Hickman State
L University

 

 

1

 

This is to certify that the

dissertation entitled

ESTIMATION OF THE THERMAL AND KINETIC PROPERTIES
ASSOCIATED WITH CARBON/EPOXY COMPOSITE MATERIALS
DURING AND AFTER CURING
presented by

Elaine Patricia Scott

has been accepted towards fulfillment
of the requirements for

Ph. D. Mechanical Engineering

degree in

 

 

Q iwwyfi’e/

Major proféssor

Date December 8, 1989

 

MS U i: an Affirmative Action/Equal Opportunity Institution 0-12771

0265/ 72,59

 

l

 

PLACE IN RETURN BOX to remove this checkout from your record.
TO AVOID FINES return on or betore date due.

I DATE DUE A DATE DUE DATE DUE I

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

MSU Is An Affirmative Action/Equal Opportunity Institution
cmmii-DJ

 

ESTIMATI

assocuri

 

ESTIMATION OF THE THERMAL AND KINETIC PROPERTIES
ASSOCIATED WITH CARBON/EPOXY COMPOSITE MATERIALS
DURING AND AFTER CURING

By

Elaine Patricia Scott

A DISSERTATION

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

DOCTOR OF PHILOSOPHY
Department of Mechanical Engineering

1 989

"'5‘ 93
”O

;’ 1:": ~ 1;}
to Qt) AI 3

ABSTRACT

ESTIMATION OI" THE THERMAL AND KINETIC PROPERTIES
ASSOCIATED WITH CARBON] EPOXY COMPOSITE MATERIALS
DURING AND AFTER CURING

By

Elaine Patricia Scott

The use of high performance composite materials in a variety of applications has
grown rapidly in recent years. A thorough understanding of these materials with
regards to thermal characteristics properties is essential in developing new processing
methods and product applications. The focus of this research is on the estimation of
the thermal properties (specifically thermal conductivity and the product of density and
specific heat) of cured carbon fiber/amine-epoxy matrix composites. and on the estima-
tion of both thermal and kinetic properties of these materials during curing. The
kinetic properties include the activation energy constants.

The thermal properties of the cured composites were first estimated using an es-
timation procedure based on the minimization of a least squares function which I
incorporates both calculated and measured temperatures. Differential scanning
calorimetry (DSC) was used in the next phase of this study for the estimation of the
kinetic parameters associated with the curing of an EPON 828 amine-epoxy system.
using several different kinetic models.

The final phase of this study related to the estimation of thermal properties
during the curing process. The sirnuitaneous estimation of thermal conductivity and
the product of density and specific heat during the curing process has not previously

been attempted. A least squares function incorporating both calculated and measured

 

immatures was

ions during curim

 

I
mists. This proc

an expenmcnt wa~
Citing the curing pl

The 111ch c
beat me dctcrmim
miniature for twq

Smlar epoxy mien

 

Mel was PRscntcc
f” the estimation of

WW! data. and

Used In Obtammg CK}

temperatures was again used. and the heat generation resulting from the kinetic reac-
tions during curing was accounted for in the analysis. using the results of the DSC
analysis. This procedure was evaluated using simulated temperature data. In addition.
an experiment was designed and evaluated for use in obtaining temperature data
during the curing process with this estimation scheme.

The thermal conductivity perpendicular to the fiber axis and the density-specific
heat were determined for the cured AS4-EPON 828 composite materials as functions of
temperature for two different fiber orientations. The kinetic parameters obtained for
EPON 828 epoxy were in good agreement with previously published kinetic data for
similar epoxy systems during the first half of the curing cycle. and a new reaction rate
model was presented for the second half of the curing process. Finally. the procedure
for the estimation of thermal properties during curing was verified using simulated tem-
perature data. and improvements were proposed for the techniques and equipment

used in obtaining experimental temperatures.

 

 

moraines,
mommies

Wilkinson

TABLE OF CONTENTS

PAGE
LIST OF TABLES ................................................... xi
LIST or FIGURES ............................ ' ...................... xvii
LIST OF SYMBOLS ................................................. xxii
CHAPTER 1. Introduction ........................................... 1
ems 2. literature Review ..................................... ’. e
2. 1 Estimation of Thermal Properties of
Composite Materials ........................................ 6
2.1.1 Estimation of Thermal Properties of Composite and
Other Anisotropic Materials ............................. 7
2.1.2 Estimation of Thermal Properties of Cured Carbon
Fiber-Reinforced Composites ........................... 9
2.2 Estimation of Kinetic Parameters in Epoxy-Based Systems ......... 10
2.2.1 Kinetics of Amine-Epoxy Systems ....................... 11
2.2.2 Differential Scanning Calorimetry (DSC) .................. 12
2.2.3 Cure Kinetics using DSC .............................. 13
2.2.4 Cure Kinetics of Neat vs. Carbon Fiber Reinforced
Epoxy Systems ...................................... 15
2.3 Estimation of Thermal Properties in Carbon/ Epoxy Composite
Systems during Curing ..................................... 16
2.3.1 Mathematical Models to Simulate Heat Transfer in
Carbon/Epoxy Composite Materials during Curing ............... 16

2.3.2 Estimation of Thermal Properties of Composites
during Curing ....................................... 17

V

 

 

2.4 Minimizat

|
Properti-

2... i

 

Vi
2.4 Minimization Methods used for the Estimation of Thermal
Properties ................................................
2.4.1 The Gauss Method ...................................
2.4.2 The Steepest Descent Method ..........................
2.4.3 The Conjugate Gradient Method ........................
2.4.4 Methods of Determining the Matrix
Derivatives .........................................
CHAPTER 3. Theoretical Considerations .............................
3. 1 Estimation of Thermal Properties in Cured
Composite Materials .......................................
3.1.1 Heat Conduction in Cured Carbon-Fiber Reinforced
Composites .........................................
3. 1.2 Estimation of Thermal Properties .......................
3.2 Estimation of Kinetic Parameters .............................
3.2. 1 Reaction Kinetics for Degree of
Cure < 50% .........................................
3.2.2 Reaction Models for Degree of
Cure > 50% .........................................
3.2.3 Selection of Kinetic Models for the Determination of
Thermal Properties ...................................
3.3 Estimation of Thermal Properties during
Curing ...................................................
3.3.1 Heat Conduction in Carbon Fiber Reinforced Composites
during Curing .......................................
3.3.2 Parameter Estimation Method ..........................
3.4 Optimal Experimental Design Criterion for the Estimation of
Thermal Properties .........................................
3.4.1 Objectives for the Experimental Design
Optimization .......................................
3.4.2 Optimal Experimental Design Criterion ..................

17
18
19
20

20
22

23

23

24

25

26

30

31

31

32
33

34

34
35

 

CW4. Expil

4.1 Experimc:
Compost
4.1.1 P:

S.

 

vii
CHAPTER 4. Experimental Procedures ...............................
4.1 Experiments for the Estimation of Thermal Properties in Cured
Composite ................................................
4.1. 1 Preparation of the Composite
Samples ............................................
4.1.1.1 Epoxy Preparation .............................
4.1.1.2 Prepreg Preparation ............................
4.1.1.3 Stacking Procedure ............................
4.1. 1.4 Consolidation ................................
4.1.2 Experimental Set-up ..................................
4. 1 .2. 1 Thermocouple Fabrication ......................
4. 1.2.2 Experimental Set-up Assembly ..................
4.1.3 Data Acquisition System and Controlled
Heat Flux ..........................................
4.1.3.1 Data Acquisition Hardware .....................
4.1.3.2 Sigral Conditioning ...........................
4.1.3.3 Data Acquisition Programs .....................
4. 1.3.4 Controlled Heat Flux ..........................
4.1.4 Transient Temperature Experiments .....................
4.1.5 Experimental Parameters ..............................
4.1.6 Fiber Volume Fraction ................................
4.2 Differential Scanning Calorimetry Experiments ...................
4.2.1 Sample Preparation for DSC Experiments .................
4.2.2 Difl'erential Scanning Calorimeter Operation ..............
4.2.2. 1 Differential Scanning
Calorimeter Set Up Procedure ..................
4.2.2.2 Initialization Procedure for
the DuPont 9 10 Calorimeter ...................
4.2.2.3 Post-Processing Procedure .....................

4.2.3 Experimental Parameters ...............................

41

41

42
42
44
52
55
57
59
6 1

22%

66
66
67
68
7O
7O
72

72

74

75
78

 

4.3 ltansicnt

 

4.3.1 SJ

4

4.

4.3.2 it

4.3.3 Da

 

5.1

5.1
5,12 ESt
He-

V111

4.3 Transient Temperature Measurements during Curing ............... 80
4.3. 1 Sample Preparation ................................... 82
4.3.1.1 Heater Preparation ............................ 82

4.3.1.2 Stacking Procedure ............................ 82

4.3.2 Experimental Set-up .................................. 84

4.3.3 Data Aquisition and Controlled Heat Flux ................. 86
4.3.3.1 The VAXlab Data Aquisition Hardware ............ 88

4.3.3.2 Signal Amplification ........................... 91

4.3.3.3 Controlled Heat Flux .......................... 92

4.3.3.4 Data Aquisition Program ....................... 93

4.3.4 Experimental Procedure for Transient Temperature

Measurements during Curing .......................... 96
4.3.5 Experimental Parameters ............................. 98
CHAPTER 5. Results and Discussion ................................ 102

5. 1 Estimation of Thermal Properties of Cured AS4/EPON 828-mPDA

Composite Materials ....................................... 104
5.1.1 Analysis of the Estimation Procedure .................... 105
5.1.1.1 The Residuals .............................. 105

5.1.1.2 The Sensitivity Coefficients of the Estimated

Parameters ................................ 108
5.1.1.3 The Sequential Estimates of the Estimated

Parameters ................................ 1 10
5. 1.1.4 Insights into the

Experimental Design ........................ 1 10

5.1.2 Estimation of Themal Conductivity and Density-Specific

Heat of Cured AS4/EPON 828 Composite Materials ........ 112
5.1.2.1 Experiments using [0°]24 Samples ............. 1 12
5.1.2.2 Experiments using [O°, i30°.i:60°,90°]

2(sym)
Samples .................................. 1 19

 

 

 

l
5.2 Estimati.

EPON 82

5.2.1 at

of
5.2.2 Pa

 

(no

5.2

5.2

5.3.2

ix
5.1.2.3 Comparison of Results from [0°]24 and

[O°,i:30°.160°.90°l Samples .............

2(sym)
5. 1.2.4 Comparison with Previously

Published Results ...........................

5.2 Estimation of Kinetic Parameters Associated with the Curing of

5.3

EPON 828/mPDA Epoxy ....................................

5.2. 1

5.2.2

5.2.3
5.2.4

Determination of the Total Heat of Reaction and Degree
of Cure ............................................
Parameter Estimation for Autocatalyzed Cure Kinetics
(no difi'usion) .......................................
5.2.2.1 Estimation of Rate constants

and Exponents .............................
5.2.2.2 Estimation of the Activation Energy Constants

and the Pre-Exponential Factors ...............
Diffusion Controlled Reactions and Parameter Estimation. .
Selection of Kinetic Parameters for Use in the Estima-

tion of Thermal Properties during Curing ................

Estimation of Thermal Properties of Composite Materials during
Curing ..................................................

5.3.1

5.3.2

Analysis of the Estimation Procedure using Simulated
Data .............................................
5.3.1.1 Description of Test Cases ....................
5.3.1.2 Results from Test Cases .....................
Analysis of the Estimation of Thermal Properties from
Experimental Data ...................................
5.3.2.1 Results from Experimental Data ................
5.3.2.2 Investigation of Possible Errors in the Estimation
of Thermal Properties during Curing ............
5.3.2.3 Improvements in the Experimental Design for the
Estimation of Thermal Properties in Composite

122

125

125

127

127

129

133
137

138

141

142

142

146

154

156

170

x

Materials during Curing ...................... 176

CHAPTER 6. Summary and Conclusions ............................. 179
APPENDIX A. One Dimensional Curing Program. CUREID ............... 183
A. 1 Summary of Program ...................................... 183

A2 Program Listing for CUREID ................................. 185
APPENDIX B. Data Aquisition Program. DATA_DA_AD .................. 2 15
B. 1 Summary of Program .................. . ........... - .......... 215

8.2 Program Listing for DATA_DA_AD .............................. 2 1 7

APPENDIX C. Parameter Estimation Results from Experimental Repetitions
using Cured Composite Samples ......................... 238

BIBLIOGRAPHY .................................................... 247

 

Table 4.1 Fabricati
hangent Er]
Table” Experim
Composite 8.
Table 4.3 Fiber Vc
Numerical V
Table 4.4 Echrin

Ripenmeni:

LIST OF TABLES

Table 4.1 Fabrication Parameters for Cured Composite Disks used in

Transient Experiments ........................

. Table 4.2 Experimental Parameters for Transient Experiments using Cured

Composite Samples ........................

Table 4.3 Fiber Volume Fractions using Acid Digestion Analysis and Optical

Numerical Volumetric Analysis (ONVfA) ........................

Table 4.4 Experimental Parameters for Difl‘erential Scanning Calorimetry

Experiments ........................

Table 5. 1 Averaged Estimated Values and 95% Confidence Intervals for
Effective Thermal Conductivity. k. Perpendicular to the Fiber Axis
and Effective Density-Specific Heat. pcp. of Cured [0°]24 AS4/

EPON 828-mPDA Composites ........................

Table 5.2 Estimated Parameters and 95% Confidence Intervals for the
Linear Regression of Both Effective Thermal Conductivity. k.
Perpendicular to the Fibers and Effective Density-Specific Heat.

pop. of Cured [0°]24 AS4/EPON 828-mPDA Composites ...........

Table 5.3 Averaged Estimated Values and 95% Confidence Intervals for
Effective Thermal Conductivity. k. Perpendicular to the Fiber Axis and

Effective Density-Specific Heat. pop. of Cured [O°.d:30° .i60° .90°]2(sym)

AS4/EPON 828-mPDA Composites ........................

Table 5.4 Comparison of Estimated Effective Thermal Conductivity. k,
Perpendicular to the Fiber Axis with Previously Published

x1

PAGE

58

69

71

81

Results
Table 5.5 Total Hr

  

EPOXY. using '
Table 5.7 Rate Con
95% Confidcr
Values, a, Le:
squares Metl
Table 5.3 Rate C01

Results ..........................

Table 5.5 Total Heat of Reaction Calculated from Dynamic Differential
. Scanning Calorimetry Experiments (Exp. 2.1) using

EPON 828/mPDA Epoxy ..........................
Table 5.6 Rate Constants. c. and o. . and Exponent. m. Determined from eq.

(3.4) and Degree of Cure Values. on. Less than 50% for EPON 828/mPDA

Epoxy. using the Ryan and Dutta (1979) Method .............
Table 5.7 Rate Constants. c. and c, . and Exponent. m. and Associated

95% Confidence Intervals. Determined from eq. (3.4) and Degree of Cure

Values. or. Less than 50% for EPON 828/mPDA Epoxy. using a Least

Squares Method ..........................
Table 5.8 Rate Constants. c| and c4 . and Associated 95% Confidence

Intervals. Determined from eq. (3.5) and Degree of Cure Values. or.

Less than 50% for EPON 828/mPDA Epoxy. using the Sourour and Kama]

(1976) Method ..........................
Table 5.9 Rate Constant. c. . and Exponent. n. and Associated 95% Confi-

dence Intervals. Determined from eq. (3.6) and Degree of Cure Values.

on. Less than 50% for EPON 828/mPDA Epoxy .............
Table 5.10 Activation Energy Constants. E. and E. . and Pre-Exponential

Factors. A. and A. . Determined from the Arrhenius Relationship (eq. 3.7)

and Estimated Rate Constants Estimated from eqs. (3.4). (3.5). and

(3.6). for Degree of Cure Values. a. Less than 50% for

EPON 828/mPDA Epoxy ..........................
Table 5.1 1 Rate Constant. c. . and Diffusion Constant. D. and Associated

95% Confidence Intervals Determined from eq. (3.16) and Degree of Cure

Values. 0:. Greater than 50% for EPON 828/mPDA Epoxy .............

Table 5. 12 Test Cases for the Estimation of Thermal Conductivity. k. and

Density-Specific Heat. pcp. from Simulated Data for Composite

Materials ..........................

 

 

Table 5.13 Esturz
the Fibers a‘
95% Confrd'
Temperature
Properties: k

Table 5.14 Estima
the Fibers ml

Confidence 1n

 

TcmPCI'aturi: i
Pmpcmcs: k =

Table 5.15 Estlniau
the Fibers and

95% Confiden.
Tantalum
Pmpflllcs; kl
PCleJ/m: °C
Table 5.16 Estimat
the Fibers 3m

95% C°nf1den
Temperature .
Pmptl'tics: k ‘
”9131.1 /m’ “C
:33}. 5'17 Compar
and Density:

e1115.6) and

at? 5'18 Estima.

1t
and Eff-ECU

xiii

Table 5.13 Estimated Efl‘ective Thermal Conductivity. k. Perpendicular to

the Fibers and Density-Specific Heat. pop. and the Associated

95% Confidence Intervals for a Simulated Composite with an Initial

Temperature of 75°C. an Initial Degreeof Cure of 0.40. and Thermal

Properties: 1: = 0.83 W/m°C and pcp = 1.84 MJ/m’ °C ............. 149
Table 5.14 Estimated Effective Thermal Conductivity. k. Perpendicular to

the Fibers and Density-Specific Heat. pop. and the Associated 95%

Confidence Intervals for a Simulated Composite with an Initial

Temperature of 125°C. an Initial Degree of Cure of 0.80. and Thermal

Properties: k = 0.83 W/m°C and pcp = 1.84 MJ/m’ °C ............. 150
Table 5. 15 Estimated Efi'ective Thermal Conductivity. k. Perpendicular to

the Fibers and Density-Specific Heat. pcp. and the Associated

95% Confidence Intervals for a Simulated Composite with an Initial

Temperature of 75°C. an Initial Degree of Cure of 0.40. and Thermal

Properties: k (W/m°C) = 0.6678 + 9.03x104’1‘ + 0.0742or. and

pcP(MJ/m’ °C) = 1.251 + 0.0045T + 0.139or ............. 152
Table 5. 16 Estimated Effective Thermal Conductivity. k. Perpendicular to

the Fibers and Density-Specific Heat. pcp. and the Associated

95% Confidence Intervals for a Simulated Composite with an Initial

Temperature of 125°C. an Initial Degree of Cure of 0.80. and Thermal

Properties: 1: (W/m°C) = 0.6678 + 9.03xio’4'r + 0.0742a. and

pcp(MJ/m’ °C) = 1.251 + 0.0045T + 0.1390: ............. 153

Table 5. 17 Comparison of Averaged Estimated Thermal Conductivity. E,

and Density-Specific Heat. FEP' Values with Values Calculated Using

eq. (5.6) and (5.7) .......................... 155
Table 5. 18 Estimation of Thermal Conductivity Perpendicular to the Fibers.

k. and Effective Density-Specific Heat. pcp. of Cured AS4/EPON 828-

mPDA Composites using Temperature Data from the Top Composite Section
Only in Exp. 3.3 .......................... 161

 

Table 5.19 we:
I. and EEOC

Table 5.20 EStuna

   

k. and 556°“
Campos“s d
section only
Table 5.21 Est‘ma‘
k and Effect:
Composites C
Composite 5
Table 5.22 Est‘mE
1L and Effect
Camposites
Sections in 1
Table 5.23 again

it and Effec

Composites
Bottom C01”
Table 5.24 Est'irr.
It and Etfec
Composites

Bottom Cor

xiv
Table 5.19 Estimation of Thermal Conductivity Perpendicular to the Fibers.
k. and Effective Density-Specific Heat. pcp. of Cured AS4/EPON 828-
mPDA Composites using Temperature Data from the Bottom Composite
Section Only in Exp. 3.3 .......................... 162
Table 5.20 Estimation of Thermal Conductivity Perpendicular to the Fibers.
k. and Effective Density-Specific Heat. pcp. of AS4/EPON 828-mPDA
Composites during Curing using Temperature Data from the Top Composite
Section Only in Exp. 3.2 .......................... 163
Table 5.2 1 Estimation of Thermal Conductivity Perpendicular to the Fibers.
k. and Effective Density-Specific Heat. pop. of AS4/EPON 828-mPDA
Composites during Curing using Temperature Data from the Bottom
Composite Section Only in Exp. 3.2 .......................... 164
Table 5.22 Estimation of Thermal Conductivity Perpendicular to the Fibers.
k. and Effective Density-Specific Heat. pop. of Cured AS4/EPON 828-mPDA
Composites using Temperature Data from Both Top and Bottom Composite
Sections in Exp. 3.3 .......................... 168
Table 5.23 Estimation of Thermal Conductivity Perpendicular to the Fibers.
k. and Effective Density-Specific Heat. pcp. of AS4/EPON 828-mPDA
Composites during Curing using Temperature Data from Both Top and
Bottom Composite Sections in Exp. 3.2 .......................... 168
Table 5.24 Estimation of Thermal Conductivity Perpendicular to the Fibers.
k. and Effective Density-Specific Heat. pep. of AS4/EPON 828-mPDA
Composites during Curing using Temperature Data from Both Top and
Bottom Composite Sections in Exp. 3.1 .......................... 169
Table 5.25 Estimation of Thermal Conductivity Perpendicular to the Fibers.
k. and Efi'ective Density-Specific Heat. pop. of Cured AS4/EPON 828-mPDA
Composites using Temperature Data from Both Top and Bottom Composite
Sections in Exp. 3.3. with Applied Heat Flux Calculated From the
Measured Power Input and the Surface Area of the Aluminum Foil

Protection Layer .......................... 1 77

 

Tabieiii Descrr;

CLREID

 

 

Table 8.1 Desert;

DATA_DA_/t'

Table C.l Estimar
the Fiber AXIS
154/EPON s_
Tempcrature c

Table C2 55mm
the Fiber Aids
54/me 82
TCmPCTaturc (

Tablec3 Estimate.
the Fiber Aids
”54/me 8:2
Tcmpefiiturg (

Table C4 Estimate.
the Fiber Aids
34/me 82

xv
Table A. 1 Description of the One Dimensional Curing Program.
CUREID ' .......................... 184

Table B. 1 Description of the Data Acquisition Program.
DATA_DA_AD .......................... 2 1 6

Table C. 1 Estimated Effective Thermal Conductivity. k. Perpendicular to
the Fiber Axis and Density-Specific Heat. pop. of Cured [0°]24
AS4/EPON 828-mPDA Composites from Experiments with an Initial
Temperature of Approximately 25°C .......................... 239
Table 0.2 Estimated Effective Thermal Conductivity. k. Perpendicular to
the Fiber Axis and DensityoSpecific Heat. pop. of Cured [0°]24
AS4/EPON 828-mPDA Composites from Experiments with an Initial
Temperature of Apprmdmately 50°C .......................... 240
Table C.3 Estimated Efi’ective Thermal Conductivity. k. Perpendicular to
the Fiber Axis and Density-Specific Heat. pep. of Cured (0°)24
AS4/EPON 828-mPDA Composites from Experiments with an Initial
Temperature of Approximately 75°C .......................... 241
Table C.4 Estimated Effective Thermal Conductivity. k. Perpendicular to
the Fiber Axis and Density-Specific Heat. pop. of Cured [0°54
AS4/EPON 828-mPDA Composites from Experiments with an Initial
Temperature of Approximately 100°C .......................... 242
Table C.5 Estimated Effective Thermal Conductivity. k. Perpendicular to
the Fiber Axis and Density-Specific Heat. pop. of Cured [0°]24
AS4/EPON 828-mPDA Composites from Experiments with an Initial
Temperature of Approximately 125°C .......................... 243
Table C.6 Estimated Effective Thermal Conductivity. k. Perpendicular to
the Fiber Axis and Density-Specific Heat, pop. of Cured

10°. 130°. 160°. 90°] AS4/EPON 828-mPDA Composites from

2(sym)
Experiments with an Initial Temperature of Approximately 25°C . . . . 244

  
   

Table C.7 Estima

Table C.8 Estima:
the Fiber An.
10'. :30', 3

I"-“13<?rllllre'nts r

xvi
Table C.7 Estimated Effective Thermal Conductivity. k. Perpendicular to
the Fiber Axis and Density-Specific Heat. pep. of Cured

[0°. 130°. t60°. 90°) AS4/EPON 828-mPDA Composites from

218W!) .
Experiments with an Initial Temperature of Approximately 50°C . . . . 245

Table 0.8 Estimated Effective Thermal Conductivity. k. Perpendicular to
the Fiber Axis and Density-Specific Heat. pcp. of Cured

[0°. t30°. 160°. 90°] AS4/EPON 828-mPDA Composites from

2(sym)
Experiments with an Initial Temperature of Approximately 75°C . . . . 246

LIST OF FIGURES

PAGE
Figure 3. 1 Dimensionless Experimental Design Criterion. AT. for Different
Dimensionless Heating Times. tr . and a Dimension-less Heat
Generation Term. g1. equal to 0.1 .................... 39
Figure 3.2 Dimensionless Experimental Design Criterion. AI. for Difl'erent
Dimensionless Heating Times. if. and a Dimension-less Heat Generation
Term. gT. equal to 1.0 (dimensionless) .................... 39
Figure 4.1 Hot Melt Prepregger (Research Tools Corporation) .......... 45
Figure 4.2a Schematic of Single Spool Mount on Hot-Melt Prepregger
............................... 46
Figure 4.2b Photograph of Single Spool Mount .................... 46
Figure 4.3a Schematic of Guide Roller. Furnace. and Tension Rollers on
Hot-Melt Prepregger ............................... 47
Figure 4.3b Photograph of Fibers Through Guide Roller. Furnace. and
Tension Rollers ............................... 47
Figure 4.4a.b Schematic (left) and Photograph (right) of Resin Pot
Assembly on Hot-Melt Prepregger ............................... 49
Figure 4.5 Detailed View of Resin Pot. Die (A-A). and Pin Guide (from
left to right) ............................... 50
Figure 4.6 Hot-Melt Prepregger Control Panel .................... 50
Figure 4.7 Stacking for Orientation for the First Six Parallel and the First
Six Quasi-isotropic Laminates for the [0°] 2 4 and the
[0°. 21:30". 160°. 90°]2(sym) Composite Disks .......... 53
Figure 4.8 Cut-Away View of Stacked Prepreg Laminate .......... 56

xvii

 

 

 

F'«iure 4.14.; Dupo.

XVIII

Figure 4.9 Prepared Thermocouple (ANSI Type E) with Close-up View of

Thermocouple Junction ...............................

Figure 4. 10 Thermocouple Placements on Composite Disks Used in fiansient

Temperature Experiments ...............................

Figure 4. 1 1 An Experimental Set-up for Transient Temperature Measurements

using Cured Composite Samples ...............................

Figure 4. 12 Schematic of Data Acquisition for Transient Experiments using

Cured Composite Samples ...............................

Figure 4. 13 DuPont Instruments 9 10 Difl'erential Scanning Calorimeter

Figure 4.14a DuPont Instruments 910 Difl'erential Scanning Calorimeter
Cell (left). Showing Bell Jar (right). Cell Cover (center). and

Ceramic Lid (on cell) ...............................

