity [=] J/kg-K

2000.0

Thermal Conductivity [=] W/m-K
2.0

Brass

Density [=] Its/ma

8500.0

Heat Capacity [=] J/kg-K

380.0

Thermal Conductivity [=] W/m-K
1 10.0

Physical System Definition

Polymer Thickness [=] cm

8.9

Initial System Temperature [=] °C

25.0

Ambient Temperature [=] °C

25.0

Convective Heat Transfer Coefﬁcient [=] W/mz-K
5.0

Microwave Power Absorption [=] W/m3
20000000

Finite Difference Parameters

Time Step [=] s

0.1

Maximum Center Temperature to be Computed [=] C
200.0 ,

Number of X-Direction Intervals in the Polymer
25

Number of X-Direction Intervals in the Brass

7

Solution Printing Frequency

300 -

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