Figure 4. 14b Close-up View of Samples Pan (right) and Reference Pan (left)

in Calorimeter Cell Shown Above ...............................

Figure 4.15 Heat of Reaction Data from One Repetition of the Dynamic DSC
Experiment using DuPont Thermal Analysis Plotting Subroutines

Figure 4. 16 Determination of Temperatures for Isothermal DSC Experiments

from a Dynamic DSC Experiment ...............................

Figure 4. 17 Stacked Composite Prepreg. Instrumented with Thermocouples and

Heaters. (Photograph taken after curing.) ....................

Figure 4. 18 Stacked and Instrumented Composite Prepreg Sealed in a Vacuum

Bag ...............................

Figure 4. 19 Mechanical Press with Composite Laminate in Place

OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO

Figure 4.20 Schematic of Data Acquisition for First Transient Experiment

during Curing ...............................

Figure 4.2 1 Schematic of Data Acquisition for Second Transient Experiment

during Curing ...............................

60

62

63

65

73

76

76

77

79

85

85

87

89

90

 

 

Figurt 4-22 Sch.

3511:2423 Th"
F1131 Tran:
Bottom t0 |
are Numbe
Figure 4.24 Them
Second Tia.
Bottom to
are Number
Boldface in:
Figure 5.1. Residue
Surface TIC
the Heated E
No. 9 (initial
F"tare 5.2. Sensith
at the Heate
Repetition 0
Teresa. Seq...“
Specific lice
lirutiai temp
$421“: 5.4. Estima
‘0 the Fiber

composites

xix

Figure 4.22 Schematic of Data Acquisition Program DATA_DA_AD

Figure 4.23 Thermocouple (T.C.) Locations in Prepreg Laminate for the

First Transient Experiment during Curing. Plies are Numbered from

Bottom to Top. and the Thermocouples used in the Data Acquisition

are Numbers 1. 2. 10 and 1 1 (shown in Boldface type) .......... 100
Figure 4.24 Thermocouple (T.C.) Locations in Prepreg Laminate for the

Second Transient Experiment during Curing. Plies are Numbered from

Bottom to Top. and the Thermocouples used in the Data Acquisition

are Numbers 1. 3. 4. 6. 7. 10, 11. 12. 13. and and 14 (shown in

Boldface type) ............................... 102
Figure 5. 1. Residuals (°C) for No Thermocouples Located at the Heated

Surface (TC #1 and TC #2) and One Thermocouple Located 3.3 mm from

the Heated Surface (TC #5) for the Second Repetition of Experiment

No. 9 (initial temperature - 100°C) .................... 106
Figure 5.2. Sensitivity Coefficients. X. (kaT/ak) and X: (pCpaT/apcp),

at the Heated Surface and 3.3 mm from the Heated Surface for Second

Repetition of Experiment No. 9 ............................... 109
Figure 5.3. Sequential Estimation of Thermal Conductivity. k. and Density

Specific Heat. pcp. from the Second Repetition of Experiment No. 9

(initial temperature ~ 100°C) ............................... 1 1 1
Figure 5.4. Estimation of Effective Thermal Conductivity Perpendicular

,to the Fiber Axis. k. of Cured [0°]24 AS4/EPON 828-mPDA

Composites ............................... 1 1 7
Figure 5.5. Estimation of Effective Density-Specific Heat. pcp. of

Cured [0°]24 AS4/EPON 828-mPDA Composites .................... 1 16
Figure 5.6. Estimation of Effective Thermal Conductivity Perpendicular

to the Fiber Axis. k. of Cured [0°. r30°, r60°. 90°]2( AS4/

s y m)
EPON 828-mPDA Composites ............................... 123

Figure 5.7. Estimation of Effective Density-Specific Heat. pop. of Cured

10‘

[0°. t30°. 160°. 90°)2 AS4/EPON 828-mPDA Composites

(sym)

Figure 5.8. Comparison of Linear Regression Curves for Efi’ective Thermal
Conductivity Perpendicular to the Fiber Axis. k. of Cured 10°)“

and Cured [0°. r30°. i60°. 90°] AS4/EPON 828-mPDA

2(s y m)

Composites ...............................

Figure 5.9. Comparison of Linear Regression Curves for Effective
Density-Specific Heat. pop. of Cured (0°)24 andCured

[0°. t30°. 160°. 90°) AS4/EPON 828-mPDA Composites

2(syml

Figure 5. 10 Arrhenius Relationship for the First Rate Constant. c. in
eqs. (3.4). (3.5). and (3.6): Ln (c. ) versus 1/R1‘a (R = Gas Const.

(kJ/mole°K); Ta = Abs. Temp. (°K)) ....................

Figure 5. 1 1 Arrhenius Relationship for the Second Rate Constant. c. in
eqs. (3.4). (3.5). and (3.6): Ln (c, ) versus l/R'l‘a (R = Gas Const.

(kJ/mole°K); Ta = Abs. Temp. (°K)) ....................

Figure 5.12 Residuals for a Simulated Test Case. using Input Temperature
Errors with a Standard Deviation of 025°C. at Four Different

Locations (total thickness equal to 4.5 mm) ................

Figure 5. 13 Temperature Measurements and Applied Heat Flux from Four

Thermocouples in the First 'IYansient Temperature Experiment during

Curing (Exp. 1) ...............................

Figure 5. 14 Temperature Measurements and Applied Heat Flux from Six

Thermocouples in the Second Transient Temperature Experiment during

Curing (Exp. 2) ...............................

Figure 5.15 Temperatures Resulting from the Simulation of the Second
Transient Experiment (3.2) using Thermal Properties Estimated from

Cured Composite at 100°C ...............................

Figure 5. 16 Residuals Associated with Three of the Five Thermocouples
(TC#1. TC#2. and TC#3) Located at the Heated Surface for the Fifth

 

Heat Flux Pulse in Exp. 3.3 ...............................
Figure 5. 17 Temperature Measurements at the End of the 2nd. 5th. 8th.

1 1th, 14th. and 17th Heat Flux Pulses in Exp. 3.3 (cured composites)

as Functions of Thermocouple Location from the Midsection of the

Heater ...............................
Figure 5. 18 Temperature Measurements at the End of the 2nd. 4th. 6th. 9th.

12th. and 15th Heat Flux Pulses in Exp. 3.2 (uncured composites:

second experiment) as Functions of Thermocouple Location from the

Midsection of the Heater ...............................
Figure 5. 19 Temperature Measurements at the End of the 2nd. 4th. 6th. 9th.

1 1th. and 14th Heat Flux Pulses in Exp. 3.1 (uncured composites:

first experiment) as Functions of Thermocouple Location from the

Midsection of the Heater ...............................

39:

ae,

Mir—A.

31"!
t; £2 .C)

t? .c;

r")

3. .r);

f.

.4"

iv

..
;.r.

U"

114

Hm p

l—o.

 

Second

Tcniarj

ae.

30;

ac:

AAA:

B
Bavg

C] re: 9C3

El 9E1 9E3

minim m
fi~

LIST OF SYMBOLS

Primary amine

Secondary amine

Tertiary amine

Pie-exponential factors used in kinetic models

Estimated parameter vector: contains thermal conductivity (W / m°C) and
density-specific heat (J /m3 °C) values

Amine to epoxide equivalency ratio

Average of binary temperature data

rate constants (s-i) in epoxide reactions

rate constants (s-i) in resident epoxide reactions

Specific heat (J / kg°C)

Epoxide consumed in kinetic reactions during curing (moles)
Initial epoxide in kinetic reactions during curing (moles)
Components of the optimal experimental design criterion
Difi‘usion constant

Regression parameters for the diffusion constant. D. as a linear function of
temperature

Epoxide

Activation Energy Constants 0“.” mole)

Heat of reaction (J)

Heat generation (constant) (J / nr3 )

Gain for ith thermocouple (°C)

Total heat of reaction (J)

Resident hydroxyl (impurities)

xxii

j

“NW

11'

W

kl 0k! vk!

m0 om!

m on]

tr

Ea gr a

To

xxiii
Hydroxide group resulting from amine-epoxide addition reaction
Thermal conductivity (W / m° C)
Efl'ective thermal conductivity (W /m°C)
Thermal conductivity of fibers (W / m°C)
Thermal conductivity of matrix (W /m° C)
Regression parameters for thermal conductivity as a linear function of tem-
perature and degree of cure
Thickness (m)
Exponential constants in kinetic models (dimensionless)
Regression parameters for the exponent. m. as a linear function of tempera-
ture
Exponential constants in kinetic models (dimensionless)
Number of experimental time steps
Regression parameters for the exponent. n. as a linear function of tempera-
ture
Marching direction in steepest descent method
Heat fiux (W /m2 )
Gas constants (mole°K/kJ)
Least squares function
Time (sec)
Time at nth time step (sec)
Total heat of reaction (see)
Time at the end of the heat flux in optimal experimental design analysis
(sec)
Temperature ( ° C)
Temperature at boundary where x = L (°C)
Maximum temperature up to time nth time step in optimal experimental
design criterion analysis (°C)
Initial temperature (°C)
Calculated temperatures (vector) (°C)

 

Measured

1‘ Adjusted
if Fiber volt.
I Weighiin.
i Position 1
I Sensitive.
l Measurec
1 'Zero‘ adj.
“ Degree oi
“D Degree 0
“9 Degree of
3 Weightiru
b Elgcni'aii
i parameti
. specific 1
i Paramet
ma] con
3 Density
x? DCDSIty
pep. .pcp, ‘ Regress
9citI “011 of 1
A Optima
3 Matrix

xxiv

V Measured thermocouple voltage (volts)

Va Adjusted thermocouple voltage (volts)

Vf Fiber volume fraction (dimensionless)

W Weighting matrix for least squares function

x Position along the x-axis (m)

X Sensitivity coefficient matrix

Y Measured temperatures (vector) (°C)

2 ’Zero' adjustment for thermocouples (°C)

a Degree of cure (dimensionless)

at) Degree of cure where diffusion becomes significant

up Degree of cure at maximum degree of cure rate (dimensionless)

g Weighting factor in steepest descent method

(3 Eigenvalues

E Parameter vector; contains thermal conductivity (W /m° C) and density-
specific heat (J /m’ °C) values

3 Parameter vector from minimization of least squares function; contains ther-

mal conductivity (W /m°C) and density-specific heat (J /m3 °C) values
p Density (kg/m3 )
pcp Density-specific heat product (J /m’ °C)
pcp. .pcp; . Regression parameters for thermal conductivity as a linear func-

pcp, tion of temperature and degree of cure

A Optimal experimental design criterion
1 ' Matrix derivative operator
Superscript:

+ dimensionless

Subscript:

T top

B bottom

 

 

 

In general. a co
Iain a new material
”431115 to 3330801 2
M applications 1
m w!“ fabricatec

witspmad (13¢ 01

Chapter 1

Introduction

In general. a composite is simply the combination of two or more materials to ob-
tain a new material with some improved or new characteristics. particularly with
regards to strength and weight. The use of composite materials in construction and
vehicle applications has been in efl'ect since the early Egyptian days when bricks and
boats were fabricated from mud fortified with reeds and chopped straw. However. the
widespread use of modern day composites. which utilize new high-performance
materials. has only come into being during the last 20 years. These high-performance
materials are commonly associated with the aeronautics and aerospace industries.
especially with regards to the military. where composites are particularly attractive due
to their generally low-weight/high-strength characteristics. One of the most notable
use of composites was in the design of the all-composite Voyager aircraft. which proved
the strength and durability of the materials in its non-stop flight around the world.

However. the use of high-performance composites is by no means limited to the
acre-related industries The automotive industry has used composites in body and
suspension components. and the boating industry has incorporated composites in hull
and rudder designs. The US. Navy has a particular interest in composites in ship and
submarine designs. Another important industry is the sporting goods industry. which
has produced a multitude of composite golf clubs. tennis rackets. racket ball rackets.
rowing oars. bicycles. skis. and poles for pole vaulting designed to give the athlete. or
would be athlete. the extra edge to hit harder, go faster. and jump higher.

 

 

mandthusbmdtc

n 43 the applk;
W the motivat;

2

Modern day composites are formulated in a variety of ways from many difi'erent
materials. Due to this variability. they are often classified as fiber. particulate. laminar.
flaked or filled (Vinson and Sierakowski. 1987). The fiber-reinforced composites are of
particular interest in this study. These composites are typically composed of strength
enhancing fibers bound together in a relatively low density matrix material. The fibrous
materials include glass. Kevlar. carbon. and boron. and the matrix materials include
epoxies. polymides. phenolics and aluminum. The focus of this study is on continuous
carbon fiber-epoxy matrix composites. The epoxies used in these composites are clas-
sified as thermoset materials since both heat and pressure are required for the epoxy to
cure and thus bind to the fibers.

It is the application of heat on composites both during and after curing that has
provided the motivation for this study. The thermal responses of cured composites to
applied heat loads induce thermal stresses. which can possibly lead to structural
failure. Accurate knowledge of the thermal characteristics of the composites are. there-
fore. very important with regards any design application in a nonisotherrnal
environment. The thermal response of composites is also pertinent with respect to the
design of the curing cycle. In the worst case scenario. a lack of understanding of the
thermal characteristics of the composite during curing could lead to an unevenly cured
product which is structurally unsafe. In the best of worlds. a thorough understanding
of these thermal characteristics would lead to the design of an optimal curing cycle
which maximizes product quality and minimizes cost.

The thermal responses of interest in this study are the result of applied heat
fluxes which lead to conductive heat transfer within the composite. These thermal
responses are characterized by the intrinsic thermal properties of the particular com-
posite material. namely the thermal conductivity and the density-specific heat product.
In addition. during curing. the epoxy matrix composites of interest undergo exothermic
reactions. in which the rate of heat generation is characterized by the kinetic properties
of the composite. It is the determination of these thermal and kinetic properties which
is the focal point of this study. The particular composite used in this study consists of
A54 fibers (Hercules Aerospace Corp.) and with an EPON 828 epoxy matrix (Shell

Charm! Comp). I
posicwith a high 5
mlyonthc dctcrmL

 

both tht anahiical
which are not nccc :
The email obv
nation of thermal
ms um and.
m d thermal prop.
1” Squirts functic
M5 Wimin the cc
mien! and both 1
"it“? Simultane
hm YYPICaHy mcas

3
Chemical Comp.). This is a moderate temperature (<150°C) high-performance com-
posite with a high strength versus weight ratio. The emphasis in this study was not
only on the determination of specific properties values. but also on the development of
both the analytical and experimental procedures for the estimation of these properties.
which. are not necessarily limited in application to the specific composite investigated
here.

The overall objectives of this study can be divided into three major areas: the es-
timation of thermal properties in cured composites; the estimation of kinetic properties
during curing; and. the estimation of thermal properties during curing. In the estima-
tion of thermal properties of cured composites. a minimization procedure involving a
least squares function is employed which utilizes experimental temperature measure-
ments within the composite. This approach is unique in that the experiments are
transient and both the thermal conductivity and density-specific heat properties are
estimated simultaneously. Previous studies (Lee and Taylor. 1975; Ishikawa. 1980)
have typically measured these two properties in composites in independent experi- ‘
ments. This approach has been used. however. for homogeneous materials (Beck and
Al-Araji. 1974: Courville and Beck. 1988: Farnia. 1976). and the emphasis in this par-
ticular instance is on developing an experimental design and on obtaining specific
property values as functions of temperature. The specific objectives are to estimate the
thermal conductivity perpendicular to the fiber axis direction and the density-specific
heat product of cured AS4/EPON 828 composite materials as functions of tempa'ature
and fiber orientation.

The second area of interest is the estimation of kinetic properties during curing.
This is shown to be fundamentally linked to (the third overall objective relating to the
estimation of thermal properties during curing. The kinetic properties are those
parameters which characterize the chemical reactions during curing. and they include
rate constants and activation may constants. These properties are estimated using
established procedures using difi‘erential scanning calorimetry (DSC). Unlike previous
studies (Sourour and Kamal. 1976) which have only focused on the first half of the cure
cycle. the efi'ort of this study was to estimate the kinetic properties which characterize

 

mmhcrmic react
char-ram these re
rimmed to {0110'
thiscasc. efforts wen
bundinthc esiirm
hammto
myfmnDSC data
M m and

4
the exothermic reactions throughout the entire cure cycle. The ldnetic properties which
characterize these reactions typically involve the determination of rate constants which
are assumed to follow an Arrhenius relationship (Barton. 1985) with temperature. In
- this case. efi'orts were directed towards obtaining specific properties values which could
be used in the estimation of thermal properties during curing. The specific objectives in
this area were to estimate the ldnetic properties associated with the curing of EPON 828
epoxy from DSC data using the various kinetic models found in the literature for similar
epoxy systems. and to select the most appropriate model(s) for this particular epoxy
system.

Previous efi'orts for the estimation of thermal properties during curing required
the interruption of the curing process and/ or separate experiments for each property
measurement (Mojovic and Mei. 1987). The focus of this investigation is new and
unique because the procedures permit the thermal properties. namely thermal conduc-
tivity and density-specific heat. to be measured simultaneously throughout the curing
process as functions of both temperature and completeness or degree of cure. The es-
timation problem is difi‘icult in both the analytical and experimental aspects.

The proposed procedure involves the minimization techniques utilized in this
study for the estimation of the thermal properties of cured composites: however. this
analysis is complicated by the exothermic chemical reactions occurring during curing.
This estimation procedure also required transient temperatures measurements. which
necessitated the development of new experimental designs applicable during the curing
process. Duetothecomplexityofthisprobleminbothanalyticalandexperimental
aspects. the focus of these efi‘orts were on the development of the estimation proce-
dures. rather on obtaining specific property values. Therefore. the specific objectives in
thiscasearefirsttodevelopandverifytheanalytiealproceduresforthe estirnationof
thermal conductivity and density—specific heat during curing. This would incorporate
the parameter estimation techniques utilized in this study for the estimation of thermal
pmperUesdcumdcanpositesandthemsunsofthekimticamlysisoftheEPONBZB
epoxysystem. Thesecondobjectiveinvolvesthedevelopmentandtestingofanap-
propriate experimental design for use in recording transient temperature measurements

duringcum The
himtiomdtanpec

 

5

during curing. The final objective is to. if possible. measure specific property values as
functions of temperature and degree of cure using this experimental design.

Chapter 2

Literature Review

A review of the literature for the estimation of thermal and kinetic properties in
composite materials is given in this chapter. In the first section. the emphasis. is on the
estimation of thermal properties in cured composites. There have been marry studies in
this area with regards to both theoretical and experimental methods. The next section
is devoted to the techniques used in estimating kinetic properties of epoxy-based com-
posite systems during curing. These methods utilize differential scanning calorimetry
to estimate rate constants which characterize the curing of the composites. The tech-
niques which have been used to estimate thermal properties during the curing process
are presented in the third section. This is followed by a final section on minimization
methods which have been used to estimate thermal properties in homogeneous
materials. These methods provide a means of coupling experimental measurements
with analytical solutions. and they allow for the simultaneous determination of both
thermal conductivity and the product of density and specific heat.

2. 1 Estimation of Thermal Properties of Composite Materials

The methods used for the estimation of thermal properties in cured composite
materials have included both experimental and analytical techniques. A synopsis of the
techniques used to estimate thermal properties in composite materials and other re-
lated anisotropic materials is first presented in Section 2.1.1. This is followed by a

W a’ previous
fiber-Matted corr

 

 

that It
calls the CE

ucuwues

7
review of previous research devoted to the estimation of thermal properties of carbon
fiber-reinforced composites. in particular.

mu: .. .1 I. m. at... t .e . cums :00 ei.’ .. xoeuo H. r...

The problem of estimating thermal properties. such as thermal conductivity. in
anisotropic composites is interesting and challenging in that not only is the directional
aspect of importance. but also because composites. by definition. are composed of at
least two different types of materials (Le.. the fiber and matrix materials). with difi‘ering
thermal properties. The existing literature on this subject indicates that research has
been concentrated in two difi'erent areas for the solution of this problem.

The first area is focused on the determination of the effective or overall thermal
properties from a mathematical analysis of the components of the composite. The es-
timation of thermal conductivity. in particular. has been the main focus of the studies
in this area. Most of these methods are based on the assumption that the thermal con-
ductivities of the individual composite elements. namely the matrix and the fiber. are
known. along with the void fraction of the fibers. These models are based on the
original theories by Maxwell and Rayleigh (I-Iasselman and Johnson. 1987). Several
such models for the efi'ective thermal conductivity of composites were proposed by
Hasselman and Johnson (1987). These models also required knowledge of the fiber
diameter and the interfacial thermal barrier resistance between the fiber and the
matrix. Ziebland (1977) presented a simpler approach in which the efi'ective thermal
conductivity was calculated using a unit—cube model with rectangular coordinates. The
model for the heat flux perpendicular to the fiber length (fiber and matrix in series) is

expressed mathematically as

k k
k ___ m f
eii‘ Vrkm+ (l-Vl.)kf

 

where kc“ is the efi‘ective thermal conductivity of the composite. km and RI are the ther-
mal conductivities of the matrix and the fiber. respectively. and Vf is the fiber volume

mica Ziebland .
at {1987) also used

 

Cmibmmfl this app
a gmaalizcd crpresll
tmions. given th_
the am 3mm}!
mama) conductivit}
l! and two-dirtCUch
We thermal concl
m analysis of 3
Ma) methOd
W With the met
N: in this pr
Wt SYSICm us

8

fraction. Ziebland also included models for cross-plied and fabric laminates. James et
al. (1987) also used the unit-cube approach. but with a cylindrical model for the fiber.
Combining this approach with a finite difi'erence model for heat flow within composites.
a generalized expression was obtained for the efi’ective thermal conductivities in ofi-axis
directions. given the efi'ective thermal conductivity parallel and perpendicular to the
fiber axis. Similarly. Mottram and Taylor (1987b) developed expressions for effective
thermal conductivity parallel and perpendicular to the fiber axis. which included both
one and tw0odirectional composite laminates. Han and Cosner (1981) analyzed the ef-
fective thermal conductivity of composites with difi‘erent void fractions. using a finite
element analysis of a unit composite cell. with heat flow perpendicular to the fibers. A
numerical method was presented by Dul‘nev et al. ( 1977) which combined Rayleigh’s
method with the method of grids to estimate the efi'ective thermal conductivity of a com-
posite. Inthisprocedure.eachblockdefinedbyagridwasanalyzedasatwo
component system using Rayleigh’s approach.

The other focus of attention for the determination of the thermal properties has
been on experimental methods. In the estimation of thermal conductivity. most of these
experimental methods are conducted at steady state conditions. Most of these methods
involve one or two thin samples which are placed in between heated or cooled plates.
Ziebland (1977) describes these methods as absolute. where thermal conductivity is
determined solely from and energy balance of the steady state conditions. or relative.
where thermal conductivity is determined by reference to a substance of known thermal
conductivity. Three of the absolute methods mentioned by Ziebland are the flat plate
apparatus with a heater guard. which uses one flat test specimen. the coaxial cylinder
cell with heater guard. which uses a cylindrical test specimen. and a two-plate ap-
paratus. which uses two test specimens and is generally preferred. Harris et al. (1982)
used the later method in estimating the thermal conductivity of Kevlar fiber-reinforced
composites as a function of temperature and fiber direction. The relative methods in-
volve apparatus similar to the two-plate apparatus. but they are more complex in their
construction. Quasi-steady state experiments. using similar apparatus. were also dis-
cussed by Ziebland for materials with low thermal conductivity.

 

Another tech.
capacity. and the
{1961). In this me
with a coated Wit
Pm”)? response
The thermal diliusi
wreath one half of
him the maximum
damned from
W was used
’htrmal diilusivuy
mam Samples as

9

Another technique for the experimental determination of thermal difi’usivity. heat
capacity. and thermal conductivity is the laser-flash method presented by Parker et al.
(1961). In this method. a laser pulse is directed on a sample a few millimeters thick
which is coated with carbon to prevent direct transmission of the beam. and the tem-'
perature response of the specimen at the rear surface is monitored by optical means.
The thermal diffusivity is then determined from the time required for the back surface
to reach one half of the maximum temperature rise. and the heat capacity is detennined
from the maximum temperature rise of the specimen. The thermal conductivity can be
determined from the thermal difi'usivity. the heat capacity. and the density. This
method was used by Brennan et al. (1982) to determine the thermal conductivity and
thermal diffusivity of thin amorphous silicon carbide fibers from composite and neat
matrix samples as functions of temperature. and by Mottram and Taylor (1987a) to
determine the thermal diffusivity of silica-phenolic composites.

Other methods have also been considered in the determination of thermal
properties. Beck and Al-Araji (1974) presented a method of estimating thermal conduc-
tivity and density-specific heat independently from a single transient experiment.
Schimmel. Jr. et al. (1977) presented a solution for the determination of the effective
thermal conductivity of a multi-layer laminate based on an analytical solution for a two-
layer composite transient system with known thermal properties of the components. A
method of estimating thermal conductivity in unidirectional materials which is
analogous to the problem of longitudinal shear loading of a unidirectional composite.
where the thermal conductivity is analogous to the average shear moduli. was proposed
by Springer and Tsai (1967). This method was compared to a thermal model in which

the filament and matrix were assumed to be arranged in series.

 

Most of the previous research on the estimation of thermal properties. such as
thermal conductivity and thermal difi‘usivity. of carbon fiber-reinforced composite
materials has been for cured composites. The methods used in the determination of

mummies int
inchidedboth expe I
The law-Has

 

along with an abso
them) diffushity '
5052 and graphite i
thermal diffusivities
Mummies tstim
tried the effects
me111311011. and Spa
We using a:
W by the libs
WI d the compo
hitter-mm titer-m;

10
these properties included some of the techniques described in Section 2.1.1.. and they
included both experimental and analytical aspects.

The laser-flash method (Parker et al. 1961) was used by Lee and Taylor (1975)
along with an absolute method. which required cylindrical samples. to determine the
thermal difi'usivity and the efi’ective thermal conductivity of carbon fiber-reinforced (PX
505) and graphite fiber-reinforced (Morganite II and Thornal 50 S) composites. The
thermal difi‘usivities of the graphite fibers were determined from the efi'ective thermal
conductivities estimated using the absolute method. Taylor and Kelsic (1986) inves-
tigated the effects of fiber fraction. fiber/ matrix thermal conductivity ratio. fiber
orientation. and specimen length on thermal diffusivity of unidirectional fiber-reinforced
composites using the laser-flash technique. They found that the conductivity ratio.
followed by the fiber volume fiaction were the major factors affecting the thermal be-
havior of the composite. Mottram and Taylor (1987a) also used the laser-flash method
to determine thermal difi'usivity of woven and unwoven carbon fiber-phenolic resin corn-
posites as a function of temperature. This work was continued (Mottram and Taylor
(1987b). and the thermal conductivity of the carbon-phenolic composite was determined
in difi'erent fiber directions using the Maxwell-Rayleigh approach.

Ishikawa (1980) analytically determined the thermal conductivity of carbon/ epoxy
composite systems under steady-state conditions using a Fourier series analysis based
on the solution of the longitudinal shear problem. This is the same approach used by
Springer and Tsai (1967). This solution was two dimensional. and. it required
knowledge of the thermal conductivity of the matrix and the fiber volume fraction. The.
analytical results were compared with experimental results using a technique called the
infrared radiation method.

2.2 Estimation of Kinetics Parameters in Epoxy-Based System
The kinetic parameters of importance in this section include the rate constants

and activation energ constants which characterize the curing of amine-epoxy based
composite systems. A discussion of the cure kinetics of amine-epoxy systems is first

 

presented. follower
mus studies in

am these studie
shimmy. Final

composite systems

1 1
presented. followed by review of difi'erential scanning calorimetry. A summary of pre-
vious studies in the estimation of kinetic parameters in amine-epoxy systems is then
given; these studies all utilize experimental results obtained using difi'erential scanning
calorimetry. Finally. a brief discussion is included on the use of neat vs. reinforced
composite systems with regards to the estimation of ldnetic parameters.

214W

The epoxy system of interest in this investigation is an amine-epoxy system in
which the initial reaction is accelerated due to autocatalyzation. while in the latter
stages. the reaction is difi'usion controlled. Due to the complexity of the system. re-
searchers have used relatively simple models over limited ranges of conversion to
describe the kineties of the reactions. '

Most of the reaction models for the initial phase of the amine-epoxy cure are
based on the mechanisms proposed by Schechter et al. (1956) and Smith (1961). The
rate determining step of the reaction scheme is assumed to be the reaction of amine.
epoxide and hydroxyl (or other proton—donor species. l-IX) (Barton. 1985). This scheme
is a two tiered process. which involves reactions of the hydroxyls formed in the amine-
epoxide reactions. Hxa. and the resident hydroxyls in the system (impurities or
additives). onz

a. «t-e-el-lx‘l

a. +e+l-Ixog' a, +l-IX

t9

a3+e+HXa a,+l-IXll

O
U.

a; +e+HXo-e a, +Hx°

In these reactions. 8. . a, . and a, are primary. secondary. and tertiary amines. e is the

epoxide. and the ciand c; values (1 = 1. 2) are the reaction rates. From this scheme. the

following overall kinetic equation was derived (Barton. 1985):

 

rideCBtheepm
mmol'HXoandth
primaryandsecon

Wali'
Diflerential

Witter d the

W in Whicti the I
3" “Mai to a (
DSC has Droven to
m as “1056 Occu
is “W 3.331
the W substa
M "1 eq. 2.1).

12

fl? = (cl Co +ci Clteo -C)(a. +33) (2.1)

where C is the epoxide consumed at a given time. t; C. and e. are the initial concentra-
tions of l-D(o and the epoxide. respectively; and. a. and a; are the concentrations of the
primary and secondary amines.

W
Difi’erential scanning calorimetry (DSC) has been defined by the Nomenclature

Committee of the International Confederation for Thermal Analysis (ICTA) as a tech-
nique in which the difi'erence in energy inputs into a substance and reference material
are subjected to a controlled temperature program (Barton. 1985). The technique of
DSC has proven to be a useful method for the investigation of exothermic reactions.
such as those occurring during the curing of epoxy resins (Fava. 1968; Rabek. 1980).
The underlying assumption in DSC kinetics is that the difi'erence in heat flow between
the reacting substance and the reference material is proportional to the reaction rate
(dC/dt in eq. 2.1).

There are two types of DSC instruments. The first operates under a power com-
pensation mode which utilizes separate sample and reference holders with individual
heaters and platinum thermometers. The second type operates in a heat flux mode; the
sample and reference are in separate containers connected by a controlled thermal
resistance with external thermocouples. The DuPont Instruments 910 Difi‘erential
Scanning Calorimeter cell. used in this study. operates in the heat flux mode. This cell
consists of a sample chamber in which the empty reference and sample pans rest on an
enclosed platform instrumented with thermocouples. In using both the power compen-
sation and the heat flux DSC instruments. it is assumed that the temperature gradients
within the sample are very small. This requires the use of small samples (on the order
of milligrams) and the assurance of good thermal contact between the sample and the
sample pan. and the same degree of thermal contact between the sample pan and the
platform. and the reference pan and the platform.

M exptrrlrr.i
kinetic modeling 0

 

areconducted at d
Slamming an 8;
hipenture studie
Imttil't can be d
their a sample is
Um referred to as
itthtscase. both
mail to the r
M Infitted to Very
“111 We? kinetics).

13

TWO experimental methods involving the use of DSC are commonly used in the
kinetic modeling of exothermic systems. In the first instance. isothermal experiments
are conducted at different temperatures. and the rate of heat flow is recorded with time.
By assuming an appropriate kinetic model. rate constants can be determined for each
temperature studied. and then the relationship between these rate constants and tem-
perature can be determined. The second method involves dynamic experiments in
which a sample is heated at a linear rate throughout the curing process. This is some-
times referred to as the Borchardt and Daniels method (Borchardt and Daniels. 1957).
In this case. both the time and temperature dependence of the conversion rate (which is
proportional to the reaction rate) may be found simultaneously. Use of this method has
been limited to very simple kinetic models in which only one rate constant is found (1. e..
nth order kineties).

W
Several researchers have used a generalized form of eq. (2.1) to identify rate con-

stants of a bisphenol-A-diglycidylether (BADGE) and meta-phenylenediamine (mPDA)
epoxy system. which is similar to the epoxy investigated here. The kinetic model con-
tains two rate constants. consistent with the mechanisms presented by Barton (1985).
The first rate constant. c. . represents nth order kinetics. while the autocatalytic nature
of the reaction is evident by the second rate constant. c; . Therefore. if no impurities
are present. c. is equal to zero. and the reaction is completely autocatalyzed. The
generalized kinetic equation is

git! a (c. + c. om)“ - a)" (2.2)

where. o is the fractional conversion. and m and n are constants. Ryan and Dutta
(1979) used this model to determine the rate constants c. and c. from maximum reac-
tion rate data from isothermal DSC experiments. assuming a second order reaction (m +
n = 2). An Arrhenius relationship was assumed between the rate constants and tem-

perature:

this relationship
illusion controllec
Equation [23
stellar epoxy set
in an ten-iglycmy}.
tied the model in
la studying Mint
“9873 used this n
d “ > 53%. Hagn
in 3th them

‘3 an earlier

14
c1 = Alexp(~E‘/Rl‘) i= 1. 2 (2.3)

This relationship was shown to follow conversion rates up to 90%. with no evidence of a
difi'usion controlled reaction.

Equation (2.2) was also used by several researchers to describe curing in other
similar epoxy systems. Mijovib et al. (1984) used this model to determine rate constants
in an tetraglycidyl-4.4’diminodiphenylmethane epoxy system. and Moroni et al. (1986)
used this model in studying BADGE and 1.4 butanedioldiglycidylether resin mixtures.
In studying curing kinetics in BADGE/ethylenediamine epoxies. Chem and Poehlein
(1987) used this model and found that difi'usion controlled the reaction rates for values
ofa > 58%. Hagnauer et al. (1983) also used this model to investigate the cure kineties
in glass fiber-BADGE epoxy systems.

In an earlier study. Sourour and Kamal (1976) assumed m = n = 1 and used

31% = (c. + e.. or)“ - ot)(B - a) (2.4)
where B is the amine to epoxide equivalency ratio. The rate constants were determined
using isothermal DSC experiments. and the activation energy constants were found
assuming an Arrhenius relationship. This relationship followed for values of a up to 40
or 50%: at that time. the reaction rate decreased. indicating a diffusion controlled reac-
tion for conversion rates greater than 50%. Likewise. Lee et al. (1982) used this model
in studying Hercules 3501-6 resin and found that the reaction was diffusion controlled
for a > 30%.

Prime (1973) implemented a simpler equation to describe curing in the
BADGE/mPDA system incorporating both isothermal DSC experiments and dynamic
DSC experiments which utilized the Borchardt and Daniels (1957) method. In this

model. and nth order reaction is assumed with no autocatalyzation (c, is zero):

‘3‘: = c. (1 -a)“ (2.5)

Doiatiom from it
trimmed to I
instigation of thi

model to investigat
Very few or
analysis. lee et

thanSO‘lhintheirs

its -

dt ‘
“43R. ct was a:
31958) DI‘Isented

3‘3 Wont-rim] t.

15

Deviations from this model were found for higher conversion rates. where the reaction
was assumed to be difi'usion controlled. Acitelli. et al. (1971) found similar results in an
investigation of the cure of BADGE/mPDA epoxy. Pappalardo (1977) also used this
model to investigate glass reinforced-BADGE epoxy prepregs.

Very few models have been developed which incorporate difi'usion in the kinetic
analysis. Lee et al. (1982) presented the following equation for extent of cure greater
than 50% in their study of Hercules 3501-6 resin:

3% = C, (l ' a) (2.6)
where. c, was assumed to follow an Arrhenius relationship (eq. 2.3). Hawley et al.
(1988) presented another model for conversion rates greater than 50%. which included

an exponential term indicative of the diffusion phenomenon. This model is

g? = c. (1 - a)e'°°‘ (2.7)
In this equation. D is a difi‘usion coefficient. and once again. the rate constant. c, . was

assumed to follow an Arrhenius relationship.

.1‘ . -Q' .. ~ . \'-_ - ., He 0'13! or -- 010.3 - ‘19,:

There have been a few studies on the comparison of the cure kinetics of neat ver-
sus reinforced epoxy systems. Mijovit: and Wang (1989) studied the effects of carbon
fibers _ in a tetra-N-glycidyldiaminodiphenylmethane (TGDDMVdianimodiphenyl-
sulphone (DDS) amine-epoxy system. and found that the presence of carbon fibers had
only a very small initial effect on the kinetics of cure. In a related study. Myovib (1986)
compared the cure kinetics of neat versus glass reinforced TGDDM/DDS epoxy. It was
found that in the reinforced system. the reaction rate constants were slightly lower. and
the time required to reach the maximmn reaction rate was slightly longer. These minor
differences were assumed to be the result of restricted mobility due to the reinforce-

merit.

timation ck
MI!

 

 

lhe estimati
pier due to the ex
tsthnate thermal
stood with regard
emulate heat tram
“"1" 0f the pi
iiiiicii to date is m

16
2.3 Estimation of Thermal Properties in Carbon/Epoxy Composite System during
Curing

The estimation of thermal properties in composite materials during curing is com-
plex due to the exothermic reactions which occur as the epoxy matrix cures. In order to
estimate thermal properties during curing. the curing process itself must be first under-
stood with regards to the heat transfer process. The mathematical models used
simulate heat transfer during the curing process are first reviewed. This is followed by
a review of the previous work in the estimation of thermal properties during curing.
which to date is quite limited.

U ., even: - = u 0.0 - . m _: . = N: e -. e .m- e an...“ om - .4.-
W

' Several researchers have developed mathematical models to simulate heat trans-
fer in cured composite materials (e.g. Golovchan and Artemenko. 1987). but few have
developed models to describe heat transfer in composites during the curing process.
Loos and Springer (1983) developed models to describe the curing process of composites
constructed from continuous fiber-reinforced. thermosetting resin matrix prepreg
materials. A computer program was developed which provides the temperature dis-
tribution. degree of cure of the resin. the resin viscosity inside the composite. the void
sizes. and the residual stress distribution given a specific process for a flat plate com-
posite. In the mathematical model for temperature. a term was added for the heat

generated by chemical reactions which was proportional to the rate of degree of cure.
More commonly. researchers have focused on modelling another process which is
mathematically related to the curing process. namely decomposition. Both processes
have added terms in the energy equation which are proportional to the chemical reac-
tion rate for the process (curing or decomposition). Tant et al. (1985) developed and
tested a mathematical model including a decomposition term for the thermochemical

expansion of polymer composites during pyrolysis. Henderson et al. (1985) developed a

model for the tilt
tion assuming 0!“
decomposition.

and Wieoek (1987*
pastor). Grillis
respome of carbo

decomposition

 

l 7
model for the thermal response of a polymer composite material undergoing decomposi-
tion assuming one dimensional heat transfer and nth order kinetics for the rate of
decomposition. Using a similar approach. Boyer and Thomas (1985) and Henderson
and Wiecek (1987) investigated chaning in composites in terms of thermochemical ex-
pansion. Griffis et al. (1981) developed a finite difi'erence model for the thermal
response of carbon epoxy composite subject to rapid heating. including the efi‘ects of

decomposition.

 

While there have been many studies on the estimation of thermal properties on
composites. there have been relatively few studies on the estimation of thermal
properties of composite materials during curing.

Mijovib and Mei (1987) designed an apparatus for the measurement of thermal
conductivity of TCDDM / DDS epoxy carbon fiber composite materials. Using this ap-
paratus. a very thin section (1 mm) composite sample was cured between two parallel
plates. allowing for the assumption of a linear temperature distribution across the
thickness of the sample. Thermal conductivity was determined as a function of time for
different isothermal curing temperatures. In a related study. Mijovit: and Wang (1988)
determined the density and heat capacity (not thermal conductivity) of the same com-
posite during curing. To measure density. samples were cured isothermally at different
temperatures for difi'erent times and then quenched to prevent further curing. The den-
sity of the partially cured samples was then determined using a density gradient
column. The heat capacity of the samples during curing was measured using difi'eren—
tial scanning calorimetry (DSC). In this procedure. the reference DSC pan contains a
substance of known specific heat (DuPont Company Instrument Systems. 1985).

2.4 Minimization Methods used for the Estimation of Thermal Properties

One means of estimating thermal properties involves the minimization of an objec-
tive function. such as a least squares function. This is a powerful estimation method.

but it has been
oater‘ah. There
om used meth
tailgate gradient
mg the appropria:

estimated. A brief

iii—Thirties;

The Gauss it
Arnold (1977) desc
minim Which are
MW Rgion C

in this metht
hm Meters

—_—_i_

Am squats film

 

18
but it has been primarily used in the estimation of thermal properties in isotropic
materials. There are several difi'erent minimization methods; three of the more com-
monly used methods are the Gauss method. the steepest descent method. and the
conjugate gradient method. In each case. the objective function is minimized by select-
ing the appropriate direction and step size for perturbations in the parameters being
estimated. A brief discussion of each of these methods is now presented.

W
The Gauss method is one of the more popular minimization methods. Beck and

Arnold (1977) describe this method as being relatively simple and effective for seeking
minima which are reasonably well-defined. assuming that the initial estimates are in
the general region of the minimum.

In this method. a least squares function is minimized with respect to the un-
known parameters (thermal properties) and the resulting expression is set equal to zero.
A least squares function given by Beck and Arnold (1977) is

s = w-runfwnr-rum (2.8)

where the vector Y contains measured temperatures. 3, is the parameter vector. the vec-
tor T contained temperatures as a function of 5,. calculated from the appropriate energy
equation for the system being studied. and W is a weighting matrix. It is assumed in
this analysis that the first derivatives of T are continuous in i and that the higher
derivatives are bounded.

When the derivative of the least squares function is set equal to zero. B. = 3,. the
true minimum of the parameters. This minimization procedure results in terms con-
taining 113,) and the derivatives of Tm) with respect to the parameters in 5,. namely.
EQTTQJIT. which is defined as x61); In the Gauss method. two approximations are made.
First. 1(3) is replaced by X(b). where the b vector contains the estimated parameters.

and secondly. TL?) is approximated by the first two terms of a Taylors series about b.

 

ma approximat
needed given an
Bock (1966)
specific heat of it
used this method .
202M351 alumin

The steepest
tidbit oi the obje

 

m tht magthdc I

fill.”

19

These approximations result is a linear equation in 3, which can be solved. iteratively if

needed. aven an initial estimate of the parameters in b.
Beck (1966) used this method in estimating the thermal conductivity and density-

speciflc heat of nickel from transient temperature measurements. and Farnia (1976)
used this method in estimating time and temperature dependent thermal properties of

2024-1351 aluminum alloy.

W

The steepest descent method is an alternate approach to the minimization
Problem (Huag and Arora. 1979). Here. the marching direction is equal to the negative
gradient of the objective or least squares function (Lamm. 1989)

,m = - visa“) (2.9)
and the magnitude of the change in Q in is found from
50“” = Em + cm?“ (2.10)
where “(kl is a weighting factor such that
= llp‘k’l lz/l la'r/anl I2 (2.11)
Where I I ' I I denotes the norm for a vector. f. with p components: it is defined by

lltlLLEei‘] ]1/2

The superscript 1:. eq. (2.9). (2.10), and (2.1 1) indicates that this is an iterative process.

 

The

coniugu.
has been used tc
1981, and Th1. 19+
and hem (1981)
posedpmblems. c<
(1979] aka used 1)

mm! Conductivi'

 

20
WW
'I‘lie conjugate gradient method is similar to the steepest descent method. and it
has been used to solve both inverse heat conduction problems (Alifanov and Kerov.
1981 . and Tu. 1988) and optimal design problems (Huang and Arora. 1979). Alifanov
and Kcrov (1981) described the method as having good characteristics for solving ill-
posed problems. compared with the method of steepest descent. Alifanov and Mikhailov
(197 9) also used the conjugate gradient method in the solution of the nonlinear inverse
thermal conductivity problem.
In this method. the marching directions. PG). for the unknown parameters in 5
are found using an iterative sequence of the form (Huag and Arora. 1979)

1"” = pa“) -y_fism“) (2.12)

The magnitude of the change in find is again found from eq. (2.10).

WWW

The minimization methods mentioned above involve the determination of the
matrix derivative of the least squares function. S. or the temperature. T. with respect to
the unknown parameters in n. 'lhese derivatives can be found directly. by using finite
Merences (or elements). or by using the adjoint method (Tu. 1988).

In the direct method. the matrix derivative of S is carried out through matrix dif-
ffirentiation. and the derivatives for T are found by difl'erentiating the appropriate
go‘nu'ning equations used to solve for temperature by n. and solving the resulting equa-

tion analytically or numerically. If finite differences are used to determine the
t’~'-ull)er‘ature derivative. a simple forward difference expression can be used. These tech-

In(Ines are discussed in detail by Beck and Arnold (1977).
The adjoint method has been used in design analysis (T 0110113111 et al.. 1989) and
in the solution of the inverse heat conduction problems (Alifanov and Rumyantsev.
1987). This method involves the solution of an adjoining set of equations. in addition to
the governing set of equations. These adjoining equations or adj oint equations. as they

at: sometimes ca
ml condition.
backwards in time

where temperature

 

2 1
are sometimes called. are solved backwards in time: that is. instead of requiring an
initial condition. a terminal condition is required. and the adjoint solution marches
backwards in time. Interface conditions. which guarantee continuity at the interfaces
where temperature measurements are made. are also included in the adjoint equations.

Chapter 3

Theoretical Considerations

In this chapter. the theoretical development of the analysis used in estimating
“Bl-trial and kinetic properties of continuous carbon-fiber/epoxy-matrix composite
materials is presented. These materials are orthogonal due to the presence of the
fibe1~s, resulting in more than one component of thermal conductivity. Due to the com-
lety of this problem. this study is limited to the estimation of the thermal
conductivity perpendicular to the fiber axis. The thermal properties estimated here are
fiective properties of the composite. and not of the individual fiber and matrix com-
Ponents.

In the first section of this chapter. the techniques used to estimate the thermal
properties of cured composite specimens as functions of temperature are discussed.
These techniques incorporate the Gauss minimization method (Section 3.4.1). The
Se(tend section is devoted to theoretical considerations used in the estimation of kinetic
Properties which characterize the chemical reactions occurring during the curing
Process. The theoretical methods used in estimating thermal properties as functions of

terIlperature and extent of cure during curing are outlined in the following section.
These methods incorporate the techniques discussed in the first section and the results
0f the analysis presented in the second section. A final section is presented on the
development of an optimum design criterion to mazdmize sensitivity in the estimation of

the thermal properties.

22

 

8.1 Militia) (_

lhedirecip

heat transfer thm

 

23
3. 1 Estimation of Thermal Properties in Cured Composite Materials

'I‘he direct problem of determining the temperature field resulting from conduction
heat transfer through a composite specimen is inherent in the procedure used for the
estimation of the thermal properties of composite materials. In this section. the direct
problem is first defined. and then the solution procedure used for the estimation of the

thermal properties is discussed.

Wanna»
One dimensional heat conduction was considered through a cured carbon-
fiber/ amine-epoxy composite flat plate. For the case where heat conduction due to an

1mI><>sed heat flux on one surface is perpendicular to the fiber axis and the temperature
at the other surface is known. the system can be expressed mathematically as

.L BI - .
ax[kmax] " pop at 0<X<L. t>0 (3.1a)

With the boundary conditions.

mag-1: = q(t) x = o; t > o (3.111)
'I‘ = TL“) x = L; t > O (3. lo)

and the initial condition.

T=T.(x) Osst; t=0 (3.1d)

Where x is the direction perpendicular to the fibers through the thickness. and L is the
nlktknees of the plate. In this case. the thermal conductivity. 1:. is the effective thermal
conductivity of the composite perpendicular to the fiber axis. It was assumed that both
the efl'ective thermal conductivity and the efi'ective density specific heat product. pcp.

were functions 01
tions mt knovml
dpositiou i

the probler.
ttttt distribution
properties. and ti.
loathed here is
W section

 

 

24

were functions of temperature: the heat flux. q. and temperature. Tr: boundary condi-
tions were known functions of time: and. the initial condition. 1‘. . was a lmown function
of position.

'l‘he problem described in eqs. (3.1a-d) can be solved numerically for the tempera-
ture distribution using the Crank-Nicolson finite difi'erence method. given the thermal
properties. and the initial and boundary conditions. The solution of the direct problem
described here is inherent in the estimation of the thermal properties discussed in the

following section.

Wm:

One method of estimating thermal properties involves the use of the Gauss mini-
mlzation method. This method requires the solution of the temperature distribution as
described in Section 3.1.1 and experimental temperature measurements within the
col171I>osite specimen. The estimation method is based on a least squares function. 8.
which can be expressed mathematically as (Beck and Arnold. 1977)

S = 5(2- rmfir - run) (3.2)

where the Y vector contains the measured temperature values. the ‘1‘ vector contains
caJittulated temperature values. and the vector 3 contains the 'true' parameter values of
thi-tr‘rnal conductivity and density-specific heat. Given the appropriate experimental
c<3nditions. the calculated temperatures can be found from the solution of eqs. (3.1a-d).
In the Gauss method. as discussed in Section 3.4.1. the least squares function in

eq. (3.2) is minimized with respect to a resulting in
V38 = 2[-X(b)l[Y -T(b)l = 0 (3.3a)
Where the sensitivity coefi'icient matrix. X(b) is defined as (VBTTMIT. and the vector 1)

Contains the estimated parameter values. This equation can be rearranged to solve for
b 11ng an iterative scheme (Beck and Arnold. 1977). where

in this equation.
protous iteration
tains values of (ex
found using a for
P33111668 is grca
would be obtained
6). l3.3b).)

Anmsimgc
Minimum.

 

 

25

bur”) = (i) + (x“1)x“) rl‘x'l‘li)" _ 1“) )l (3.31))

In this equation. 1 indicates the iteration number. b“) contains the parameters at the
previous iteration. bu”) contains the new estimates for the parameters. and ‘1‘” con-
tains values of temperature calculated from hm. The sensitivity coefficients in i" are
found using a forward difi'erence approximation for X(b) about b“). (If the number of
parameters is greater than about three. the solution of the normal equations. eq. (3.3a).
would be obtained by solving eq. (3.3a) directly rather than using the inverse implied in
“L (3.3b).)

An existing one dimensional parameter estimation program based on this proce-
dure- PROPID (Beck. 1987). was used in the estimation of thermal properties in
carbon/epoxy composite materials. This program utilizes the concept of sequential
estimation (Beck and Arnold. 1977). in which the parameters are evaluated at each
(measurement) time step. This has one advantage in that one can observe the effects of
addltlonal data on the estimates of the parameters to evaluate the adequacy of the ex-
Pel‘imental design. Ideally. at the conclusion of an experiment. there should be no
Change in the parameter estimates with time: that is. any additional data would not
efi‘ect the parameter estimates.

This program was used specifically to estimate the effective thermal conductivity
perPendicular to the fiber axis and the efi'ective density-specific heat of cured carb-
°nl epoxy composites at difi'erent temperatures. The measured temperatures. Y. were
c’btained from experiments conducted at different temperatures as described in the ex-
Pel‘imental procedures in Chapter 4. The resulting estimates of these efi‘ective thermal

Properties at the difi'erent temperatures were then used to determine relationships be-
tWeen the thermal properties and temperature.

3.2 Estimation of Kinetic Parameters

In order to estimate thermal prOperties during curing. the heat generation due to
the exothermic reaction of the epoxy must first be characterized. This analysis is based

on the assumptio
(ate. thus enabitn
illicit parameter:
prom several
separated into (Wt
one less than 5C
greater than 50%).

Would;
Wilmette)

9m at! present

 

26
on the assumption that the heat of reaction rate is proportional to the degree of cure
rate. thus enabling the use of difi‘erential scanning calorimetry (DSC) in determining the
kinetic parameters which characterize the reaction. Due to the complexity of the curing

process. several difi'erent kinetic models were investigated. these models can be
separated into two groups: 1. limiting reaction rates due to autocatalyzation (degree of
cure loss than 50%): and. 2. limiting reaction rates due to diffusion (degree of cure

greater than 50%).

W

The kinetic models investigated for the initial curing phase of an amine-epoxy
System are presented in eqs. (2.2). (2.4). and (2.5). and are again listed below.

'3? = (c. +c.or"‘)(I-ui)n m+n=2 (3.4)
g: = (c. +0. a)(1 - ans - a) (3.5)
f}: = c. (1 -01)“ (3.6)

In all cases. the rate constants. c‘. are assumed to follow an Arrhenius relationship with

tenlperature:

c1 = Aiexp(-E1/R(T+273.15)) i= 1. 2 (3.7)

Where. in this case. the temperature. T. is °C.
An autocatalyzed reaction was assumed in the models given by eqs. (3.4) and
(3-5). and an nth order reaction was assumed in the model shown in eq. (3.6). By re-
qmring the exponents m and n in eq. (3.4) to sum to two. a second order reaction is
knIllicitly assumed in this model. The parameters to be estimated in each case are the
activation energy constants. Ei. and the pre-exponential factors. A . and in addition. the
two exIDOnents. m and n. are to be determined for the models shown in eqs. (3.4) and

(3.6. respectively.

 

typically measure:
toupentures.

The underly
the cumulative he
discussed above r
tine (assuming 3:

dia as follows:

ghofcure

27
(3.6). respectively. In the analysis of each of these models. the heat of reaction data are
typically measured using difi'erential scanning calorimetry (DSC) for several isothermal
tmperatures. '

The underlying assumption in utilizing the DSC data for kinetic analysis. is that
the cumulative heat of reaction is proportional to the reaction rate. The kinetic models
discussed above require the fractional conversion (or degree of cure). a. as a function of
tine (assuming an isothermal cure). which can be obtained from the heat of reaction
data as follows:

- _9___ (3.8)

where H(t) is the heat of reaction at time. i. measured during isothermal DSC experi-
ments. and llt is the total heat of reaction from dynamic DSC experiments. given by

it“
Ht = oHindi (3.9)

Where t. is the total curing time. Dynamic DSC experiments are those in which the
temperature is increased linearly with time throughout the entire cure. It is assumed in
“Sing this formulation that the total heat of reaction is independent of the curing tem-
Pel’ature. even though at lower curing temperatures complete curing may not be
acl’lle-ved.

Kinetic parameters were determined for the first model (eq. 3.4) using two dif-
ferem methods. In both procedures. a second order reaction was assumed. so that
rn + 11:2. and the parameters estimated were AI . A. . E. . E. . and m. The first
method to be described is that used by Ryan and Dutta (1979). In this case. the rate
conStant. cI . was found by the initial degree of cure data; at the initial time. t. the de-

gree of cure. a. is assumed to be equal to zero. and eq. (3.4) reduces to

I
dtir

 

The second

matmcmaxin‘

lb: mission {0

mm time. a:

(2m)

5W eq. (3.11) f

 

one: the rate

28
g? t-O = c. (3.10)
The second rate constant. c. . and the exponent. m. were found from the degree of
cure at the maximum degree of cure rate. where the maximum is defined by
5111!
d 3 °
The expression for the degree of cure rate given in eq. (3.4) was differentiated with

respecttotime. andthe resulting expressionwas setequalto zero. with a aapasfol-
lows:

(2-m)c.¢11;°m + o. (2ap-m) = o (3.11)

Solving eq. (3. 1 l) for 0, resulted in

(2-m)c, 01”“

o. (3. 12)

m-2a
P

By substituting eq. (3.12) into eq. (3.14) with a = up and rearranging terms. Ryan and
Dutta obtained the following implicit relationship for m:

a’ (2-m)c. al'm
mln(ap) = 1n ——-—9—-u-a )2_m - c. - 1n ——P—m_2ap (3.13)
P

where a; is (dd/dt) evaluated at a = up . This equation was solved numerically for m.
given values for c, . up. and a; from DSC experiments at different temperatures.

Once the rate constants. cl and c. were determined for a few difi‘erent isothermal
cure temperatures. the activation energy constants and pre-exponential factors were

found from eq. (3.7) using linear regression.

Another 1m
in. The initial '
maid {mm c

cureratc data. E

“-3

WWI (3.14) 11
Wed values
c413.13) using 1
WWW: hz
“5 the disadvan
anduandda/dta

In 313860011
“1 . E. . and
WWW Sich.

ammmhrtc

 

29
Another method of determining the kinetic parameters in eq. (3.4) is presented
here. The initial procedure is the same as that used by Ryan and Dutta (1979): m was
estimated from c. and up using the initial degree of cure and the maidmum degree of
cure rate data. Equation (3.4) was then rearranged as

M = c. +0, 01'" (3.14)

(l-a) 2-m

Equation (3. 14) was solved for c. and 0. using linear regression and the previously
estimated values for m. This is an iterative procedure where m was recalculated from
eq. (3.13) using the new value for c. . and eq. (3.14) was again solved for c. and c. .
This procedure has the advantage that is uses all of the degree of reaction data. but it
has the disadvantages in that all the parameters are not determined simultaneously
and a and mum are treated as if they are independent of each other.

In the second kinetic model (eq. 3.5). there are four parameters to be determined:
A. . A. . E, . and E. . The rate constants were first determined using the procedure
described by Sichina (DuPont Publ. No. TA-93): the kinetic equation was rearranged in
a matter similar to that shown in eq. (3. 14).

M. " C] +Cz a (3.15)

( l-a)’

and. linear regression was used to determine 0. and c. . The activation energ con-
stants and pre-exponential factors were then estimated using linear regression from the
rate constants at different isothermal temperatures.

In the third model (eq. 3.6). an nth order reaction was assumed, and the
parameters estimated were A. . E. . and the order of reaction. r1. These parameters
were estimated by first using linear regression to find the order of reaction. 11. and the
rate constant. c. . for each temperature investigated. Then. A. and E. were found from
the rate constants at different temperatures using linear regression.

mm

)iawleyet1
Wm T?
themnnberofc

 

 

 

fore. the rcacttor

tn the model

We the diffus)

W 3 nOtllinea
“30' an Arrhent

C):

M A: and
mm”. 1987
Amodificat

in ”mm bee

Maine is dcs‘é

30
WW
Hawley et al. (1988) proposed the following model for a degree of cure greater than
fifty percent. The basis of this model is the assumption that. as the epoxy cures and
the number of cross-linkages increases. the flow of the molecules is restricted: there-
fore. the reaction is diffusion controlled. This is evident by the exponential decay term
in the model:

a: = C3 (bale-Du (3.16)

where the difi'usion coefi‘icient. D. and the rate constant. c, . were found numerically
using a nonlinear parameter estimation scheme. The rate constant was assumed to
follow an Arrhenius relationship with temperature:

c, = A. e'E’ [m (3.17)

where A, and E, were found using linear regression (Scientific Programming
Enterprises. 1987).

A modification of eq. (3.17) is presented here. It was based on the assumption
that diffusion becomes significant at some value of extent of cure. say at equal to 50%.
This value is designed as an as shown in the equation below.

-D(a -a)
g? = [3801-01 “3’ (“lime D (3'18)

D

Again. the unknown parameters. c, and D. were found using a nonlinear parameter
estimation scheme (Scientific Programming Enterprises. 1987). and the activation
energy constant. E, . and the pre-exponential factor. A, . were estimated from c,
evaluated at difi'erent temperatures.

It should be noted that technically. neither of these models are true kinetic
models. since by definition. diffusion is negligible in kinetic analysis: however. for these

moses!!!“
theywerekinet

initiation
Skull“

reaction of the c ;
ing the heat gel
Mowing sectiox .
“Walls c‘
To determ :'

 

3 1
purposes. it is convenient to treat the reactions for degree of cure greater than 50% as if
they were kinetic reactions.

EWWWWW

Since it was assumed that the degree of cure rate was proportional to the heat of
reaction of the composite during curing. the kinetic parameters were used in determin-
ing the heat generation due to the exothermic reaction of the epoxy. As shown in the
following section. the amount of heat generated is necessary for the estimation of ther-
mal properties during curing.

To determine the degree of cure rate throughout the curing process. one model
was selected from the three models given for degree of cure less than 50%. and a second
- model was selected from the two models were presented for degree of cure greater than
50%. The selection of the models was based on the minimization of the 95% confidence
intervals of the estimated parameters.

3.3 Estimation of Thermal Properties during Curing

The estimation of thermal properties of a composite during curing is similar to
that described in Section 3.1 for estimating the thermal properties of cured composites.
Several important difi'erences must be noted. however. First. since the curing process is
a result of an exothermic reaction. a heat generation term must be added to the energy
equation shown in eq. (3. la). This results in an additional differential equation.
namely. the appropriate kinetic model (eq. 3.4). which must be solved along with the
energy equation. Also. the thermal properties may be a function of degree of cure as
well as temperature.

‘ In this section. the direct problem of determining the temperature distribution
resulting from heat conduction through a composite specimen during curing is defined.
followed by a discussion of the solution method for the direct problem and the proce-
dure for the estimation of the thermal properties.

 

WLC 1

One din
epoxy reinfon
botmdmy ant
mmdm
posite plate a
1983). A mail

art

“13) the bound ,

L'kf‘l‘.at 7':

.—

 

 

One dimensional heat transfer was considered through a flat carbon-fiber/amine-
epoxy reinforced composite plate during curing. with an imposed heat flux on one
boundary and a known temperature history on the other boundary. It was assumed
that the chemical reactions due to the curing of the epoxy generate heat within the‘com-
posite plate at a rate proportional to the rate of extent of cure (Loos and Springer.
1983). A mathematical statement of this problem is

3;.[kfl‘aln] + Ptht a pc paling? O<x<L: t>O (3.19s)
with the boundary conditions.
-m.a)§§ = q(t) x = o; t > 0 (3.1910)
T = Tth) x = L: t > O (3. 19c)
and the initial condition.
T=T.(x) Osst; t=O (3.19d)

where x is the direction perpendicular to the fibers through the thickness. L is the
thickness of the plate. p is density. and I-It is the total heat of reaction. In this case. the
efl'ective thermal conductivity (perpendicular to the fiber direction). k. and the efi'ective
density-specific heat. pep. were assumed to be functions of temperature. T. and extent of
cure. on. Once again. the heat flux. q. temperature, TL. and the initial condition. T. .
were assumed to be known. The act/at term was obtained from the kinetic models
selected for a < 50% and a > 50%. as discussed in Section 3.2.3.

One method of solution involves the use of finite differences. At each time step.
the appropriate kinetic equation (eq. 3.4 or eq. 3.18) can be solved using a forward dif-
ference approximation for the time derivative of o. and then the energy equation (eq.

3.19a-d) an
derhathu To
oountfor the te
A one dim
tube the prob)
above. In the so
both the kinetic
Wiles were

one. for thermal

“1!. +1

i

“Mi: contix
Manes of)

«cum. “1' Equa
o...

 

33

3.19a-d) can be solved using Crank-Nicolson approximations for the temperature
derivatives. To simplify the solution. a quasi-linear approximation can be used to ac-
count for the temperature and extent of cure dependence of the thermal properties.

A one dimensional finite difference program. CUREID (Appendix A). was written to
solve the problem described by eqs. (3.19a-d) using the solution method described
above. In the solution. eq. (3.4) and eq. (3. 18) were selected for the kinetic models. and
both the kinetic and thermal properties were assumed to be known. The thermal
properties were assumed to be piecewise linear functions of temperature and extent of
cure. for thermal conductivity. k:

(3.20)

k: k°i+k'iT+k‘ia T‘<T<T“l. a‘<a<a.”l

where k is continuous at T‘ and 01‘. and k. . k. . and k. are constants obtained from
known values of the thermal conductivity for a given temperature. T‘. and a given extent
of cure. 01‘. Equation (3.20) was written in terms of k(q .1; ). Moi +1 .T ). and km“1 .‘l‘u ) as

follows

k = Maid“) + [k(aHl.T‘)-k(a1.T‘)l(a - at)

(T - T) (3.21)

l

+ [Mai-91' Ti +1) ““111 '11)]

T1 +1 'Ti

A similar expression was obtained for the density-specific heat product. Using the

quasi-linear approach. these parameters (k and pcp) were assumed constant over each

time step.

W
The parameter estimation procedure was similar to that described in Section

3.1.2 with the exceptions that the calculated temperatures in eq. (3.3) were obtained

1
from L

PROP)

{3.19:1

techno

peratu

thoofth
the optin

‘neotthe

 

 

34

from the solution of eqs. (3.19a-d) instead of eqs. (3.1a—d). and the measured tempera-
tures were obtained from a composite specimen during curing.

An updated version of the previously mentioned parameter estimation program.
PROPIDMA (Beck. 1989). was modified to include the heat generation term shown in eq.
(3. 19a) due to the chemical reactions occurring during the curing of the epoxy. The
modified program is called PROPID_CURE. The same kinetic equations and solution
technique used for the direct problem in CUREID were also used in solving for the tem-
perature distribution. T. in PROP1D_CURE.

3.4 Optimal Experimental Design Criterion for the Estimation of Thermal
Properties

This section deals with the optimization of the experimental design for the estima-
tion of the thermal properties of the composite material during curing. The objectives of
the optimization are first presented. followed by a discussion of the development and
use of the design criterion.

1W

One objective of this study was to estimate thermal conductivity and density-
specific heat of carbon-epoxy composites during curing. with the assumption that these
parameters are functions of temperature and extent of cure. In order to estimate ther.
mal conductivity and density-specific heat independently. a heat flux boundary
condition. such as that shown in eq. (3.19b). is required (Beck and Arnold. 1977). To
simplify the estimation problem. it was desired to have short heat flux intervals with
small temperature and extent of cure variations. compared to the overall temperature
and extent of cure changes during the cure cycle. to justify the assumption of constant
thermal properties over each heating interval. The objective here is to incorporate a
number of these intervals at different degree of cure and temperature values in the ex-
perimental design so that a relationship between the thermal properties and degree of

cure and temperature values can be obtained.

 

35

The recommended cure cycle for the AS4/EPON 828-mPDA composite system
used in this study is for the specimen to be heated for two hours at 75°C and two hours
at 125‘C. Therefore. one objective for the experimental design was to have the heat
flux interval be short compared to the overall curing time. and another was to have the
temperature rise due the the heat flux be small compared to the temperature difi‘eren-
tials in the curing cycle (e.g.. 50°C). The third objective was that the heating interval be
chosen to minimize the variance of the estimated thermal properties.

W
The optimal experimental desigi criterion was based on the third objective listed

above. with the first two objectives as constraints. In other words. a criterion was
sought in which the heating interval duration time was optimized for the estimation of
the thermal properties with the lowest variance. with the constraints that the interval ‘
be short compared to the total curing time and that the temperature rise during the
heating time be small compared to temperature differentials in the recommended curing
cycle. Practically. these constraints involved a fixed temperature rise and a fixed num-
ber of temperature measurements.

Many criteria have been proposed for the optimal design of experiments. and most
of these involve the XTX matrix. where x is the sensitivity matrix. The criterion used
here is recommended by Beck and Arnold (1977) because it is equivalent to the mini-
mization of the hypervolume of the confidence region (for more than one parameter and
for the standard statistical assumptions. Beck and Arnold. 1977). This criterion is the
maximization of the determinant of the xTx matrix.

A = )XTXI (3.22)

Beck and Arnold (1977) presented an expression for A at time. t". where n is large and
fixed. using uniformly spaced measurements. and two parameters. The expression for

A for one sensor in dimensionless terms is

their.

 

In these oquati:
Dimmer to be
“Willy of t
The beam
““8 was det
”3371ng I
mmthis:

36

* = c. tot-cs (3.23a)

where.
(r t‘“) QI"
=4“)? Io.)§'(tlxj'(t)dt )g' (t) . 31315 (3.23b.c)
d T” - In“ (3 23d)
an ' max " qL/k °

In these equations. Tm is the maximum temperature up to time. t“. B1 is the ith
parameter to be estimated. q is the heat flux. L is the thickness. and k is the thermal
conductivity of the sample.

The heating time interval used in the transient temperature experiments during
curing was determined using this criterion. The following hypothetical problem in
dimensionless terms was formulated to simulate one heat flux interval during the ex-

perirnents:

:3: + g" = if: 0 < x" < 1 t" > O: (3.248)
with boundary conditions.

- if: = {(1) 1:: ‘ t? x” = O: t+ > 0: (33413)

'1‘” = O x“ = 1 t+ > O: (3.24c)
and initial condition.

’1‘+ -.- 0 O s x+ s l t+ > O; (3.24d)

where.

 

The heat getter.
Note that the b
chosen because
allmtlmate is:

37

where.
'1‘" = $12. 8” = gL/q. J“ = X/Lo
and
t“ = ot/L’. (3.24e-h)

The heat generation term. g’. was assumed to be constant over the heating interval.
Note that the boundary condition at x+ = l is ‘1“ = O. This isothermal condition was
chosen because the tested composite materials have low thermal conductivity and an
approximate isothermal condition is relatively easy to simulate: it is done by attaching
the specimen to a high thermal conductivity material. This is in contrast to estimating
thermal properties of metals. which have relatively high thermal conductivity values. In
such cases. the case of an insulated boundary condition at x = L has been used (Beck
and Arnold. 1977).

The temperature solution for the conditions given in eqs. (3.24a-d) was found for

time. t*< tf. as.

T‘tx‘.t*) e chin-x” ) + (1-2?) - 2 m2

3 +

3
1" m coswmflwm-g’ (-1)"')/il:n (3.25a)
and for t*> tf .

- ' t+ - ‘ (ff-tr)
)5 c Bm I]

r*(x*.t*) = 0.5g*(1-x*= ) -2m21cos(pmx*)[(om-g*t-1)“‘)e m - om /B’

m

(3.25b)

The sensitivity terms in eq. (3.23b) were found by differentiating eqs (3.25a.b) with
respect to the parameters thermal conductivity. k. and density-specific heat. pep.
Difi'erentiating eq. (3.25a) first with respect to k resulted in the following expression for

t+< tT .

 

an“

OH

 

 

and for t3 if,

 

1C = ‘05]

Equations (3,252
K. For t‘< t: .

and [01‘ r) t: .

38

+ a -B’ t+
xf = 1(3):: = -.5g*(l-x”)-(l-x*) +2m21e m cosmmx‘)

xth-g"(- 1)“‘)(1+ts;nt*)/t5;u (3.26a)
and for t+> tf .

-’t+

x.‘ = -o.53”(1-x") +2mglcostpmxt)[(1+p;nt*)(1-g*(-1)'“/Bm)e m

a + + -B;(t+-tr)] a
- (1+Bm(t -t. l) e um (3.26b)

Equations (3.25a.b) were then differentiated with respect to pc to obtain expressions for

P
Xf. For t*< t1).
x:- __ fl+ _ E + - Zn+ + +
- pcpak - -2m.,t e costhx )(Bm-g (-1)”‘)/iim (3.27a)

and for t*> t.+ .

B’ t+ -B’ (t+-t.+)]

X: = -2mglfcoswmf)[(l-iftollmlfimle m -(t*-tT)(e m
(3.2%)

To calculate the criterion A+ in eq. (3.23a). the maximum temperature rise. Tm.
was first determined from eqs. (3.25a.b) at x+ = 0. The Ci) components in eq. (3.23b)
were then found using numerical integration from TL“. and the sensitivity coefficients.
X? and )6. calculated using eqs. (3.26a.b) and (3.27a.b). Finally. the criterion. A+ . was
found and compared for different heating intervals. No different values of the dimen-
sionless heat generation term. g’. were used in analysis.

Results for the criterion of A+ as a function of time for six different dimensionless

heating times are shown in Figure 3.1 for g’ = 0.1 and in Figure 3.2 for g” = 1.0. The

PM (A
I l 0 De i 0
MW nm AU nu
finmbfiuhofiwahn0n-whvh
c. Justus-the 23.}...th --~sb~5¥0-efltfl
.9

0 00(
qun
Dimens

 

0.004

01
000)

v.
aflflU-A-ngnvgn

AV .flaOtnva—LHV cut-Isles nuke-nonstaag
-

Dime

39

 

 

 

 

 

so u - 1
+ 0.0201 F5 m an 1:. - z
<. .4 m D. . A A-A H. I a
g .. . E14: 'A 000 o-o t." = 4
0 0.016— . o .' b '5 s W! t; .. a
E“ o -' A' .2. 'o X-X hes-10
Ea 0.012~ Vv
r3 V
D 0
£35 0.005—
E
E. 0.00“

o.ooo~oLaL e

(dimensionless)

Figure 3.1 Dimensionless Experimental Design Criterion. A“. for Different
Dimensionless Heating Times. t. . and a Dimensionless Heat Generation Term.
g”. equal to o. 1.

 

00 t,’ =- 1

0.020“ E] .3 {‘9 - 2
. F331. m t: = a

00-0 L?) D AA so if = 4

0.016- -' if 5» v-v t1 - e
(5 B 0'51 59°00 X-X t,’ a 10

1

p
O
_a
N
l

P
o
o
m
l

 

L

0004-1

Experimental Design Criterion, A+
(dimensionless)

 

o.ooo- '

 

G

 

(dimensionless)

Figure 3.2 Dimensionless Experimental Design Criterion. U. for Difi’erent
Dimensionless Heating Times. t. j and a Dimensionless Heat Generation Term.
g . equal to 1.0.

optinium heat

 

40

optimum heating interval was chosen from the curve with the highest value of 13*. In
both figures. the curves with the highest value of A+ resulted from a dimensionless heat-
ing time. t’. of 2. Further analysis of the case with g’ = 0.1. revealed that the optimum
heating time was closer to 2.125. An analysis of the case with g” = 0.0 indicated that
the optimum heating time was approximately 2.25: therefore. the case for g+ = 1.
resulted in only a 10% decrease in the optimum heating time. compared to that with g+
= 0. .

The dimensional optimum heating time was determined from thermal property
values based on results obtained for the cured composite samples from the analysis
presented in Section 3. l and a fixed thickness for the composite sample. The required
heat flux was then calculated from eq. (3.23d). assuming a maidmum temperature rise
of no more than 5‘C.

Chapter 4

Experimental Procedures

The focus of this chapter is on the experimental procedures used in the experi—
ments used to estimate thermal and kinetic properties in carbon/epoxy composite
materials. The first section is devoted to the procedures used in the transient tempera-
ture experiments for the estimation of thermal properties in cured composite materials
(Section 4.1). and the second section is concentrated on the difi’erential seaming
calorimetry experiments used to estimate kinetic properties during curing (Section 4.2).
In the final section. an outline of the methodology used in the transient experiments for
the estimation of thermal properties during curing (Section 4.3) is given

4.1 Experiments for the Estimation of Thermal Properties in Cured Composites

The experiments conducted for the estimation of thermal properties of cured com-
posites involved temperature measurements within carbon/ epoxy composite samples
subject to an imposed heat flux at one boundary. These experiments are described in
the following subsections. Included in these descriptions are the sample preparation
techniques. the experimental set-up design. the data acquisition system, the procedures
for the transient temperature experiments. and finally, a discussion of the experimental

parameters.

41

 

 

and. 4. consolida‘

the urban fibers
two phases. stac
tacked and con
hbr‘ration of lo

dfished in deta

““3 same

P0811: materials

T‘n'

“Wm Expert

42

W
The composite samples used in this study were composed of carbon fibers
suspended in an epoxy matrix. All of the samples were prepared using the facilities in
the M.S.U. Composite Materials and Structures Center. The preparation procedure can
be divided into four-phases: l. epoxy preparation; 2. prepreg preparation: 3. stacking;
and. 4. consolidation. In the first two phases. the two part epoxy was mixed. and then
the carbon fibers were impregnated with epoxy (prepreg preparation). During the final
two phases. staclnng and consolidation. the prepreg plies or impregnated fibers were
stacked and cured to form a composite plate. Each experimental set-up required the
fabrication of four composite plates. The phases of the preparation procedure are

described in detail in the following subsections.

4.1.1.1 Epoxy Preparation

The same type of epoxy was used in the transient experiments with cured com-
posite materials. in the differential scanning calorimetry (DSC) experiments. and in the
transient experiments during curing. The epoxy consisted of Shell’s EPON 828 Resin
with 1.3 Phenylenediamine (mPDA) as the curing agent. mixed in a one to one equiv-
alency. The curing agent is a suspected carcinogen, so safety was a number one
priority in the preparation procedures. The crystalline form of the mPDA is the most
critical with regards to safety. Extreme caution was taken so that the crystalline form
of the curing agent was not inhaled. and all skin contact with the mPDA was avoided.
Therefore. a respirator. protective clothing. such as a laboratory coat, and disposable
gloves were worn at all times during the preparation procedure. The protective clothing
also served to protect one's personal clothing. since contact with the curing agent
resulted in permanent stains. The detailed step by step procedure used to prepare the
epoxy is given below.

1. The mPDA was set out at room temperature from cold storage at least one half

hour before use.

2. Two large disposable beakers were cleaned with acetone. and then wiped dry

with a disposable tissue.

 

 

alency. 0)

measure (

the lid on

 

1h: jar
“M on
5. The be
agent wa
Odlcaily,
7. While
scale. ti
cleaned
Placed
9. The
minim
Drehg;
the rr
The e
1410
The:
dirm
rem
10.

or.

43
3. One of the beakers was tarred on a scale: the appropriate amount of resin was
poured into the beaker (approximately 0.200 kg resin for prepreg preparation or
0.050 kg for DSC experiments); and. the weight of the resin was recorded.
4. The resin was then heated in an oven to approximately 70°C.
5. In the mean time. the second beaker was tarred. and the appropriate amount
of mPDA was determined from the weight of the resin. For one to one equiv-
alency. 0.0145 kg mPDA are required for every 0.100 kg epoxy (Rich. 1987). To
measure out the appropriate amount of mPDA. the jar of mPDA was shaken with
the lid on and placed on its side on some foil that was spread out in a fume hood.
The jar was then opened. and the appropriate amount of mPDA was carefully
scoped out into the second beaker using a clean spatula.
6. The beaker with mPDA was placed into the oven with the resin until the curing
agent was completely melted (about 40 minutes). The mPDA was checked peri-
odically. and stirred with a clean glass stirring rod if needed.
7. While mPDA was melting. the foil in the fume hood was removed. and the
scale. the spatula. and the surfaces around where the curing agent was used were
cleaned thoroughly with acetone. Both the foil and the cleaning cloths were
placed in a sealed plastic bag for disposal.
9. The beaker with the resin was removed from the oven approximately five
minutes before the mPDA was completely melted. and a vacuum oven was
preheated to 60°C. Once the curing agent was melted. the resin was poured into
the mPDA beaker. and the mixture was stirred until it appeared homogeneous.
The epoxy mixture was placed into the vacuum oven (VWR Scientific. Inc. Model
1410). and the pressure was reduced to the limits of the instrument (-99 kPa).
The mixture was held under a vacuum until bubbles on the side of the beaker had
diminished (approximately 5 to 10 minutes): at that time. the epoxy mixture was
removed. and the oven was turned off.

10. The epoxy mixture was then ready for immediate use in prepreg preparation

or in DSC cxperirhents.

 

 

    
  

the prepregs WEI"
it). Ahot-melt
he resin to to ..
hoogh a small
mitgnated with
Mm) resin.
{med Diocedu
WCIDOrattor
mating the ;
alamprotecttv.
1- While
Prepared
mount a
What,

SCIt’ws.

44

4.1. 1.2 Prepreg Preparation

A prepreg consists of a single layer of fibers impregnated with uncured resin. In
this study. the prepregs were prepared from EPON 828/mPDA epoxy resin and con-
tinuous A84 carbon fibers with 12,000 fibers per tow (Hercules Aerospace Corp.); all of
the prepregs were prepared using carbon fibers from the same lot number (Lot No. 708-
4C). A hot-melt prepregger (Research Tool Co.) was used to impregnate the fibers with
the resin to form a prepreg. In this procedure. a tow of continuous fibers was fed
through a small vat of epoxy and then through a die. so that the tow was completely
impregnated with fibers. The tow was then wrapped around a drum to form a prepreg.
The epoxy resin was prepared using the procedure described in Section 4. 1.1.1. and the
detailed procedure to impregnate the fibers with the resin is given below. The Research
Tool Corporation Prepregger is shown in Figure 4.1. Additional views of the prepregger.
indicating the parts referenced in this section. are shown in Figures 4.2-4.6. Once
again. protective clothing and disposable gloves were worn.

1. While the mPDA was melting (step 7.. Section 4.1.1.1), the prepregger was

prepared. First. a spool of A54 carbon fibers was placed on the single spool

mount and secured in place using the large spool holder. the side mounting

bracket. and the end nut. The side bracket was held in place with two thumb

screws. The single spool mount with the spool of fibers in place is shown in

Figures 4.2a.b.

2. A tow. or strand of fibers. was then fed up over the first guide roller. through

the furnace (which was not used). and around the tension rollers. as shown in

Figures 4.3a.b.

3. The resin pot chamber was fastened in place with two thumb screws. The tow

was then guided through the resin pot. and the exposed layer of the fiber on the

spool was pulled off and thrown away. Masking tape was used around the ends

prior to cutting.

4. The large end of the die appropriate for this fiber (die A-A) was placed under

the resin pot. and secured in place with a thumb screw. The control panel of the

prepregger was then turned on. and full tension was applied to the fiber. The tow

45

 

 

 

 

Figure 4.1 Hot Melt Prepregger (Research Tools
Corporation).

 

 

 

 

 

 

 

 

 

4
3
2 \ l
1. Large spool holder
2. Side mounting brackets
3. End nuts
4. Thumb screws

 

 

Figure 4.23 Schematic of Single Spool Mount on Hot-Melt
Prepregger.

 

 

 

 

Figure 4.2b Photograph of Single Spool Mount.

 

47

 

 

 

1. Guide roller (a)
2. Furnace (not used)
3. Tension rollers

 

Figure 4.3a Schematic of Guide Roller. Furnace. and Tension
Rollers on Hot Melt Prepregger.

 

 

 

 

 

Figure 4.3b Photograph of Fibers Through Guide Roller. Furnace.
and Tension Rollers.

 

 

then tut-i

follows;

6. Two
Pm'ent

thllroi
7. Or
Was r
was '

gUld.

48

was pulled taut against the die opening so that all of the fibers were in the die
opening. The small end of the die was then put in place with a thumb screw.
taking care to keep the tow taut. No large thumb screws were then used to
secure both ends of the die. Extreme care was taken not to pull on the dry tow
once the die was in place. since this would break the fibers. The die. mounted on
the resin pot. with the fibers in place is shown in Figures 4.4a.b. and a close-up
view of the resin pot. die and pin guide (step 7) is shown in Figure 4.5.
5. The resin pot. flattening pin. and guide roller heaters (Figures 4.4a.b) were
then turned using the control panel (Figure 4.6). The set points were adjusted as
follows:

Resin pot = 125°F (69.4°C)

Flattening pin = 95°F (52.8°C)

Guide roller = 95°F (52.8°C)
6. No sheets of plastic. 193 cm by 33 cm. were cut. The plastic was used to
prevent the impregnated tow from sticking to the drum and to protect it from out-
side contamination. One end of one sheet was taped to the prepregger drum
(Figures 4.4a.b). and the drum was rotated so that the plastic lay smoothly on the
drum. The remaining end was then taped down. and then the plastic was cleaned
thoroughly with acetone.
7. Once the epoxy mixture was ready (see step 10. Section 4.1.1.1). the mixture
was poured into the resin chamber until it was about two-thirds full. Some epoxy
was poured over the fibers above the resin pot to ease the placement of the pin
guide. The pin guide was then inserted so that the fibers ran in front of the top
pin. behind the second. and in front of the bottom two pins. The guide was placed
so that the top pin lay just below the surface of the resin pot. and it was secured
in place with a side mount and two thumb screws (Figure 4.4a.b).
8. Once the pin guide was set in place. the resin chamber was filled up to the rim
with epoxy.
9. The tension on the fibers was released. and the wetted tow of fibers were

pulled down through the die. between the flattening pins. and around the guide

49

 

 

 

 

' _—
7“. es! user a

riff)"

 

 

Resin pot chamber

 

Die (A-A)
Guide pin
Flattening pin
Guide roller (b)
Drum

(b) photograph

l
2.
3.
4
5
6.

(a) schematic

 

Figure 4.4a.b Schematic (left) and Photograph (right) of Resin Pot Assembly
on Hot Melt Prepregger.

 

 

 

 

 

Figure 4.5 Detailed View of Resin Pot. Die (A-A). and Pin
Guide (from left to right).

 

 

 

 

 

Figure 4.6 Hot-Melt Prepregger Control Panel.

 

 

 

roller.
one ha‘;
10. The

adjuste

The ltbt

 

camag
Dmiou
11. The
rESlIi it
12. W)
tion a.
to the
13. T
Place
Dias
the
14.
bag
15

Ca

51
roller. The tow was then taped to the drum. and the drum was rotated at least
one half revolution to ensure that the tow stayed in place.
10. The drum rotation. fiber tension. and drum carriage movement controls were
adjusted as follows:
Drum rotation - 1.0 rpm
Fiber tension - 0.5 to 1.0
Drum carriage movement - approximately 23 (scale: 0 to 100)
The fibers were watched carefully for the first three to four revolutions. and the
carriage movement was adjusted so that in each rotation. the tow lined up the
previous rotation with no gaps between the two lines.
1 l. The resin pot was checked and refilled every five to ten minutes to keep the
resin level above the top guide pin.
12. When the fiber had wound to the left hand side of the drum, the drum rota-
tion and drum carriage movement were turned off. and the fiber was cut off close
to the drum.
13. The remaining sheet of plastic was cleaned with acetone. and one end was
placed on the surface of the exposed fibers. The drum was rotated slowly. and the
plastic was wrapped smoothly around the drum. The prepreg was cut away from
the drum along a grooved cutting line using a utility knife with a new blade.
14. The prepreg was then carefully rolled or folded. and placed in a sealed plastic
bag in a freezer until needed.
15. The clean up process was critical in this procedure since the epoxy mixture
can eventually cure and harden at room temperature. The clean up procedure is
given below.
a. The fiber around the resin pot was cut off and discarded. The resin pot
was then removed by unscrewing the two thumb screws on the side of the
pot. and the remaining resin in the pot was drained into the resin beaker
under the fume hood.
b. The prepregger was cleaned thoroughly with acetone. especially around

the resin pot chamber. The flattening pins were cleaned in plate. but the

52
guide roller was removed and cleaned under the hood. The drum was also
cleaned with acetone. and excess tape was removed.
c. All switches on the control panel were turned off. along with the power
switch to the control panel.
d. The spool of fiber was wrapped in foil. labeled. and put away.
e. The screws. pin guide. and die were removed from the resin pot under a
fume hood. All pieces were carefully washed twice with acetone. and
placed back on the prepregger.
f. Finally. the fume hood area was checked and cleaned with acetone. and
the soiled cleaning cloths and empty beakers were sealed in a plastic bag
and disposed of properly. The epoxy in the resin beaker was allowed to

cure in the hood at room temperature before discarding.

4.1.1.3 Stacking Procedure

The prepreg plies were stacked and prepared for curing in this phase. Twenty-
four pieces or plies were cut from the prepreg and stacked for each composite plate. In
stacking the prepreg plies. two fiber orientations were used: [0°]24 (parallel fibers) and
[0. 30. ~30. 60. -60. 9001mm) (quasi-isotropic laminate). The two stacking orientations
are shown in Figure 4.7. One prepreg was required for each of the [0°]24 composite
plates. and one and a half prepregs were required for each of the quasi-isotropic plates.
A total of eight [0°]u and four [0. 30. -30. 60. -60. 90°12!“ plates were prepared. After
stacking. the prepreg laminates were placed on an aluminum plate covered with a piece
of release ply. which is a plastic sheet used to protect the aluminum plate from the
epoxy. A dam was built up around the composite using a gummy tape or a cork tape to
maintain the shape of the laminate during consolidation. Pieces of porous TeflonR cloth
and bleeder cloth were then placed on top of the composite. The bleeder cloth is an
absorbent material used to absorb the excess epoxy during consolidation. and the
Porous TeflonR cloth was placed between the prepreg laminate and the bleeder cloth to
Protect the surface of the composite. A second aluminum plate. covered with a release

ply was laid on top of the bleeder cloth. and then the entire assembly was sealed in a

53

 

 

~60°
90°

 

a. Parallel b. Quasi-isotrOpic

 

Figure 4.7 Stacking for Orientation for the First Six Parallel and the
First Six Quasi-isotropic Laminates for the [0°]2‘ and the

[0°.30°.-30°,-60°.60°,90°]2(.ym) Composite Disks.

 

ncuuin bag .

 

  
 

 

 

 

54

vacuum bag for curing. The steps used in the stacking process for one plate are as fol-
lows.

1. One 23 cm by 23 cm square piece was cut from the porous TeflonR cloth and
from the bleeder cloth. and two 23 cm by 23 cm squares pieces were cut from the
release ply. In addition. a small piece of thick bleeder cloth (approximately 8 cm
by 10 cm) and sheet of vacuum bag material at least 40 cm by 70 cm were cut.

2. The prepreg was taken out of the freezer and allowed to cool just enough to
become pliable (approximately 5 minutes): during this time. approximately 3 kg of
dry ice was obtained and chopped into small chunks.

3- The prepreg. which measured 193 cm by 30.5 cm. was spread out on a sheet

of foil. and a piece of poster board was placed under the section to be cut. Using

a carpenters square as a guide and a sharp utility knife. twenty-four 15.2 cm by

1 5-2 cm plies were cut from each prepreg for the [0°]24 composite plates. Since

the prepreg had to be cut at difierent angles for the quasi-isotropic plates. one and

a half prepregs were required for each of these plates.

4- One piece of the release ply was secured using to a smooth 23 cm by 23 cm
Square aluminum plate using small amounts of Tacky TapeR (Schnee-Morehead.
Inc.) at the corners.

5. The dry ice was used to facilitate the removal of the release ply from the
Prepreg. Several prepreg plies were laid out on the foil, and the remainder plies
were placed back in the freezer until needed. Using thermal gloves. the dry ice
was spread over the prepreg plies. and the release plies on one side of each ply
were peeled off as they stiffened and separated from the prepreg. Particular atten-
tion was taken not to peel off any fibers with the plastic. After any accumulated
frost on the exposed surface had evaporated off. two prepreg plies were stacked
together with the desired fiber orientation ([0°12‘ or [0. 30. -30. 60. -60. 90°)2 ( . y m) ).
A roller was used to press the plies together in an effort to remove air trapped
between the plies.
6. Step 5. was repeated with all 24 plies. resulting in a 24 ply laminate with two
release plies still in tack.

7. One 1;

 

placed la
8. Elthe“
2.54 cm
Sales. In
it was lex
9. The l
was cove
second a
10. The '
material.
vacuum
tube to“
folded ow II

the bag.

 

Where “1
Prepreg 1

Sofidatio

4.1.14 cm

The p;
9531qu (Frc
alll

5030px p18

55
7. One of the last two release plies was then removed. and the laminate was
placed face down on one of the aluminum plates covered with release ply.
8. Either gummy tape (Air Dam I. General Sealants) or 0.32 cm (or 0.16 cm) by
2.54 cm cork tape (Pressure Sensitive Backed Cork Dam. Northern Fiber Glass
Sales. Inc.) was used to as a dam around the sample. The dam was built up until
it was level with the laminate thickness.

9. The last release ply was removed from the sample. and the exposed surface
was covered with the porous TeflonR cloth. followed by the bleeder cloth and the
second aluminum plate protected with release ply.

10. The entire assembly was centered on one half of the sheet of vacuum bag
material. A rubber hose was held in place with Tacky TapeR at one end of the
vacuum bag. with the piece of thick bleeder cloth placed over the mouth of the
tube towards the inside of the bag. The remaining vacuum bag material was
folded over the assembly. and then Tacky TapeR was used around the edges to seal
the bag. The assembly was checked visibly for leaks. and extra tape was added
where needed. especially around the rubber tube. A diagram of the stacked
PI‘eIJreg is shown in Figure 4.8. The prepreg laminate was then ready for con-

Sondation.

4. 1.1.4 Consolidation

The prepreg laminates were consolidated either in an hydraulic press with heated
Platens (Fred S. Carver. Inc.. Hydraulic Equipment. Model SP-F-6030) or in an
autoclave (United McGill Corporation). The eight [0°]24 plates and two of the quasi-
isotropic plates were cured in the press; the remaining two quasi-isotropic plates were
Cured in the autoclave. (The autoclave was not available at the time the first ten
SamDles were made.) The same curing cycle was used in both cases. Pressure was
applied so that the pressure over the surface of the laminate equaled 689 kPa (100 psi).
and the laminate was heated to 75°C at 5°C/min. During this time. the hose on the
Vacuum bag was attached to a vacuum pump. and pressure inside the base was

reduced to -99 kPa. This was done to further remove any air which might have been

 

56

 

Vacuum bag

Tacky TapeR

Bleeder cloth

Porous TelflonR cloth

Prepreg laminate

Cork tape

Thick bleeder cloth

  

Cut-Away
View

Rubber hose

 

Figure 4.8 Cut-Away View of Stacked Prepreg Laminate.

 

trapped beta
twashcaie'
theretomme‘
Alter c
0m to mow
autoclave. n
51 this time

ilu‘clmess. u

mm

No es
and a third
907% dis
tomposite pl.

The die
to Mid any
m was me.
“crate 1h lcl
Table 4.1. T.
“It most 111
Wed aCCu
3mmallied

“‘5 immany

57
trapped between the prepreg plies. The laminate was held at 75°C for two hours. then
it was heated to 125°C at 5°C and held at 125°C for another two hours. according to
the recommended curing cycle for this composite (Rich. 1987).

After completion of the cure cycle. the composite laminate plates were cooled
down to room temperature. and then they were taken out of either the press or the
autoclave. removed from the vacuum bag. and separated from the stacking materials.
At this time the plates. measuring approximately 15.2 cm by 15.2 cm by 3 to 4 mm in

thickness. were ready for use in the experimental set-ups. described in the next section.

W

No experimental set-ups were assembled using the eight [0°]u composite disks.
and a third experimental set-up was assembled using the four [0. 30. -30. 60. -60.
90°]2l-yml disks. Each experimental set-up was composed of four disks cut from the
composite plates. a resistance heater. thermocouples. and two aluminum cylinders.

The disks. measuring 7.6 cm in diameter. were cut from the center of each plate
to avoid any end effects resulting from the stacking process. The thickness of each
disk was measured at various locations around the disk and recorded: a record of the
average thickness. consolidation method. and fiber orientation for each disk is given in.
Table 4. 1. The variations in the thicknesses between disks cured in the hydraulic press
were most likely a result of differences in the applied pressure. which could not be con-
trolled accurately. (Due to the hydraulic control on the press. the pressure was
maintained at a constant value throughout the curing process; however. the pressure
was initially set with an accuracy of only :i: 2 kPa.) The greater thicknesses reported for
the samples cured in the autoclave were most likely due to the use of the cork tape as a
damming material instead of the gummy tape. The cork tape proved to be a better con-
tainment material for the epoxy than the gummy tape due to its more rigid form. (The
cork tape was not available at the time the first eight samples were fabricated.)

The resistance heaters were thin (0.25 mm). 7.6 cm diameter. 9.4 ohm
'I'herrnofoilR heaters (Minco Products. Inc.) and were used to provide the heat flux

boundary condition shown in eq. (3.1b). The thermocouples were fabricated from

 

 

 

 

58

Table 4.1 Fabrication Parameters for Cured Composite Disks used in Transient
Experiments.

 

 

 

Disk Thickness(mm) Consolidation Fiber
No. mean st.d. Method Orientation
1 2.96 0.04 hydraulic press [0°]24
2 3.32 0.05 hydraulic press [0°]24
3 3.56 0.02 hydraulic press [0°124
4 4.17 0.10 hydraulic press [0°]24
5 3.02 0.11 hydraulic press [0°124
6 3.01 0.11 hydraulic press [0°]24
7 3.68 0.08 hydraulic press [0°]24
8 3.66 0.03 hydraulic press [0°]24
10 3.56 0.09 hydraulic press [0,30,-30,60,-6O,90°]2(gym)
11 4.61 0.04 autoclavea [0,30,-30,60,-60,90°]
2(sym)
12 4.93 0.02 autoclavea [0,30,-30,60,-60,90°]
2(sym)

 

 

 

 

 

 

 

a. Cork tape used as a dam instead of gummy tape.

 

59
chrome] and constantan wires (ANSI Type E). The two aluminum cylinders. each with a
diameter of 7.6 cm and a height of 3.7 cm. were used to as heat sinks to approach a
constant temperature for the boundary condition shown in eq. (3.1c). The procedures
used in fabricating the thermocouples and in assembling the experimental set-up are

given in the following subsections.

4.1.2. 1 Thermocouple Fabrication

Fine thermocouples were used in both the transient temperature experiments on
cured composite samples and the experiments on composite samples during curing.
The thermocouples were prepared from 0.08 mm (AWG 40). chromel/constantan (ANSI
Type E). Teflon'VNeoflonll duplex insulated wires from Omega Engineering. Inc. The
duplex insulation consisted of an inner insulation layer around each individual wire.
plus an outer insulation layer around both wires.

To prepare each thermocouple. the thermocouple wire was cut into pieces ap-
proximately 94 cm long. and the outer insulation layer was stripped off 40 cm from one
end. The inner insulation layer around each wire was stripped off about 8 cm from the
same end. and pieces of overbraid insulation. approadmately 32 cm long. were slipped
over each wire to cover the portion of the wire with only one layer of insulation. The
exposed wires were flattened to approximately 0.05 mm by placing them between
smooth. flat. steel dies and applying pressure using a hydraulic press. The flattened
ends of the wires were over-laid and welded using a portable welder (Black 8: Webber.
Model 848). The other end of the thermocouple was stripped about 1 cm from the end.
and instrumented with a male. Type E. subminiature thermocouple connector from
Omega Engineering. Inc. A prepared thermocouple with a close up view of the ther-
mocouple junction is shown in Figure 4.9.

In addition. extension thermocouple wires were used to connect the instrumenta-
tion thermocouples to the data acquisition system. PFA TeflonR Coated ANSI Type EX
extension grade wires with 304 stainless steel shielding (Omega Engineering. Inc.) were
used for this purpose. Female subminiature connectors were placed on one end for

connection with the instrumentation wire. and the other end was either stripped for

71.

 

 

 

 

 

 

 

 

 

 

Close-up view of Thermocouple
Junction

 

Figure 4.9 Prepared Thermocouple (ANSI Type E) with Close-up View of
Thermocouple Junction.

 

l

direct cont

llCCIOl'S.

4.1.2.2 EX]

The c
plate. The t
center line I

the center-ll

 

 

 

 

mm on om
disks were p
on the adjac
60, goelm
adjacent Sur
comWind. j
Plates and b
Side of each
Silicon was :
“Imps.

A resis
21“mocoum.

We“) the

61
direct connection to the data acquisition system or fitted with male subminiature con-

nectors.

4.1.2.2 Experimental Set-up Assembly

The composite disks were stacked with two thermocouples laid in between each
plate. The thermocouples were placed across the disk about 6 mm on either side of the
center line of the disk perpendicular to. the fibers. with the thermocouple junction on
the center-line of the disk parallel to the fibers. A diagram of the thermocouple place-
ment on one disk is shown in Figure 4.10. For the set-ups with parallel fibers. the
disks were placed on top of one another with the fibers on one disk parallel to the fibers
on the adjacent disk. In the case of the set-up using the disks with the [0. 30. -30. 60.
-60. 90’1“.“ fiber orientation. the disks were arranged so that the fibers on the two
adjacent surfaces were parallel to provide symmetry. Silicon grease (Silicon Heat Sink
Compound. Dow Corning Corp.) was used to provide good thermal contact between the
plates and between the thermocouples and the plates. This grease was applied to one
side of each disk before placing the thermocouples using a small. thin. fiat board. The
silicon was spread out down until its thickness appeared uniform. with no gaps or
clumps. I

A resistance heater was placed between the middle two disks. and two additional
thermocouples were placed on each side of the heater. again using the silicon grease
between the heater and the composite plates. The disks were stacked so that the total
thickness on either side of the heater was approximately equal. The stacked composite
disks instrumented with thermocouples and the resistance heater. were then placed
between the two aluminum cylinders. and two thermocouples were placed at each of the
composite disk-aluminum cylinder interfaces. The entire assembly was then pressed
together between two 14 cm by 14 cm by 4 mm aluminum plates. Pressure was applied
through threaded rods which ran through the corners of the aluminum plates. A

photograph of the experimental set-up is shown in Figure 4.11.

 

 

Com

62

 

 

Thermocouple junctions

 

Center-line

 

 

 

 

 

 

 

j .
Perpendicular
6min _|,/ \ f to Fibers
l l
Thermocouples
Composite disk ‘— Center-line
Parallel to
Fibers

 

Figure 4.10 Thermocouple Placements on Composite Disks Used in Transient
Temperature Experiments.

 

 

 

 

 

Figure 4.11 An Experimental Set-up for Transient Temperature
Measurements using Cured Composite Samples.

 

4.1

64

WW

Temperature measurements from the thermocouples were recorded using a PDP
1 1/05 microcomputer (Plessey Peripheral Systems) running under an RT-l 1/V4 operat-
ing system. The signal from the thermocouples was conditioned through the use of
amplifiers prior to being read by the RT-l 1/V4 system. The data acquisition hardware.
signal conditioning. and data acquisition software are described in the following sec-
tions. A schematic of the data acquisition system and power supply for the controlled

heat flux is shown in Figure 4.12.

4.1.3.1 Data Acquisition Hardware

The PDP 11/05 was equipped with an analog-to-digital (A/D) converter (Model
D’I’2764) and a real-time clock/counter unit (Model DT2769). Both of these units were
supplied by Data Translation. Inc. The full scale input voltage on the A/ D converter
was configured to operate from 0 to 10 V. with a gain of one and with eight difi'erenti'al
ended input channels. The clock unit was a programmable unit which determined the

intervals of count events. Details of these devices are given by Osman (1987).

4.1.3.2 Signal Conditioning

Since the signal produced by the thermocouples was on the order of millivolts and
the A/D operated on a scale from 0 to 10 V. a signal conditioning unit was required to
amplify the thermocouple signals prior to processing by the A/D converter. This unit.
also supplied by Data Translation. Inc.. consisted of a :15 V power supply (Model
DT7692). a backplane board (Model DT750). and eight amplifiers (Model DT6705E).

Each amplifier was instrumented with an electronic cold junction compensation
for ANSI Type E thermocouples. In this case. the extension thermocouple wires were
connected directly to the channel trip on the DT750 backplane. The gain of the
amplifiers was set at 1000. and before each experiment. the zero calibration was ad-

justed using the calibration screws on the front panel of each amplifier.

65

 

Microcomputer

 

 

backplane

 
 
 
  
 
 
 
 
  
  
 
 
  
  
  
   
  
 

Amplifiers (8)
(Data Translation)

PDP 11/05

 

(Plessey
Peripheral
Systems)

    

 

 

Real Time Clock

  

 

 

  

 

 

 

 

 

 

composite
samples
heater
Experimental _/
Set-up
oven
to 1
Power Supply heater
(The Superior
Electric) -
multimeter

    
 

(amp-meter)

  
  
 

  

multimeter
(volt-meter)

  

 

 

 

Figure 4.12 Schematic of Data Acquisition for Transient
Experiments using Cured Composite Samples.

 

 

4.1.3.3

and th.
package
used to
'sueep' :
clock in
1987),
Ar

output 1';

4.1.3.4
Th
Em. A
Supply t
Ebcted u
Sippiy “
“as 31111:
To

3535 ure

66

4.1.3.3 Data Acquisition Programs

An existing program. DATACQ (Osman. 1987). was used for controlling the A/D
and the real-time clock-counter units. The program utilizes a real-time software
package. DTLIB/RT. provided by Data Translation. Inc. (1981). The real-time clock was
used to trigger the A/D converter at user designated intervals. The A/ D operated in a
’sweep' mode. in which the input channels 0 through 7 were read sequentially on each
clock trigger. The time required for each sweep was less than 0.002 seconds (Osman.
1987).

An adjoining program. DATFIT (Osman. 1987). was used to convert the binary

output from the thermocouple readings to temperature values.

4.1.3.4 Controlled Heat Flux

The resistance heater was controlled independently from the data acquisition sys-
tem. A Powerstat" Variable Autotransforrner (The Superior Electric. Co.) was used to
supply the power requirements for the resistance heater. The power supply was con-
nected to a timer: during operation. the timer was switched on manually. and the power
supply was activated for a preset interval on the timer. The maximum heating interval
was limited to 60 seconds using this timer.

To measure the applied heat flux. two multimeters (Keithley Co.) were used to

measure the current to the heater and the voltage drop across the heater (Figure 4.12).

W

The transient experiments using the cured composite samples consisted of plac-
ing the experimental set-up in an oven at the desired temperature until steady-state
conditions were obtained. and then activating the heater and recording the resulting
temperature response using the data acquisition program. The procedures used for
each of these experiments were as follows:

1. The experimental assembly was placed in an oven: then the thermocouples

cured
ambit:
heat a

total n

[020:

67

were connected to the extension cables wired to the backplane of the data acquisi-
tion system; the heater was wired to the power supply: and. the. assembly was
heated to the desired testing temperature (this usually took two to three hours).

2. The data acquisition program was started. and the experimental parameters.
such as sampling rate. total experimental time. and number of thermocouples.
were entered. The total heating time was set on the timer connected to the power
supply.

3. Data acquisition commenced upon an external input to the data acquisition
program. At the same time the data collection was started. a stop watch was ac-
tivated. The heater was manually started 20 seconds after each experiment was
begun. and the voltage and amperage readings were recorded every five seconds
for the duration of the heating interval. Data collection continued on the data

acquisition program until the end of the total experimental time.

4 n r r

The experimental parameters in the transient temperature experiments using
cured composite materials included total experimental time. data sampling interval.
ambient temperature. heating time interval for the resistance heater. magnitude of the
heat flux. fiber orientation with respect to heat flux. number of themiocouples and the
total number of repetitive experiments for each set of parameters.

The total experimental time was set at 200 seconds. with a one second data sam-
pling interval for all cases. The experiments were conducted at five different ambient
temperatures: 25 (or room temperature). 50. 75. 100. and 125°C. The ambient tem-
perature conditions above 25°C were conducted using the oven as described in Section
4. 1.4. These temperatures served as the initial temperatures for each run since the
resistance heater heated the composite samples above these temperatures.

The heating time interval was set at 40 seconds. using the timer connected to the
PowerstatR power supply. and the output power from the power supply was adjusted to
about 28-29 watts. This resulted in a maximum temperature rise of approximately 15

to 20°C above the initial temperature.

as dh
prapc:
FKTPC
result
mned
the r

Voluri

moco
the a
there
ones
IEoco

Very '

68

The fiber stacking angles investigated were [0°]24 and [0. 30. -30. 60. -60. 90°12‘07“).
as discussed previously. This study was limited to the estimation of the thermal
properties transverse to the fiber direction. so in all cases the heat flux was assumed
perpendicular to the fiber axis. Therefore. it was anticipated in this case that similar
results would be obtained for the two different fiber stacking angles. It should also be
noted that the orientation of the fibers is only pertinent with respect to the estimation of
the effective thermal conductivity. since the density-specific heat product is a
volumetric term. and should not depend on fiber orientation.

Twelve thermocouples were used in each experimental set-up. with two ther-
mocouples at each disk interface with the aluminum cylinders. the heaters. and with
the other disks. Only six amplifiers were available on the data acquisition system;
therefore. only six thermocouples were actually used for data collection. the remaining
ones were installed as a precautionary measure in case one or more of the other ther-
mocouples broke. (Due to the small diameters of these thermocouple wires. they broke
very easily.)

Each experiment was repeated at least six times using the same experimental
parameters. The experimental parameters for each set of experiments. including fiber

stacking angle. stacking order for the disks. temperature. and number of repetitions are

shown in Table 4.2.

4.1.5 Elbe; Vglgmg £139;ng
The fiber volume fractions of disk numbers 5. 6. and 8 (in Table 4.1) were

measured by personal at the Composite Materials and Structures Center using an
Optimal Numerical Volumetric Analysis (ONVfA) technique developed by Waterbury
(1988). In this technique. the density of the samples were first measured by an immer-
sion weighing technique. The samples were then cut through the thickness
perpendicular to the fibers and polished. The polished surface was placed under a
microscope. and the image was digitized. The fiber volume fraction was calculated from
the digitized image. the measured density. and the published densities of the individual
fiber and epoxy components. using the ONVfA software developed by Waterbury. These

Table
Compo

 

 

 

 

 

 

n1 [‘44 Hr

Table 4.2

Experimental Parameters for Transient Experiments using Cured

Composite Samples.

 

a.
b.

69

 

 

Exper. Initiala Fiber Disk No. of
No. Temp. (°C) Orientation Numbers Repetitions
1.1 29 - 32 [0°]24 3,2,1,4 9
1.2 53 - 54 [0°]24 3,2,1,4 6
1.3 76 - 78 [0°]24 3,2,1,4 6
1.4 101 - 103 [0°]24 3,2,1,4 6

.5 126 — 128 [0°]24 3,2,1,4 9
1. 25 - 29 [0°124 1,4,3,2 6
1.7 49 - 52 [0°]24 1,4,3,2 12
1.8 75 [0°124 1,4,3,2 6
1.9 100 [0°]24 1,4,3,2 6

1.10 126 [0°]24 1,4,3,2 6
1.11 26 - 27 [0°]24 5,8,7,6 6
1.12 52 - 53 [0°124 5,8,7,6 6
1.13 75 - 77 [0°]24 5,8,7,6 6
1.14 100 [0°]24 5,8,7,6 6
1.15 125 [0°124 5,8,7,6 6
1.16 25 - 29 [0,:30,i60,90°]2(sym) 10,11,12,9 12
1.17 52 - 53 [0,:30,160,90°]2(sym) 10,11,12,9 6
1:18 78 [0,130,r60,90°]2(SYm) 10,11,12,9 6
1.19 25 - 31 [0,r30,i60,90°]2(5ym) 11,10,9,12 6
1.20 49 - 53 [0,i3O,i60,90°]2(sym) 11,10,9,12 6

76 - 77 [O,:t30,:t60,90°] ll,10,9,12 6

1.21

 

 

2(sym)

 

 

 

to bottom.

Initial laminate temperatures in experiments.
Disk numbers refer to Table 4.1.

They are listed in stacking order,

 

from top

results it:
the matm
results for

mlume fra

4.2 Differ

Dillez
estimate th
resin. The
3-2- The es
Obtain the l
Periments t.
Calculate 1H

Parameters

 

 

Sampl
tau“ the re
audit “'OUld
momogenet
PitsenCe 0f t
bl the ObSEI‘
fibers had on

The $211

mpmeter

70
results were compared with those obtained using an acid digestion technique. in which
the matrix material is dissolved away. so that the remaining fibers can be weighed. The
results for both the ONVfA analysis and the acid digestion analysis. including the fiber

volume fraction. the void volume fraction. and the density. are shown in Table 4.3.

4.2 Differential Scanning Calorimetry Experiments

Differential scanning calorimetry (DSC) experiments were conducted in order to
estimate the kinetic parameters associated with the curing of EPON 828/mPDA epoxy
resin. These parameters are those shown in the kinetic models described in Section
3.2. The estimation of the parameters required several isothermal DSC experiments to
obtain the heat of reaction at difi'erent temperatures. and dynamic or ramped DSC ex-
periments to determine the total heat of reaction. The heat of reaction data was used to
calculate the total heat of reaction and the degree of cure. from which the kinetic
Parameters were estimated as described in Section 3.2.

Samples of neat resin rather than prepreg were used in these experiments. be-
cause the required sample size for DSC experiments was small (on the order of 10 mg).
and it would have been very difficult to insure consistent sample composition due to the
inhomogeneous nature of the prepreg. In using the neat resin. it was assumed that the
Presence of the fibers had no effect on the cure kinetics. This assumption was justified
by the observations of Mijovib and Wang (1989) who found that the presence of carbon
fibers had only a very small initial effect on the cure kinetics.

The sample preparation method. the experimental procedures and the experimen-

tal Parameters for the DSC experiments are presented in the following sections.

We

Sample preparation for the DSC experiments involved the preparation of the
epoxy resin. the placement of the resin in small special pans. and the closure of the
pans With lids. These lids and pans were especially designed for use with the DSC

e(11.11p‘rnent operated in this study.

 

ruse 413
Numerical '

 

 

 

 

 

P.a:e*se
ho a (

5 1

6 1

3 1
10b 1

N‘

‘ 9;.te

b . CNYJ fA

 

 

 

 

71

Table 4.3 Fiber Volume Fractions using Acid Digestion Analysis and Optical
Numerical Volumetric Analysis (ONVfA).

 

Acid Digestion ONVfA

 

Plate Density Fiber Matrix Void Fiber Matrix Void
No. (kg/m3) Vol. Fr. Vol. Fr. Vol. Fr. Vol. Fr. Vol. Fr. Vol. Fr.

 

5 1,538. 66.2% 28.6% 5.1% 65.7% 29.3% 5.0%
6 1,531. 64.9% 30.2% 5.0% 66.4% 29.4% 4.2%
8 1,563. 64.7% 32.9% 2.4% 68.5% 28.0% 3.5%

1ob 1,570. 64.3% 32.4% 3.3% -- -— --

 

 

 

 

 

 

 

 

 

 

a. Plate number corresponds to disk number in Table 4.1.
b. ONVfA analysis not appropriate for the quasi-isotropic plates.

72

The pans and lids designated for use with liquid samples were initially weighed
and separated according to weight. and the epoxy was prepared according to the proce-
dures in Section 4.1. Once the epoxy was mixed. several 3cc Becton Dickson single use
syringes were filled with the epoxy and placed in a freezer until needed. To prepare a '
sample for a DSC experiment. apprordmately 10 mg of epoxy was placed in the center of
a tarred pan using the syringe. The weight of the epoxy was then measured and re-
corded. A lid was placed on the pan. and the pan and the lid were crimped and
hermetically sealed together using a crimper for liquid DSC pans (DuPont Instruments.
Inc.). Each experiment required a reference pan containing no resin. so a second pan
and lid of approximately the same weight as the empty sample pan and lid were also
crimped and sealed together. The syringe was placed back in the freezer until further

use.

W
A DuPont Instruments Thermal Analyzer System 9900 coupled with a 910

Differential Scanning Calorimeter were used for these experiments. The DuPont 910
Calorimeter is shown in Figure 4.13. The operation procedure consisted of a set up
procedure. in which the experimental parameters for each experiment were entered. an
initialization procedure. in which the experiment was begun. and after the experiment
was completed. a post-processing procedure. These procedures are discussed in the

following sections.

4.2.2.1 Differential Scanning Calorimeter Set Up Procedure
The DuPont System 9900 software is menu driven. and the experimental
parameters for each experiment were entered using two of these menus: the Sample
Info menu and the Select Method menu.
The Select Method menu contains a number of difi‘erent time-temperature profiles
which can be edited to fit the users needs. (The program stores approximately 17 dif-
ferent methods. and each of these methods is designated by a number.) In the Sample

—§iulh.‘l| —

73

 

 

 

 

Figure 4.13 DuPont Instruments 910- Differential Scanning Calorimeter.

 

74
Info menu. the output data file name is entered. along with the sample weight. the user
name. and the method number from the Select Method menu.

In setting up a method in the Select Method menu. several difi'erent segments for
the time-temperature histories were used. These included INITIAL. lSOTI-IERMAL.
SAMPLING INTERVAL. and RAMP. In the INITIAL segment. the calorimeter cell is
heated rapidly to some designated temperature. and it is held at that temperature until
prompted by an external trigger by the user to the thermal analysis program: at which
time the program proceeds to the next segment. In the ISOTI-IERMAL segment. the cell
is held at its existing temperature for a designated time. The SAMPLING INTERVAL
segment is used to set the sampling rate for data collection. In the last segment. RAMP.
the cell was heated at a designated rate (i.e.. 5°C/min) to a designated temperature.

Using these different segments. the method for the dynamic or ramped tempera-
ture experiments was set up as follows:

1. INITIAL: 25°C

2. SAMPLING INTERVAL: 1.0 second

3. ISOTHERMAL: 1 minute

4. RAMP: 5°C/minute to 220°C
In this case. the DSC cell was first initialized at 25°C. and the sampling interval was set
to collect data at one second intervals. Upon external trigger to the thermal analysis
program. the cell was held at 25°C for one minute. then the temperature was increased
linearly at a rate of 5°C/minute until it reached 220°C. at which time the experiment
was completed and the cell cooled down to the room temperature.

Similarly. the isothermal experiments were set up using the INITIAL. SAMPLING
INTERVAL. and lSOTI-IERMAL segments to define the time-temperature history.

4.2.2.2 Initialization Procedure for the DuPont 910 Calorimeter

The dynamic experiments were begun after all the necessary parameters had been
entered in the Thermal Analysis program. To start the initialization procedure. the bell
jar. the cell cover. and the ceramic lid were removed from the DSC cell. and the sample

pan and the reference pan were placed on the calorimeter cell as shown in Figures

 

4.14%. T}

   
 
  
  
  
  

were replac
cc/mlnute.
through the

The i
achieve iso
mize the l
preheated

for the m1

75
4.14a.b. The pans were covered by the ceramic lid. then the cell cover and the bell jar
were replaced. and a nitrogen atmosphere was induced into the bell jar at a rate of 50
cc/minute. The data acquisition and controlled heating cycle was then initiated
through the thermal analysis program.

The isothermal experiments were initiated differently. Since it was desired to
achieve isothermal conditions throughout these experiments. it was important to mini-
mize the heating time for the sample. To do this. the calorimeter cell was first
preheated to the desired temperature using the INITIAL segment. and the rate was set
for the nitrogen atmosphere into the bell jar. The ISOTI-IERMAL segment was then
started. and the bell jar. the cell cover. and the ceramic lid were removed. The reference
- and samples pans were then placed in the calorimeter cell. and the jar. cover. and lid
were replaced as quickly as possible. The time required to complete these maneuvers

after the ISOTHERMAL segment was started was approximately 30 to 40 seconds.

4.2.2.3 Post-Processing Procedure

After the completion of each experiment. several post-processing programs as-
sociated with the Thermal Analysis System were utilized.

In each repetition of the dynamic DSC experiment. the area under the heat of
reaction rate cure was integrated. and the total area under each curve or the total heat
of reaction for that experiment. was determined using the Thermal Analysis software.
The heat of reaction rate data and the integrated data (the cumulative heat of reaction)
were then plotted using the Thermal Analysis plotting subroutines. The resulting plot
for one repetition of the dynamic DSC experiment is shown in Figure 4.15.

In each of the isothermal experiments. the associated baseline was subtracted off
the heat of reaction rate curve. and the results were plotted using the plotting sub-
routine. A baseline consists of the heat of reaction data for a sample after it has been
fully cured. assuming the same curing method: it serves as a reference line for the heat
of reaction rate data of the samples during curing. After the baseline was subtracted
off. the modified heat of reaction rate data was then transferred to a floppy disk in ASC

format for later processing.

76

 

 

 

 

Figure 4.1V4a DuPont Instruments 910 Differential Scanning Calorimeter
Cell (left). Showing Bell Jar (right), Cell Cover (center), and Ceramic
Lid (on cell)

 

 

 

 

Figure 4.14b Close-up View of Sample Pan (right) and Reference Pan
(left) in Calorimeter Cell Shown Above.

 

 

 

ZHI\Um has DONNIQOMN UCOEEOQ

No ”v“ mm\m0\mo ”lulo c3! ZHI\U~ m U QNWIQ mm ”bu—«uh!
bhoum ”LOUILIQO DE 0000 .m« nannm
No.mlwlr.b u< "luau. U m S no .omw‘dnutlh 2:355

77

859895 $583 sum—«.2 352:. .
68.5 as»: 38:83 own 3885 05 3 :2 25 88.. Sun 28:985.. 28: B a. flaw...—

 

 

 

 

 

 

oomm accuse <m.~> noeocom Leda. was»
ow mm on mm om ma 0“ n a
O . b L b . b p b t , F b L p N. O!
. a.
. u\am.mvv .
c"e«6.m« xxx .
\\\\\
cm 1 I o . 0
th\\
\Ii‘lv‘ll \\ o 1
1:. .. \.\\\ \\ _.
_ xxx 4;
n \\ _
._. .\. n .
2: 1 .\. .r umd
m x .\ m.
a . n cow 1.
J \ a
B .u
M , C
a on” t \ t #6
a \\ x l .
.. t m“
m \ w .
.t .\\\\\\ xxx
\\ \\\\\ 00V 1
com 1 \ttttttttt rm. o
. \s
\ 1
. css.m.mm .
cum 0.0
szxum hq oommtuomm nucerou
mouva mo\m0\mo "cu-o cam 2H:\u_ m u ommuo mw ”nonuox
Chum "Louncono U m D as good“ 33m
8.33:» 2. 82m «o.ommtmmmxp 835m

(G/Mi Hora 3'9H

 

F on
were con
ments we
experimer
minutes.

The
CCdUI‘C d:

Procedure

 

10°C belo
heated at
the taupe

Stlected is

 

 

 

 

rtaction c
isomer-ma]
were com

DcralUre.

78
Wm

Four difi’erent sets of isothermal experiments and one set of dynamic experiments
were conducted using the DuPont Thermal Analyzer System 9900. The dynamic experi-
ments were conducted using the segment methods discussed in Section 4.2.2. These
experiments were repeated eight times. with each experiment running for about 40
minutes.

The four temperatures for the isothermal experiments were chosen using the pro-
cedure described by Sichnia (DuPont Applications Brief No. TA-93). Using this
procedure. the isothermal temperatures were chosen from an interval defined between
10°C below the onset of cure and a point midway to the peak maximum of a thermoset
heated at a rate of 5°C/min. Since the dynamic experiments followed this heating rate.
the temperatures were selected using these thermosets. Sichnia recommended that the
selected isothermal temperatures be 5°C to 10°C apart. Figure 4.16 shows a heat of
reaction curve from one of the dynamic DSC experiments with the interval for the
isothermal temperatures indicated. Based on this interval. the isothermal experiments
were conducted at 60°C. 70°C. 100°C. and l 10°C. with three repetitions at each tem-
perature. ‘

Preliminary experiments were run to establish the required curing time at each
temperature. The resulting time-temperature segments used for each set of isothermal
experiments are given below.

‘ 60°C: INITIAL: 60°C

SAMPLING RATE: 2 seconds

ISOTHERMAL: 300 minutes
70°C: INITIAL: 70°C

SAMPLING RATE: 2 seconds

ISOTHERMAL: 240 minutes
100°C:INITIAL: 100°C

SAMPLING RATE: 1 second

ISOTHERMAL: 90 minutes
110°C:INITIAL: 110°C

 

2Hz\‘ur. F41 ccnnlrx. t“ .‘ail‘l-Irl‘rh

No ”v“ mm\.mO\mo ”coco cam 23‘}: m U omm.-u mm ”Datum:
bboom "LOuILoao DE 0000 .mn eanm
No.mlmII... "4 ”Oman. U m D no .ONNIONEIh ua~nsem

79

.855 own 283.5 a Spa. 38:55 own 388:8»— 8.. acafionnue .8 Eugene 2 a. any...

 

 

 

 

 

 

comm ucomaa <m.~> aoeocoo “no. oeauoemneok
omw . omm . om" . an: t Wm r o .

m or

no.0

im.o

W
TIII. 3:85 .358 I1

Iv.o

Im.o

0.0
sz\om P4 uommruomm "ucoseoo
mo”v« mm\m0\no "anon cam 2H2\o~ m u ommru mm ”nocumz
Phoum "Lou-cone U m D a... 88.3 83m
mo.mmmmzhn« "want «o.ommtmwmxh “mananm

(5/M) “Old 199H

Each of ti

 

sample pal

ASL

ber of repr
4.3 Trans

The
those desc
co[liposite
were taken
With them
corlducted
data aCQUIs
mistauom

acquismon

 

amplifleis a
Winning
one adding
3'2 Urlder Sir
In eacl
Curing Droce
So that e ac}
Shorter e.

er
l.- r-
:15 sub-seem:

80
SAMPLING RATE: 1 second
ISOTHERMAL: 60 minutes
Each of the above isothermal segments was also repeated once using a previously cured
sample pan to establish a baseline for each temperature regime.
A summary of the DSC experiments. including the time. temperature. and num-

ber of repetitions is shown in Table 4.4.
4.3 Transient Temperature Measurements during Curing

The transient temperature experiments discussed in this section were similar to
those described in Section 4.1 for the transient temperature experiments using cured
composite samples; however. in these experiments. the temperature measurements
were taken during the curing process: therefore. the prepreg laminate was instrumented
with thermocouples and heaters during the stacking process. Two experiments were
conducted using the procedures described in this section. In the first experiment. the
data acquisition procedure was similar to that described in Section 4.1.3. except that a
VAXstationII/GPX microcomputer was used instead of the Plessey system. A new data
acquisition procedure was developed for the second experiment which included new
amplifiers and a new computer controlled power supply for the heaters. These two ex-
periments will be referred to as Exp. 3.1 and Exp. 3.2 in subsequent chapters. Finally.
one additional experiment (Exp. 3.3) was conducted using the cured samples from Exp
3.2 under similar experimental conditions.

In each experiment. the heaters were activated at regular intervals throughout the
curing process. In the analysis. procedure. these intervals were considered separately.
so that each curing experiment could be considered as a combination of 15 to 20
shorter experiments. The procedures used for these experiments are given in the follow-

ing sub-sections.

Table 4
Erpertme

 

Expe.
NC I

 

 

 

 

81

Table 4.4 Experimental Parameters for Differential Scanning Calorimetry
Experiments.

 

 

Exper. Experiment Temperature (°C) Time No. of
No. Type (Heating Rate)a (min) Repetitionsb
2.1 Dynamic 25°C-220°C (5°C/min) 39 8
2.2 Isothermal 60°C 300 3
2.3 Isothermal 70°C 240 3
2.4 Isothermal 100°C 90 3
2.5 Isothermal 110°C 60 3

 

 

 

 

 

 

 

a. For dynamic experiments only.
b. Does not include experiments to establish baselines.

 

    

proced'

outliner

heaters

the pro

and the

4.3.1.1

TI
tion in (I
Inc.) wei
and hell

coming

 

Sheet of

distfibut

4.3.12 5
The
du‘mg ti
and then
themed :
prepreg
Smocth E
Placm 11".
described

no” of t:

82

W

Prepregs. composed of EPON 828/mDPA epoxy and continuous AS4 carbon
fibers. were also used in these experiments. The epoxy was prepared according to the
procedures in Section 4.1.1.1. and the prepregs were prepared following the procedures
outlined in Section 4.1.1.2. The prepreg plies were stacked with thermocouples and
heaters embedded between the plies: these thermocouples were prepared according to
the procedure given in Section 4.1.2.1. The preparation procedure for the the heaters

and the stacking process are discussed in the following sub-sections.

4.3.1.1 Heater Preparation

Thin. flat. resistance heaters were used to supply the heat flux boundary condi-
tion in the transient experiments. No 7.6 ohm ThermofoilR heaters (Minco Products.
Inc.) were laid on top of each other so that the heating elements were not overlapping
and held in place using a very thin layer of the Silicon Heat Sink Compound (Dow
Corning Corp). The heated surface of the heaters measured 12.4 cm by 9.8 cm. A
sheet of aluminum foil was cut and adhered to each side of the heaters using the silicon
compound to protect the heaters from the epoxy resin and to insure a uniform heat

distribution.

4.3.1.2 Stacking Procedure

The prepreg was instrumented with the thermocouples and the resistance heaters
during the stacking procedure. Forty-four plies were cut from each prepared prepreg.
and then they were stacked with the fibers parallel to each other ([0°I“) and instru-
mented symmetrically with the thermocouples and the heaters. The instrumented
prepreg laminate was set between porous TeflonR cloths. bleeder cloths. and flat.
smooth aluminum plates covered with release ply, and then the entire assembly was
placed in a vacuum bag for curing. This procedure is very similar to the procedure
described in Section 4.1.1.3, with some important differences. such as the instrumenta-

tion of the laminate with the thermocouples during the stacking process. and the

F
hmhs

 

mhpr

off

0f]

lur

83

inclusion of the bleeder and TeflonR cloths on both sides of the laminate. An outline of

this process follows. -
1. No 15.2 cm by 17.8 cm pieces were cut from the porous TeflonR cloth. the
bleeder cloth. and the release ply plastic. In addition. a small piece ‘of thick
bleeder cloth (approximately 8 cm by 10 cm) and piece of vacuum bag material at
least 40 cm by 66 cm were cut.
2. The prepreg was taken out of the freezer. and allowed to cool just enough to
become pliable (apprordmately 5 minutes). and dry ice was obtained and chopped
into small chunks.
3. The prepreg was spread out on a sheet of foil. and a piece of poster board was
placed under the section to be cut. Using a carpenters square as a guide. forty-
four 10.2 cm by 12.7 cm plies were cut from the prepreg using a utility knife with
a sharp blade.
4. One piece of the release ply was adhered to a smooth 23 cm by 23 cm
aluminum plate. using small amounts of Tacky TapeR at the comers.
5. Several prepreg plies were laid out on the foil. and the remaining pieces were
placed back in the freezer until needed. Using thermal gloves. the dry ice was
spread over the pieces of prepreg. and one of the release plies on each prepreg ply
was removed. taking care not to peel off any fibers with the plastic.
6. After any frost which had accumulated on the exposed surface had evaporated
off. two prepreg plies were stacked together with the fibers parallel to each other.
This process was repeated with the other plies. except that some plies were instru-
mented with thermocouples. and a resistance heater was placed at the mid-plane
of the plies (step 7.).
7. The plies were instrumented with thermocouples by placing the thermocouple
junction at the center of the exposed prepreg surface. with the exposed portion of
the wires oriented perpendicular to the fibers. A second ply was stacked as
described above. taking care to keep the thermocouple in place. The resistance
heater was placed in between the middle two plies. A roller was used in an effort

to remove air trapped between the plies or between the prepreg and the heater.

84
8. Once the prepreg laminate had been stacked and instrumented. one of the last
two release plies was removed. and a piece of porous TeflonR cloth was centered on
the exposed surface. The sample was then placed on top of one of the bleeder
cloths placed on the aluminum plate covered with release ply.
9. Cork tape was used to as a darn around the sample. The dam was built up
until it was level to the sample thickness: special care was taken not to disturb
the thermocouple wires and the heater connectors.
10. The last release ply was removed from the sample. and the exposed surface
was covered with the second pieces of porous Teflon“ and bleeder cloth. Finally.
the aluminum plate covered with release ply was placed on top of the bleeder
cloth. A stacked composite laminate. with the porous TeflonR and bleeders cloths
cut away to expose the thermocouples at the surface. is shown in Figure 4.17.
(This photograph was taken after curing.)
l 1. The entire assembly was placed on one half of the vacuum bag material and
the other half of the sheet was folded over the assembly. Tacky TapeR was used to
seal the bag. and a short rubber hose (approximately 12 cm long) was placed at
one end. The remaining piece of thick bleeder cloth was placed over the mouth of
the tube and held in place with Tacky Tape“. The assembly was checked visibly
for leaks. and extra tape was added where needed. especially around the rubber
tube. A stacked and instrumented laminate. sealed in a vacuum bag is shown in
Figure 4.18.
12. The vacuum bag with sample enclosed was placed in the freezer until ready

for curing.

W

The curing process for thermoset epoxy composites requires both elevated tem-
peratures and applied pressure. The required temperature regime was obtained by
curing the stacked prepreg laminate in a laboratory oven (Matheson Scientific). and a
mechanical press was designed and built to fulfill the pressure requirements. The re-

quired pressure for the type of epoxy used is 689 kPa (100 psi). and the composite

85

 

 

 

 

 

Figure 4.17 Stacked Composite Prepreg, Instrumented with Thermocouples
and Heaters. (Photograph taken after curing.)

 

 

 

 

 

Figure 4.18 Stacked and Instrumented Composite Prepreg Sealed in a Vacuum Bag.

 

the m
held :1
nuts.

vacuu
spring
spring |
weight
quired
1am1na
0f Engii

 

 

I
in Sec
device:
Sides ‘
hYdral

CQuipn

94;

lure d
“he a:
A/D c

Come

pQSite

86
samples were approximately 12.7 cm by 10.2 cm (see Section 4.3.3). resulting in a re-
quired load on the sample of 890 N.

The press was designed using two 28 cm by 28 cm by 1.2 cm thick steel plates
and eight 5 cm compression springs with springs constants of 175 N /cm. The design
called for eight 1.3 cm holes to be drilled in each plate. four at each corner. and four at
the rnidplane of each edge. at a location 1.9 cm from each edge. The springs were to be
held in place at these locations using 1.3 cm by 12.7 cm long bolts with the appropriate
nuts. The press was designed such that the stacked prepreg laminate sealed in the
vacuum bag lay between the plates. and the bolts fastened the plates together with the
springs on top of the plates. Pressure was applied through the deflection of each
spring 0.75 cm. which resulted in a total force from the eight springs of 890 N. (The
weight of the upper plate was subtracted from the required load in determining the re-
quired deflection for each spring.) A photograph of the press design with the composite
laminate in place is shown in Figure 4.19. The press was machined at the MSU College
of Engineering Machinery Shop.

It should be noted that preferably the autoclave or the hydraulic press discussed
in Section 4.1.1.4 would have also been used in this consolidation process. These
devices are preferable because they allow for uniform pressure and temperature on both
sides of the composite throughout the curing process. Neither the autoclave or the
hydraulic press could be used. however. because the data aquisition system and this

equipment were situated in different locations. and none of these could be moved easily.

WWW
The VAXlab system (Digital Equipment Corporation) was used to record tempera-

ture data during both the transient temperature experiments. In the first experiment.
one analog-to-digital (A/D) converter was utilized. and in the second experiment. two
A/D converters and one digital-to-analog (D/A) converter were incorporated. The A/D
converters were used to record temperature data from the thermocouples in the com-

posite sample. and in the second experiment. the D/A converter was used to control a

87

 

 

 

 

Figure 4.19 Mechanical Press with Composite Laminate in Place.

 

pow

moc

 

amp
men
expe
the i!
to It.

SECOI

inth

expc
4.3.3.

Eqm;
board
Verter
AXV1
(comm
0mm
the fi
which
be Us.
tachec
Signal
D/A c.

88

power supply which supplied the heat flux for the resistance heater. The ther-
mocouples produced a signal on the order of millivolts. and amplifiers were used to
amplify the signal for the A/D input. New amplifiers were used in the second experi-
ment. while the Data Translation amplifiers (Section 4.1.3.2) were used in the first
experiment. Two different computer programs were employed for these experiments. In
the first case. the programs used on the Plessey system (Section 4.1.3.3) were modified
to run using the VAXlab system subroutines. A new program was developed for the
second experiment. using both the A/ D and D/A converters.

The details of the data acquisition systems used for the two experiments are given
in the following sub-sections. Schematics of the data acquisition systems for the two

experiments are given in Figures 4.20 and 4.21.

4.3.3.1 The VAXlab Data Acquisition Hardware

The VAXlab system was run on a VAXstationII/GPX microcomputer (Digital
Equipment Corporation). The system hardware included a KWVI 1-C real time clock
board and two AXV 1 l-C combination boards with one A/ D converter and two D /A con-
verters each. Each A/D converter was set for a maximum of eight channels. The
AXVl 1-C boards were equipped to run using synchronous (sequential) or asynchronous
(continuous) input/output (I/O). however the asynchronous I/O was limited to using
only the A/D or the D/A. but not both. Therefore. the asynchronous I/O was used in in
the first experiment. and the synchronous I/O was used in the second experiment
which included a controlled power supply. The KWV l 1-C is a clock module which can
be used to trigger the AXV11-C devices. In the second experiment. the clock was at-
tached to both boards so that there was a total of sixteen available A/D channels. The
signal range for the A/D converters was set from 0 to 10 volts. and the range for the

D/A converter used in the second experiment was set from ~10 to 10 volts.

 

 

 

 

 

 

89

 

   
  

Amplifiers (8)
(Data Translation)

VAXStationII/GPX
Microcomputer
(Digital Equipment
Corp.)

 

   

 

 

 

     
 

stacked

  

KWVl l-C
real time clock

 

 

 

(in vacuum ., _ '
bag) , ' x '
J A
0V6“
Experimental
Set-up
to 1
Power Supply heat"
(The Superior
Electric) multimeter

(amp-meter)

 

   

multimeter

-—>
(Volt-meter)

 

 

 

 

Figure 4.20 Schematic of Data Acquisition for First Transient
Experiment during Curing.

 

 

 

VAXStationII/GPX
Microcomputer
(Digital Equipment
Corp.)

   
  

  

AXVll-C

 

 

 

 

 

  

A Amplifiers

(12)
(Ectron)

g.

 

 

 

 

 

 

 

 

 

 

 

 

AID Converter
D/A Converter

 

  

KWVI l-C
real time clock

 
   
      
 

 

:-’- . é. . . . . ...' .I._ ”:5.

    

DC Power
Supply

stacked

prepreg

laminate
(in vacuum

has)

 

1 MD

b

thermocouple wires

 

 

 

 

 

 

Experimental :

Set-up

 

 

(Hewlett-

Packard

Co.)

to heater '

 

Figure 4.21 Schematic of Data Acquisition for Second Transient
Experiment during Curing.

 

4.3.

 

 

amplifie}
aniphflex

mined.

The
DI] the an

potentiorr

91
4.3.3.2 Signal Amplification

Since the signal from the thermocouples was on the order of millivolts and the
range of the A/D converters was from 0 to 10 volts. amplifiers were used to amplify the
signals.

The eight Data Translation amplifiers discussed in Section 4.1.3.2 were also used
for the first experiment on the composite samples during curing. and twelve new Model
687 DC Amplifiers (Ectron Corporation) were used in the second experiment. These
Model 687 amplifiers were equipped with set gains of 10. 20. 50. 100. 200. 500. and
1000. with a vernier adjustment to obtain values between the set points. and the fre-
quency response of these amplifiers was on the order of 3 kHz. A gain of 1000 was
used in this case. The remainder of this sub-section focuses on the Model 687
amplifiers.

No optional accessories. Model 683. Universal Thermocouple Adaptor (UTA). and
Model 684. Ambient Temperature Compensator (ATC). were installed on each Model 687
amplifier for operation with thermocouple inputs. The Universal Thermocouple
Adaptors were attached to the input connector at the rear of each amplifier. which
served as the reference junctions for the thermocouples. The ATC is a card which was
attached to the amplifier board and operated in conjunction with the UTA to simulate
an ice point reference junction for the particular type of thermocouple used. which was
in this case Type E. chromel-constantan.

Prior to use. the gains and zeros of the amplifiers were checked according to the
instruction manual for the amplifiers (Ectron Corporation. 1982). To check the actual
gain of each amplifier. a power supply with set voltage outputs was connected to each
amplifier. bypassing the UTA. 'lhen. one and ten millivolt signals were sent to the
amplifier. and the output signals were recorded with the set gain at 1000. From these
readings. the fractional differences between the set gain and the actual gain were deter-
mined.

The amplifiers were zeroed with the UTA switched off. There are two zero controls
on the amplifiers: a referred to input (RTI) potentiometer and a referred to output (RTO)

potentiometer. To zero the amplifiers. a jumper was first connected across the signal

1an
I‘CIll
was

taint

adju:
were
0°C t

adjus

4.3.3.

 

the l"
regular

heater

led us
The M.
to 60 t
Supply
sollrce
The ac
Source
vhas re;
““3 W.
the Dov

“'33 3C“

C

92
input terminals. then the gain was adjusted to 10. and any offset (from zero) was
removed using the R10 potentiometer. Then the gain was set to 1000. and any offset
was removed using the RTI. This process was repeated until a zero reading was ob-
tained from both RTO and RTI potentiometers.

The actual reference junction temperature was also measured and adjusted. After
adjusting the RTO and RTI potentiometers. the UTA was turned on. and thermocouples
were connected to the amplifiers. The junctions of the thermocouples were placed in a
0°C electronic ice bath (Omega. Inc.). and the output signals from the amplifiers were

adjusted to read 0.0 mvolts using the potentiometer on the UTA card.

4.3.3.3 Controlled Heat Flux

The same power supply apparatus described in Section 4.1.3.4 was also used in
the first transient experiment during curing. The timer was operated manually at
regular intervals throughout the curing cycle to provide the required heat flux to the
heaters.

In the second experiment. the input voltage to the resistance heaters were control-
led using a DC power supply (Model 6024A. DC Power Supply. Hewlett-Packard Co.).
The Model 6024A is an autoranging 200 W power supply. with an operating range of 0
to 60 volts and 0 to 10 amps. and with remote programming capabilities. The power
supply was operated in a constant voltage. voltage control mode. where an input voltage
source from 0 to 5 volts produced a proportional output voltage from zero to full scale.
The actual output voltage of the power was measured remotely. The input voltage
source was obtained from the VaxStationII/GPX D/A converter. and the output voltage
was read by one of the A/D converters. Since the voltage ranges for both the A/D and
the D/A on the VaxStationII/GPX were set at 0 to 10 volts. the input voltage source to
the power supply was scaled to 0 to 5 volts using a 2:1 divider. and the output voltage
was scaled from 0 to 60 volts to 0 to 10 volts using a 6: 1 divider.

4.3.;

bine

 

Th5

111°C?
to (11
0pc

age

I”Toce
bkun
data

the F

Eddy
that
tme

hEat

D 0“)

93
4.3.3.4 Data Acquisition Program

The data acquisition programs. DATACQ and DATFIT (Osman. 1987). were com—
bined and modified to run on the VaxLab system for the first experiment during curing.
This new program. DATAAQ. utilized the Vaxlab LabStar I/ O Routines (LIO) (Digital
Equipment Corporation. 1986) to read thermocouple data using one of the A/ D convert-
ers. The operating mode of this program was similar to that described for DATACQ and
DATFIT in Section 4.1.3.3. The remainder of this section is devoted to the discussion of
a new data acquisition program. DATA_DA_AD (Appendix B) . which was used in the
second curing experiment.

DATA_DA‘AD was written for the VaxStationII/GPX to read from the ther-
mocouples and the power supply using the two AXV 1 1-C A/D converters. while writing
to the power supply using an D/A converter on one of the AXV1 l-C devices. These
operations required the use of the LIO routines. The program converts the binary volt-
age readings from the thermocouples to temperature using the LabStar Signal-
Processing Routines (LSP) (Digital Equipment Corporation. 1986). and it converts the
binary voltage readings from the power supply to heat flux. The program also sets up a
data file for use in the parameter estimation program. PROP1D_CURE. A schematic of
the program and its subroutines is shown in Figure 4.22. and a detailed description of
the program is now presented.

The parameters required for data acquisition are first entered using the sub-
routines SETUP_DATA. SETUP_POWER. and SETUP__AMP. In the first subroutine.
SETUP_DATA. the user enters the sampling interval. the total number of A/D channels
(maximm of twelve). and the total number of data points points for each channel. In
addition. the program allows the user to average data within each sampling interval. so
that the number of data points to be averaged over each sampling interval is also
entered. In the second subroutine. SE'IUP_POWER. the parameters for the controlled
heat flux are entered. These values include the surface heated area of the sample in the
transient experiments. the resistance of the heater. and the input voltage data for the
power supply. namely, the times at which the power supply is activated. and the

desired voltage input corresponding to that time. The D /A is set up in such a way that

94

 

 

Program
DATA_DA_AD

Initialization:

Set-up sampling
interval. number of
data points. etc.

   

       

Subroutine
SETUP_DATA

. Subroutine
SETUP_POWER

   

Input voltage
values for
controlled heat flux

  
     
   

Subroutine

Convert volts to
binary numbers

     

Set-up amplifiers;
adjust gain. zeros,
and weights

 

Set-up data
acquisition hardware

(KWVll-C. AXVll-C)

Start data acquisi-

Close files and l
shutdown hardware

 

Data Processing:

Find zeros for .
intial data values ' Subroutine
AUTO_ZERO

Convert binary data S b t'
to temperatures and u rou me
calculate heat flux PROCESS_DATA

Convert binary Subroutine
numbers to volts

 

 

Figure 4.22 Schematic of Data Acquisition Program DATA_DA_AD.

 

 

95
once it was activated. it continued to send the same signal until the signal was
changed. These data are converted to binary format in subroutine TRANS_DAC. where
the input 0 to 10 volts are scaled to approximately 0 to 2048 in binary format.

Finally. the 0°C reference junctions. the gains and the weights of the amplifiers
can be adjusted in subroutine SETUP_AMP. The 0°C reference junctions may be ad-
justed by the user. or the program will calculate the 'zeros’ by averaging the first few
data points over a time in which no heat flux is applied. assuming the conditions are
isothermal. The gains of the amplifiers can also be adjusted. and a weighting factor is
assigned to each amplifier. These values are either set to one or to zero. for the case of
a ’bad' amplifier or a broken thermocouple.

Once the initial set up procedure has been completed. the program begins the
data acquisition hardware setup. which utilizes the LIO routines. First the devices are
attached. including the KWVII-C clock. and the two AXVll-C boards (AXVll-Ca and
AXVI 1-Cb). The clock is then set up to read data at the desired rate. and both AXVll-
C devices are set up to read the desired number of channels from the A/D converters.
The AXVll-Ca device is set up to sweep through all channels on each clock trigger.
while the AXV 1 1-Cb device is set up to trigger on the 'READ' statement of the AXV 1 1-Ca
device. The AXV 1 l-Cb is also set to write to the power supply using the D/A converter.

The clock is started by a trigger from the user. and data is read from the A/D con-
verters and written to the D/A converters synchronously. The data is immediately
written to a file after it is read. in case of unexpected program interruption. Once the
desired number of data points have been read. the program begins the shut down pro-
cedure. in which a terminating zero voltage is sent to the power supply. and the clock is
stopped. and the devices are detached.

The analysis portion of the program then begins. If indicated. the 0°C reference
junctions are calculated in subroutine AUIU_ZERO. Here the binary reading from the
thermocouples are averaged over the first few time step. and the 'zeros' for each channel

are calculated from

Z! = (Bug - B‘w‘g‘)/l'lt

96

where z‘ is the 'zero’ adjustment for the ith channel. B“" is the average of the binary
data over all the channels and over nt time steps. 13“" is the same average. but just for
the ith Channel. and gt is the gain for the ith channel.

The subroutine PROCESS_DATA is used in the rest of the analysis. The data are
first converted to voltages using ’I‘RANS_ADC (0 to 4095 ~ 0 to 10 volts). and the data
from the power supply are adjusted to the actual range of the power supply (0 to 60
volts). The data readings from the thermocouples are then adjusted to account for the

gains and the 0°C reference junctions as follows:
vi. = Vi gt + 21

where V: is the adjusted voltage for the ith channel and V‘ is the unadjusted voltage.
These values are converted to °C using one of the signal processing subroutines.
LSP$THERMOCOUPLE_E. in the Vax LabStar system. which is specifically for Type E.
chromel-constantan. thermocouples. The heat flux associated with the resistance
heaters is calculated from the heated area. the resistance of heater. and the voltage
measurements. Finally. the time. heat flux. and thermocouple readings in °C are writ-

ten to a file for use in the parameter estimation program. PROP1D_CURE.

4 ‘car -11' _ -_ ’ u «u --fr n: .i'; -mo‘ ._ ur urm ‘ u 4°, 11'
The procedures for the two experiments to measure temperature in the carb-
on/epoxy composite during curing were similar. except that some of the procedures
used in the first experiment were modified in the ’second experiment due to the pur-
chase of new amplifiers and a new powers supply. In both experiments. the stacked
prepreg laminate (Section 4.3.1.2) was cured in the press as described in Section 4.3.2.
using the standard cure cycle recommended for this type of composite (Rich. 1987).
This cycle included a heating up period from room temperature to 75°C. a two hour
isothermal period at 75°C. a second heating period to 125°C. and a two hour isother-
mal period at 125°C. (This is the same cure cycle used for the disks prepared for the

transient experiments using cured composite samples. described in Section 4.1.1.) The

97
heaters embedded in the laminate were activated at regular intervals throughout the
the curing cycle. The data acquisition programs were set up to record temperature data
continuously throughout the entire cure cycle.

At the beginning of each experiment. the laboratory oven (Section 4.3.2) was first
preheated to 75°C. and the appropriate data acquisition program for each experiment
was set up. The prepreg laminate. sealed in a vacuum bag. was then taken from the
freezer in the Composite Materials and Structures Center and set out at room tempera-
ture for about five minutes. A valve was attached to the rubber hose on the vacuum
bag. and then it was connected to a vacuum pump. Pressure was reduced in the bag to
-99 kPa for ten minutes. At this time the valve was closed. and the vacuum bag. includ-
ing the short hose and valve. was disconnected from the vacuum pump and transported
to Rm. A-32. Research Complex / Engineering as quickly as possible. The vacuum bag
was placed in the press. then the press was assembled. and pressure was applied to the
laminate through the depression of the springs on the press. Due to the large mass of
the press. extra heaters were attached to the top and bottom extremities of the press to
reduce the heating time required to heat the press from 25°C to 75°C and from 75°C to
125°C. (These heaters were disconnected during the short heating intervals
throughout the isothermal periods.)

The press was then placed in the preheated oven. and the thermocouples were
connected to the extension thermocouple cables. and the heaters were connected in
parallel to the power supply as quickly as possible. The oven door was closed. and the
data acquisition program was started. along with a separate stop watch.

In the first experiment. the same power supply discussed in Section 4.1.3.4 was
used. The power supply was disconnected from the timer during the initial heat up
period in the curing cycle and during the heat up period from 75°C to 125°C. During
these periods. power was continuously supplied to the internal heaters in the stacked
laminate and to the heaters fixed on the surface of the press in order to reduce the time
required to heat the sample and the press. At the end of the first heat up period. when
the laminate temperature was close to 75°C. the power supply was disconnected from

the heaters on the press and reconnected to the timer. Using the stop watch to record

98

events with respect to overall experimental time. The heaters embedded in the laminate
were activated at regular intervals using the timer. throughout the isothermal curing
period at 75°C. At the end of the first isothermal period. the timer was disconnected.
and the oven temperature was set at 125°C. The heaters on the press were recon-
nected. and power was supplied to the heaters in the same manner as in the first
heating up period. Once the laminate was close to 125°C. the heaters on the press
were disconnected. and the timer was reconnected. and the heater embedded in the
laminate was activated at regular intervals as before. The output temperature
responses resulting from the imposed heat fluxes were recorded continuously
throughout the whole experiment using the data acquisition program. DATAAQ.

In the second experiment. the power supply was computer controlled so that the
complete history. including the heating up periods and heating intervals during the
isothermal portions of the cure cycle. for the power supply output was entered through
the data acquisition program at the beginning of the experiment. Tire stop watch was
activated at the time the sample was placed in the oven. in case of an unplanned inter-
ruption in the data acquisition program. and the external heaters on the press were
connected during the initial heating up period to 75°C. At the end of the first isother-
mal period. the oven temperature was increased to 125°C. and the external heaters on
the press were once again connected during the heating up period from 75°C to 125°C.
The output responses from the thermocouples and from the power supply were read
continuously using the data aquisition program. DATA_DA_AD.

In both experiments. after the curing cycle was completed. the oven was turned
off. and the data processing portions of the data acquisition programs were completed.
Once the cured samples were completely cooled. they were cut apart at the ther-
mocouple junctions. perpendicular to the thermocouples. using a diamond circular saw.

and then the exact thermocouple locations at the cut surfaces were measured.

W
The experimental parameters in the transient temperature experiments on the
carbon/epoxy composite materials during curing included the curing cycle. the data

99
sampling interval. the heating interval for the embedded heaters during the isothermal
curing periods. the frequency of the heating intervals. the magnitude of the heat flux.
fiber orientation with respect to the heat flux. the number of plies in the laminate. and
the position and number of thermocouples.

In both experiments. the recommended curing cycle discussed previously was
followed. Due to the mass of the press. the heating up periods took much longer than
the usual 20 minutes recommended. Therefore. the isothermal periods were shortened
slightly to account for the longer heat up time. The fibers were oriented perpendicular
to the heat flux. and forty-four plies were used in both experiments. resulting in two
cured laminate plates approximately 4 mm thick for each experiment. (There was one
plate on either side of the heater.) Due to the type of press used in the experiments. it
was felt that an attempt to cure a thicker sample might result in an increased number
of voids.

In Exp. 3.2. which was the first experiment. the data sampling interval was
limited to six seconds. due the data acquisition program used. The heating interval was
set at forty seconds. consistent with that used in the experiments with the cured com-
posite samples. The heating intervals were repeated every ten minutes with the
exception of the first two which were repeated at six minute intervals. This resulted in
a total of fifteen heating intervals throughout the curing cycle. The output power from
the power supply was adjusted to approximately 50 watts for the first half of the heating
intervals and 100 watts for the second half. A total of twelve thermocouples were em-
bedded in the laminate during the stacking process. but only four of the data
translation amplifiers were working at the time of the experiment: therefore. two ther-
mocouples on either side of the heater. and the two thermocouples next to the top and
bottom plates of the press were used for data acquisition. as shown in Figure 4.23.

In the second experiment. the data acquisition program was adjusted so that data
was recorded every second. The duration of the heating intervals was determined using
the optimal experimental criterion discussed in Section 3.4. First. the dimensional
heating time was determined from eq. (3.24h). using the optimal dimensionless heating

time of 2 based on the assumption that the dimensionless heat generation term in eq.

 

100

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

ply 44
1 TC 2
plies 31-43
TC 4
plies 29-30
Note: TC 6
Thermocouple .
junctions are plies 26-28 TC 8
located at the
mid-plane of .
the laminate. plies 23'” ' TC 12
TC 10
heater 4
(plane of _/ TC 11
symmetry) TC 9
plies 20-22
TC 7
plies 17-19
TC 5
plies 14-16
TC 3
plies 2-13
TC 1
ply 1

 

 

Figure 4.23 Thermocouple (TC) Locations in Prepreg Laminate for the First
Transient Experiment during Curing. Plies are Numbered from Bottom to
Top. and the Thermocouples used in Data Acquisition are Numbers 1. 2, 10.
and 11 (shown in boldface type).

 

101

(3.241). g’. was equal to l. a thermal diffusivity value determined from the cured com-
posite at 100°C. and the thickness was chosen as a fixed value equal to the maximum
of the values shown in Table 4.1. This resulted in an optimum heating interval of ap-
proximately 135 seconds. (Altematively, one could also optimize the thickness of the
sample for a given heating rate.) In order to reach steady state conditions after each
heating interval. 500 seconds were required between each interval.

The heat flux used in this experiment was determined from eq. (3.24e). with a
maximum temperature rise of 5°C. The dimensionless temperature. T’. in eq. (3.24e)
was determined from eq. (3.25a) using the optimum heat time. t.’ . and g’ equal to 0.1.
This resulted in a value of T’ equal to 1.045. and the required heat flux was calculated
to be approximately 800 W/m’ .. The results of the DSC experiments (Section 4.2) were
used to justify the assumption of g’ equal to 0.1. The heat generation term in eq.
(3. 19a) is proportional to the heat of reaction rate (tha/dt). and the heat of reaction
rate was determined from the maximum heat of reaction rate data from the DSC experi-
ments conducted at 70°C. The mardmum heat of reaction was approximately 50 W/kg
at this temperature. Using this value as the heat of reaction rate. and the density
values shown in Table 4.3. the calculated dimensionless heat generation term was ap-
proximately 0.4. which is on the order of 0.1. supporting the assumption of g’ equal to
0.1. (Note that there was very little difference (<6%) between the optimal heating times
reported Section 3.4.2 for g‘ = 0.1 and g’ = 1.0.)

A total of sixteen thermocouples were embedded in the laminate during stacking.
but since there were only twelve Ectron amplifiers. only twelve thermocouples were ac-
tually used for data acquisition. The locations of these thermocouples are shown in
Figure 4.24.

One additional experiment was conducted using the cured samples from Exp. 3.2
under similar experimental conditions; however. the heat flux during the heating inter-
vals was increased from 800 W/m’ to 1500 W/m’ . This experiment is referred to as
Exp. 3.3 in Chapter 5. The purpose of this experiment was to provide a means of

evaluating the experimental procedures using during the curing experiments. since

102

 

 

  
  
    
   

 

 

 

 

 

 

TC 12
TC 16
plies 35-44
TC 10
plies 29-34
TC 7
Note: TC 8
Thermocouple
junctions are plies 23-13
located at the
mid-plane of
the laminate. TC’s 3, 4, &
l4
heater TC’s l, 13
(plane of & 2
symmetry)
plies 17-22
TC 5
TC 6
plies 11-16
TC 9
plies l-lO
TC 11
TC 15

 

Figure 4.24 Thermocouple (TC) Locations in Prepreg Laminate for the Second
Transient Experiment during Curing. Plies are Numbered from Bottom to
Top. and the Thermocouples used in Data Acquisition are Numbers 1. 3. 4. 6.
7. 8. 9. 10. ll. 12. 13, and 14 (shown in boldface type).

 

103
properties evaluated from this data could directly be compared with the results from the

experiments discussed in Section 4.1.

Chapter 5

Results and Discussion

The results of the analysis for the estimation of thermal and kinetic properties are
presented and discussed in this chapter. In the first section. the results for the estima-
tion of thermal properties in cured composites from the experiments discussed in
Section 4.1 are given. These results include estimates of thermal conductivity perpen-
‘ dicular to the fiber direction and density-specific heat as functions of temperature for
two different fiber orientations. The results for the estimation of kinetic properties of
the EPON 828 epoxy are presented and discussed in Section 5.2. Section 5.3 is devoted
to an analysis of the estimation procedure and experimental design used in the estima—

tion of thermal properties during curing.

5.1 Estimation of Thermal Properties of Cured AS4/EPON 828-mPDA Composite
Materials

The thermal properties. effective thermal conductivity perpendicular to the fiber
axis and effective density-specific heat. were estimated using the parameter estimation
program PROPI D from the transient experiments using the cured AS4/EPON 828 com-
posite samples (Section 4.1). Three different aspects of the estimation process were first
analyzed to provide insight into the experimental and analytical procedures. These

results are presented in Section 5.1.1. The parameter estimates for the experiments

104

105
shown inTable 4.2. in which [0° 124 and [0 ° .130 ° .160 ° .90 ° 12(5)“) composite samples were

tested. are presented and discussed in Section 5.1.2.

544W

Three important aspects of the estimation procedure were analyzed in detail in
this section. The first of these pertains to the residuals. which give insight into ex-
perimental errors. This analysis is presented in Section 5.1.1.1. The second aspect of
the estimation procedure investigated here relates to the sensitivity coefficients: these
results are presented in Section 5.1.1.2. Finally. a discussion is presented on the se-

quential estimation of the parameters in Section 5. 1. 1.3.

5.1.1.1 The Residuals

The residuals are defined as the differences between the experimental tempera-
tures and the calculated temperatures using the estimated parameters at corresponding
times and locations. Close examination of the residuals can give insight into the ade-
quacy of the experimental design.

The residuals for one typical experiment are presented and discussed here. These
results correspond to the second repetition of Exp. 1.9. shown in Table 4.2. In this par-
ticular experiment. six thermocouples were used. Three were located at the heated
surface (TC #1. TC #2. and TC #3); the fourth and fifth thermocouples (TC #4 and TC
#5) were located 3.0 mm and 3.3 mm from the heated surface. respectively. and the
sixth thermocouple was used to deterrrrine the second boundary condition. which was
7.1 mm from the heated surface. The heat flux was activated ten seconds after the in-
itialization of the experiment. The duration of the heat flux interval was approximately
40 seconds. and the total experimental time was 200 seconds. The residuals of the first
two thermocouples. TC #1 and TC #2. and the fifth thermocouple. TC #5. along with the
time interval in which the heat flux was activated. are shown in Figure 5. 1.

Several observations can be made from this figure. First. the magnitude of the
residuals associated with the thermocouples located at the heated surface (TC #1 and

TC #2) are highest at the beginning and end of the heat flux interval. The maximum

106

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1.00 ,
. I — TC,“
m 0.50- a j
'8 . "VNVV
SA 000 # 5“»th . i
%8 WV] ’I/V " V“ 10’ U :3 ”(\[VJJVVUWVJV
9’ — 50- | r/
m .
‘ IL~ll-Ieat Flux j]
-1.00- '
1.00
1 _ - m #2
m 0.50“ g ,:"\'| ‘
. 1 shake "1| |' I“,
:38 0.00 I: 1‘??? E" h ‘1 2,114,; .1“ 1" “ Lg‘gll- I I 'Ia ,r‘
a e j \ 1" ' 1" I V) II I
0) .50d , i
m ' .III
I
100 bfleatfiux j
' l
1.00
-4 m #5
In 0.50- .
'8 ‘ 3 ' ..'. .2.. . : '. . i
SA 000 3. :"ir’f. :' ""1: {1: A , "w .. f i 9.3.45-"
"9.9 - :"..i:2.'°-'--':--3'.-' ' '2 Ir- 31." °' . Ju- .-'
.mv 1: ...:..0.~ ::: .- u ‘e "Z": 'I t
a? -.50- 35'3" 5 ' i"
. C ' I
100 I Heatflux 1
I I I
0.0 50.0 100.0 150.0 200.0
Time (sec)

Figure 5.1. Residuals (°C) for Two Thermocouples Located at the Heated
Surface (TC #1 and TC #2) and One Thermocouple Located 3.3 mm from the Heated
Surface (TC #5) for the Second Repetition of Experiment No. 9 (initial

temperature - 100°C).

. 107

residual is approximately 1 °C. which is approximately 5% of the maxirnum temperature
rise during the experiments. The increase in the magnitude of the residuals at the
beginning and end of the heat flux interval was not as pronounced for TC #5. which was
located away from the heated surface. Second. the residuals for TC #1 and TC #2 were
both biased. but opposite in sign during and immediately after the heating interval.
(The residuals for TC #1 are predominantly negative. and the residuals for TC #2 are
predominantly positive over this time period.) Also. small biases were observed for the
residuals associated with all three thermocouples shown in Figure 5.1 (TC #1. TC #2.
and TC #5) and for the two thermocouples not shown in this figure (TC #3 and TC #4)
during the second half of the experiment. In all three cases shown in Figure 5.1. these
biases were negative. but the biases were observed to be positive for the residuals as-
sociated with TC #3 and TC #4.

The high residuals at the beginning and end of the heat flux interval could have
resulted from errors in determining the exact time in which the heat flux was activated.
As discussed in Section 4.1.3.4. the heater in this experiment was activated manually
through use of a timer. Due to the diffusive nature of heat conduction. these errors
were not as pronounced for the thermocouple located some distance from the heater
(TC#5). Also, finite difference errors tend to be the greatest during step changes in the
boundary condition.

The residual bias over the heat flux interval for the heaters located near the sur-
face was most likely due to the placement of the thermocouples with respect to the
heater. The heating element in each of the heaters used in these experiments traced
back and forth across the heater in rows. leaving a small gap between each row.
Therefore. a thermocouple placed directly beneath the heating element would be ex-
pected to read higher than the average surface temperature. and a thermocouple placed
between the element rows would be expected to read lower than the average surface
temperature. This would result in biased residuals. depending on where the ther-
mocouples were located.

The small biases observed for each thermocouple towards the end of the experi-

ment may have been due to improper zeroing of the thermocouples. These experiments

108
were completed using the Data Translation amplifiers (Section 4.1.3.2) which were very
difficult to zero accurately.
No important points from these observations are that the thermocouples next to
the heated surface were the most susceptible to measurement errors and that one

major source of errors appeared to be related to the heat flux initialization time.

5.1.1.2 The Sensitivity Coefficients of the Estimated Parameters

The sensitivity coefficients are defined in this case as

XI = kgi] and X, = pcp [3:53;]

Monitoring each of these values as a function of time can also provide insightful infor-
mation regarding to the experimental design and the estimated parameters. These
sensitivity coefficients are shown in Figure 5.2 for the first and fifth thermocouple loca-
tions for the experiment discussed in Section 5.1.1. The first thermocouple location
corresponds to TC #1. which was located at the heated surface. and the fifth ther-
mocouple location corresponds to TC #5. which was located 3.3 mm from the heated
surface.

The sensitivity coefficients are useful in determining whether or not the estimated
parameters are correlated: if the parameters had been correlated. the sensitivity coeffi-
cients would have been proportional to each other. From Figure 5.2. the two curves for
X. and the two curves for X, cross over each other and change signs at different times.
indicating that the curves are not proportional. and thus implying that the parameters
are not correlated.

By definition. the sensitivity coefficients indicate the sensitivity of temperature
with respect to small changes in the thermal property values. Thus. relatively higher
values of the sensitivity coefficients are desired for parameter estimation. From Figure

5.2. the sensitivity coefficients are. apprordmately zero for the first ten seconds of the

109

.m .ozugogabggonumgbuém 623: snags—Em.»
65 895m 638: 05 a .. at: as .x is 32.8.: .x 55.880 53:23 In m paw...—

Aommv 055.

 

 

com on: oofi on o
a — n F p P p o." I.
[0'
.I I
~ I I
a /
~ I loll
~ //
e I
\ \t \ \I. II I / .\K .IX
\ \ \ / x. rml )9
\\ \ .7 on
x
\ \ \ x / {Op
\\ \ .x / INII
tttt . x x
\r z
a. /
\\ /
/
.v. - it o
...... . . v.\.\ / / .\ \ f
l I i i I .i.i. .t. .................... .\. .\ \ vocuuflm “guano: EOE Sand .ux
ttttttttttt \ suntan poses: an ax- ... rm
suntan posse: Bob 5:5. m. “x l
rectum pours: as. x I.

 

 

 

1 10

experiment since there was no heat flux over this period. The sensitivity coefficients
corresponding to TC #1 (at heated surface) increased in magnitude immediately with
the onset of the heat flux. while the responses of the sensitivity coefficients associated
with the interior thermocouple (TC #5) were dampened due to the diffusive effects of
conduction heat transfer. The magnitudes of the sensitivity coefi'icients at the heated
surface were higher throughout the heating interval (from 10 to 50 seconds). at which
time they experienced a sharp decrease in magnitude. The sensitivity coefficients at the
interior location also diminished after the heating. interval. but in this case. the
response was delayed.

From these observations. the sensitivity coefficients at the heated surface

provided the most information with regards to the estimation of the parameters.

5.1.1.3 The Sequential Estimates of the Estimated Parameters

Investigation of the sequential parameter estimates is also useful in evaluating an
experimental design. The sequential estimates of thermal conductivity. k. and density-
specific heat. pop. for the experiment discussed in Section 5.1.1 are shown in Figure
5.3. The heat flux was activated ten seconds after the beginning of the experiment. The
estimated parameters fluctuated greatly over this period. with some values off the scale
shown in Figure 5.3. These values can be disregarded since no information was avail-
able from the measurements. (As discussed in the the previous section. the sensitivity
coefficients equaled zero during the first ten seconds.) The estimates for both thermal
conductivity and density-specific heat were constant after approximately 130 seconds.
indicating that additional data would have provided little additional information for the
estimation of the parameters; also it indicates that the heat conduction model is satis-

factory.

5.1.1.4 Insights into the Experimental Design
The observations discussed in Sections 5.1.1.1. 5.1.1.2. and 5.1.1.3 are sum-
marized here. One important observation was that the thermocouples closest to the

heated surface provided the most information for parameter estimation. however. these

111

Y—
p0,,

(MJ/rn3°C)

Estimated Densit
Specific Heat

a 40.03 .. 8332—88 ESE: m dz «cg mo convene”— vacuum
95 age . 2. as: 8....on bacon e8 .0. 538588 aabfi. co gonna—emu assuage an 2:?

 

 

 

 

 

60V we;

.oom om” cod pm 0 .
o H a x Jam ..| H m 0
on .amm . I _

_
_
_
_

3 1 _ ed
_

_ a

j
_
_
_

9N IIIIIIIIIIIIIIIIIIIII till: rqfi
x:
:
___

am- hi .3
f
_

#4

 

 

(Dom/M)
>1 ‘eronpuoo
remnant peaemnsa

1 12
thermocouples were also the most sensitive to experimental errors. In addition. at least
two thermocouples were required at the heated surface to account for the variations in
temperature due to the location of the heating element within the heater. Finally from
the analysis of the sequential parameter estimates. the total experimental time of 200

seconds appeared to be satisfactory.

no. or 0 Iron. no. as .90 0'1 - 0' i r. o .10
AS4ZEPQN 825 ngpgsitg Materials

The thermal properties. thermal conductivity and density-specific heat were es-

timated using the procedures presented in Section 3. 1 for each repetition of the

experiments discussed in Section 4.1.5 and shown in Table 4.2. The results of this

estimation procedure are presented in this section. First. results for the experiments

using the [0°]24 samples are given and discussed in Section 5.1.2.1. and then the

results for the experiments using the [0°.:30°.d:60°.90°] samples are presented

2(symi
and discussed in Section 5.1.2.2. A comparison of these results is given in Section
5.1.2.3. and finally. the results from this study are compared to previously published

data in Section 5.1.2.4.

5.1.2.1 Experiments using [0°]2 4 Samples

The results for the thermal properties. namely the estimated effective thermal
conductivity perpendicular to the fiber axis and the estimated effective density-specific
heat product, of the cured [0°]2 4 AS4/EPON 828 compositesamples are given for the
experiments with approximate initial temperatures of 25°C. 50°C. 75°C. 100°C. and
125°C in Tables C.1-C.5 (Appendix C). respectively. These results include the root
mean squared errors. which were based on the difference between the experimental and
calculated temperatures using the estimated thermal properties. In each of these ex-
periments. five or six thermocouples were used. and at least two were located at the
heated surface.

The results from each repetition of each experiment were averaged. and the

averaged property estimates are given in Table 5.1. In each case. the total number of

113

Table 5.1. Averaged Estimated Values and 95% Confidence Intervals
for Efl'ective Thermal Conductivity. k. Perpendicular to the Fiber Axis
and Efl'ective Density-Specific Heat. pc . of Cured [0°] AS4/EPON 828-
mPDA Composites. " 2 ‘

 

 

 

 

 

 

 

 

 

 

 

Exper. No. of Avg. Temp. k p:
P
No.a Repet. Range (°C) (W/m°C) (kJ/m’ °C)
1.1 9 25 - 52 0.79710.014 1,580110
1.6 6 27 - 46 0.79710.oo7 1.570130
1.11 6 26 - 46 0.74510.014 1.560110
1.1.1.6
1 11 21 26 - 48 0.78210.013 1.570110
6 53 - 75 0.79310.006 1.650120
1.7b 12 50 - 68 0.86810.010 1.680120
1.12 6 52 - 72 0.79010.021 1.600130
1.2,

12 12 52 - 72 0.79110.009 1.630120
1.3 6 78 - 98 0.81010.012 1.690120
1.8b 6 75 - 91 0.9181o.009 1.810130
1.13 6 77 - 95 0.79710.020 1.760115
1.3.

1 13 12 77 - 95 0.80310.010 1.730130
1.4 12 102 - 120 0.83210.015 1.900150

.9b 6 100 - 115 0.97110.015 1.960130
1.14 6 100 - 115 0.81610.008 1.830130

.4.

1 1‘ 18 101 - 117 0.82610.010 1.880140
1.5 9 127 - 145 0.88010.04o 2.050180
1.10b 6 126 - 141 1.01110.010 2.090130
1.15 6 126 - 140 0.87410.007 1.930130
1.5.

1 15 15 126 - 142 0.87710 020 2.000150

 

 

 

 

 

 

a. Experiment numbers referoto those in Table 4.2.
b. Samples preheated to 150 C; not used in boldface average values.

 

l 14
repetitions. the average temperature range of the experiments. the average effective
thermal conductivity perpendicular to the fiber axis. and the average effective density-
specific heat are given. along with the 95% confidence intervals of the estimated
parameters. The 95% confidence intervals were calculated using the estimated stan-
dard deviation of the averaged values and the t-distribution (Walpole and Meyers.
1978).

The averaged parameter estimates within the different temperature levels were
then compared. In the experiments initially at approximately 25°C (Exp. 1.1. 1.6. and
1.1 l in Table 5.1). the 95% confidence intervals for the estimated thermal conductivities
of Exp. 1.1 and 1.6 overlapped one another. but the 95% confidence interval for Exp.
1.1 1 was slightly lower. The 95% confidence intervals for density-specific heat over-
lapped each other for all three cases. indicating that statistically. the means were
equivalent.

The composite samples in Exp. 1.7. 1.8. 1.9. and 1.10. with corresponding initial
temperatures of approximately 50°C. 75°C. 100°C and 125°C. were accidentally
heated to 150°C for two hours prior to the experiments. This is above the glass transi-
tion temperature of the epoxy. which is approximately 135°C to 150°C. depending on
the heating rate. (The glass transition temperature increases with increased heating
rates.) The glass transition temperature is the temperature at which the molecules in
the amorphous phase (epoxy) begin to rotate. and at this temperature. the material
begins to change from a glassy structure to a rubbery structure (Sichina. 1989). (There
is. however. no visible change in the epoxy.) It is evident that these changes effected
the epoxy even after the samples were cooled. when comparing the 95% confidence in-
tervals of the estimated parameters at each temperature interval. In all cases. the 95%
confidence intervals of the effective thermal conductivities estimated for Exp. 1.7. 1.8.
1.9. and 1.10 were higher than the other two corresponding experiments at each tem-
perature interval. One explanation for the increased thermal conductivity is that the
rotation of the molecules facilitated better bonding along the matrix fiber interface after

the composite had cooled. and therefore, increased the contact conductance between

1 1 5
the fiber and the epoxy matrix. Although the mean estimates of the effective density-
specific values were higher for Exp. 1.7. 1.8. 1.9. and 1.10. than for the other
experiments at the same temperature intervals. the 95% confidence intervals were
higher only for Exp. 1.8. in which the initial temperature was approximately 75°C.

Comparing the parameter estimates for the experiments not preheated to 150°C
at each of the four temperature levels above 25°C (50°C. 75°C. 100°C. and 125°C). the
95% confidence intervals of the estimated thermal conductivity values overlapped one
another within each temperature level. The 95% confidence intervals of the density-
specific heat estimates did not overlap, however. for the experiments with initial
temperatures at 75°C and 125°C.

The estimated thermal properties for Exp. 1.1 , 1.6. and 1.11 (~25°C)..Exp. 1.2
and 1.12 (~50°C). Exp. 1.3 and 1.13 (~75°C). Exp. 1.4 and 1.14 (~100°C). and Exp. 1.5
and 1.15 (~ 125°C) were combined at each temperature level. and the combined results.
along with their respective confidence intervals are indicated by the boldface type in
Table 5.1. Note that the results from the samples preheated to 150°C were not in-
cluded in these values. The combined results clearly indicate that both thermal
conductivity and density-specific heat estimates increase with temperature.

The experiments conducted above 125°C approached the glass transition tem-
perature. which may have affected some of the estimated parameters. even though the
sample achieved these temperatures for only a few seconds near the surface of the
sample. From Table C.5 (Appendix C). the estimates for thermal conductivity perpen-
dicular to the fibers increased with each successive repetition in Exp. 1.5; however. this
did not hold true for the repetitions in Exp. 1.15. Therefore. it is difficult to conclude
whether or not the glass transition temperature had any effect on the parameter es-
timates based on these observations.

It was desired to associate the estimated parameters values with discrete tempera—
ture values. but this was difficult since each experiment was conducted over a
temperature range rather than at single temperature. and because of the temperature

variation within the experimental samples. For convenience. the estimated parameters

1 16
were associated with the average surface temperature next to the heater from each ex-
periment, and the estimated thermal conductivity and density-specific heat values given
in Tables C. l-C.5 are plotted against these temperatures in Figures 5.4 and 5.5. respec-
tively.

A linear regression curve was fit through the estimated thermal conductivity
values from Exp. 1.1-1.6. shown in Figure 5.4. The regression line and the 95% con-
fidence band of this curve are indicated by the solid lines. A second regression curve
was determined for the estimated thermal conductivity values from Exp. 1.7-1.10 in
which samples were preheated to 150°C for two hours; this regression line and the as-
sociated confidence band are indicated by the medium dashed lines. These two
regression lines are independent of one another since their respective confidence bands
do not overlap one another. A third regression line (short dashed line) is shown for the
data from Exp. 1.1-1.4. 1.6. and 1.11-1.15; this curve does not include data from the
experiments initializing at approximately 125°C. in which the sample may have ex-
ceeded the glass transition temperature for a few seconds in each experiment. The
confidence band for this curve overlaps the confidence band for the first regression line
which included the experiments above 125°C. indicating that there is no significant
difference between these two curves.

The parameter estimates for density-specific heat were analyzed and plotted in a
similar matter. and the results from the linear regression analysis are also shown in
Figure 5.5. The 95% confidence bands of the linear regression curves associated with
Exp. 1.1-1.6 and 1.11-1.15 (solid lines) and Exp. 1.7-1.10 (medium dashed lines) did
not overlap. indicating that these regression lines are independent, as was found for the
thermal conductivity curves. The confidence bands for the regression line which did
not incorporate the data above 125°C (short dashed lines) overlapped the bands which
did include this data (solid lines). indicating that. as for the case for thermal conduc-
tivity. the regression lines were the statistically the same.

The estimated effective thermal conductivity perpendicular to the fiber axis and
the efi‘ective density-specific heat values can be expressed as functions of temperature

from the linear regression curves in Figures 5.4 and 5.5 as:

117

8303 a. 32. be... 05 3 Egon 10.59850 3885 258mm .3 egg in 85w...—

.8=8aaoo 31818 29.8%... a. .o.

Gov ongoeomfime

 

 

o3 o3 2: on so 9. om
_ _ h . _ p _ L _ . NHO
o lad
m of. ......... .
o .. .
o .... <Q$ ..
u m o \\\\\H\m
O \ \\\“\“ \ \ s I .
O \<“\““ \ a O
O \\\\\“\““
o \Hkmnux .
\xnmuxq
«unuu.- . . .. . . -os
v x x 2 T: ea 655 ES Now a. :8:
1 2.7.5 .ea Eoe 82:9 s 8383me
3.7.2464 ”VT"; .96 been .300 Now a. use: .
nflglz." 65L.“ dam 683 .300 New on coo:
94):.“ "oAI—A .9me 80.: mos—g x c3953.”—

 

 

 

«A

(Dom/M). ‘>i ‘AiiAnonpuoo
IBmJaui peiemnsa

118

o¢fi

50:89:00

A
Eugene 295?? are. case do . ca 58: 8.88-538 058E a :3ng .3 8:9..—

omfi

Aoov Eggnomfiob

owfi om ow ov

_ — - (— p — _

om

 

 

 

8.75 .me 653 .38 x8 a :8:

8.7: .90 Eat ace in

3.72.3: 6.73 .mxm 655 .E8 x8 x =8:
2.7:; 6.73 .96 ”2:3 .38 Nos a. a8:
9.7:; ”QT: dam 89: seq sum

v4

 

 

ds

109
-z<usueo paiemnsa

ooew/m)
49911 on.

.dod(

119

k(T) = k0 1k1 T (5.18)
for thermal conductivity. and.
pcpm = (pcpk +(pcp). T (5.1b)

for density-specific heat. The regression parameters (k. . k| . (pcp). and (pop). ) and
their associated 95% confidence intervals are given in Table 5.2.

5.1.2.2 Experiments using [0°.130°. 160°. 90°] Samples

2(sym)
Results. including the estimated effective thermal conductivity values perpen-
dicular to the fibers and the estimated effective density-specific heat values. for each

repetition of the experiments using cured (0° .130° . 160°. 90°] samples with initial

2(sym)
temperatures of approximately 25°C. 50°C. and 75°C are given in Tables C.6-0.8
(Appendix C). respectively.

Averaged repetition values for each experiment are given in Table 5.3. In each
case. the average thermal conductivity and density-specific heat values are given. along
with the 95% confidence intervals of the averaged parameters. the total number of
repetitions averaged. and the average temperature range of the experiments.

The average parameter estimates for each experiment within each different tem-
perature level were again compared. At each temperature interval. the confidence
intervals of the density-specific heat overlapped one another. but the confidence inter-
vals of the thermal conductivity values did not. The difi‘erences in the values for
thermal conductivity may have been the result of the different processing methods used
to cure the samples. Since. from Figure 5.2, the sensitivity coefficients evaluated
closest to the heater were the highest in magnitude. the thermocouples at these loca-
tions had the greatest influence on the estimates of the thermal conductivity.
Therefore, in Exp. 1.16, 1.17. 1.18. the estimates of the parameters had the greatest
influence from the samples cured in the hydraulic press. since these sample plates were

closet to the heated surface. Likewise. the samples cured in the autoclave had the

120

Table 5.2. Estimated Parameters and 95% Confidence Intervals for the
Linear Regression of Both Effective Thermal Conductivity. k. Perpendicular to
the Fibers and Efi'ective Density-Specific Heat. c. with Temperature of Cured
[0°] 2 4 AS4/EPON 828-mPDA Composites.

 

 

 

 

 

 

 

 

 

Thermal. Conductivity k - k. + k. '1'

Figure Experiment k. k.

Number Numbers (W/m°C) (W/m(°C)’)
5.4 1.1-1.6; 1.11-1.15 0.74210.015 <0.90210.173)10‘3
5.4 1.1-1.4; 1.6; 1.11-1.15 0.75810.014 (0.6151o.183)10’3
5.4 1.7-1.10 0.76010.016 (1.94310.175)10‘3
5.6 1.16-1 21 0.68910.038 (0.87710.642)10’3

Density-Specific Heat pcp - (pep). + (pcp).T

Figure Experiment (pcp ) . (pc:p ).

Number Numbers (MJ/m’ °C) (MJ/m’ (°C)3 )
5.5 1.1-1.6; 1.11-1.15 1.39010.034 (0.45010.039)10'2
5.5 1.1—1.4; 1.6; 1.11-1.15 1.40710.033 (0.41910.043)10'2
5.5 1.7-1 10 1.36210.038 (0.5621o.042)10’2
5.7 1.16-1.21 1.58810.092 (0.18710.154)10’2

 

 

 

 

 

121

Table 5.3. Averaged Estimated Values and 95% Confidence Intervals
for Effective Thermal Conductivity. k. Perpendicular to the Fiber Axis

 

 

 

 

 

 

 

and Effective Density-Specific Heat. of Cured
[0° .130°.160°.90°]2 ( . . AS4/EPON 828 mPDA Compoéites.

Exper. No. of Avg. Temp. 'E 33;
No.3 Repeti. Range (°C) (W/m°C) (kJ/m?°C)
1.16 11 27 - 41 0.68710.017 1.680170
1.19 6 28 - 51 0.767:0.037 .1,600i40
1.16,

1 19 17 27 - 46 0.715:0.025 1,660t50
1.17 6 53 - 71 0.716:0.009 1,650i20
1.20 6 52 - 75 0.786i0.020 1,690i20
1.17,

1 20 12 52 - 74 0.75110.025 1,670i20
1.18 6 76 - 97 0.744:0.008 1,710i20
1.21 6 78 - 97 0.767i0.061 1.8201220
1.18,

1 21 12 77 - 97 0.756i0.026 1,770i90

 

 

 

 

 

 

a. Experiment numbers refer to those in Table 4.2.

 

122

greatest influence on the parameters estimates for Exp. 1.19. 1.20. and 1.21. At each
temperature level. the estimated thermal conductivities which had the greatest in-
fluence from the autoclave cured samples were higher than the estimated parameters
which had the greatest influence from the samples cured in the press. One explanation
for the higher thermal conductivity values is that the autoclave provided a more
uniform pressure distribution with time throughout the curing process, thus providing
better bonding between the epoxy matrix and the fibers. It is expected that these dif-
ferences would be greater when the fibers are mum-directional. as they were in this
case. since during the experimental stacking procedure. it was observed that the fibers
tended to side over each other when they were stacked at different angles.

The average results for each experiment were combined at each temperatures
interval. These values. along with the associated 95% confidence intervals are indicated
by the boldface type in Table 5.3. As was observed for the results using the [0°]2 4 . both
the estimated thermal conductivity and density specific heat values increased with tem-
perature. The estimated thermal conductivity and density-specific heat values are
shown versus the average surface temperature in each experiment in Figure 5.6 and
5.7. respectively. A linear regression curve was fit through the estimates in each of
these figures using PLO'I‘itR (1989) software. and these curves. along with their as-
sociated 95% confidence bands are indicated by the solid lines. The estimated
parameters associated with these linear regression curves (see eq. 5.1a.b) are also
shown in Table 5.2.

5.1.5.3 Comparison of Results from [0°]2 4 and [0°.130°. 160°. 90°] Samples

2 (s
The linear regression curves indicated by the solid lines in Figuresy:4 and 5.6 for
thermal conductivity. and by the solid lines in Figures 5.5 and 5.7 for density-specific
heat are compared in Figures 5.8 and 5.9. respectively. The curves in Figure 5.8 for
thermal conductivity are clearly independent; one explanation for these differences is
that there is better bonding between the laminates stacked with the plies oriented in the

same direction than when the ply angle is changed. Poorer bonding would decrease the

123

 

 

 

 

 

0.90
0 Eat. k values from Exp. 1.16-l2! o
1 —- Mean 8: 95% conf. band: Exp. 1.16—1.21

A 0.85~

0
-—0° 1
00 E
E; 0.80- 60 3
my
£1“
11 " 0.75~
.8?
0.2.
E...)
5 g 0.70-
‘3'0

1: i o

O o o

O 0.65“ o

0.80 . 1 . . . . .
20 40 60 80 100

Temperature (°C)

Figure 5.6. Estimation of Effective Thermal Conductivity Perpendicular to

 

 

 

 

 

 

the FiberAxis. k. of Cured [0°. 130°. 160°. 90°)2hym) AS4/EPON 828-mPDA
Composites.
2.3
0 Est. pc values from Exp. 1.18-1.21
2 2‘ — Mean 1 957: cont. band: Exp. 1.18-1.21 °
l 2.1 p
>50; 1
an)
.2 QW 2.0
.0.-‘0 "
8 89. 1.91
‘UmE * o
3.2—a 1.8- 3
“:2 < /
E 0" o
.,.. <1) 1.7-« __ __9
+3 Q. °o o o o
‘:30) ‘ 0° 0
1.8+ o
. o
00
1.51
1-4 ' I ' I ' I ' I .‘ I fi I ' I '
20 30 40 50 60 70 80 90 100

Temperature (°C)

Figure 5.7. Estimation of Efl'ective Density-Specific Heat. pcp. of Cured

[0°. 130°. 160°. 90012037111) AS4/EPON 828-mPDA Composites.

Estimated Thermal
Conductivity. k. (W/m°C)

y-

Estimated Densit

124

 

 

 

 

 

0.90
— loan a: 95% cont. band: Exp. 1.1-1.0: 1.11-1.15 [0'1“

q - - "can t 95% cont. band: hp. 1.16-1.21 [0' . t30'. 360'. 901“,“
O.85~
0.80-
0.75-1
0.70-
0.65 . , . . . , . , fl , .

20 40 00 80 100 120 1410

Temperature (°C)

Figure 5.8. Comparison of Linear Regression Cum for Effective Thermal
Conductivity Perpendicular to the Fiber Aids, )1. of Cured [0°]24 and Cured
(0°. 130°. 160°. 90°12“, m) AS4/EPON 828-mPDA Composites.

 

 

 

 

 

— Hun & 063 cont. band: Exp. 1.1-1.0: 1.11-1.16 10']
- - Hun & 95X cont. band: In. 1.18-1.21“). . 150'. 290-11“
2.04
3- .
0
Q
“a
330 1.6-
:r: E
a} .
552
UV
8.
U) 1.6~
1-4 I I I I m I ' I ' I i
20 40 80 80 100 120 1410

Temperature (°C)

Figure 5.9. Comparison of Linear Regression Curves for Effective Density-

Specific Heat. pop. of Cured [0°]24 and Cured 10‘. 130°. 160°. 90‘1“” m)
AS4/EPON 828-mPDA Composites.

125
thermal contact conductance between the ply interfaces and reduce the resulting effec-
tive thermal conductivity perpendicular to the fiber direction. The differences between
the density specific heat curves in Figure 5.9 are not as evident since the two curves
overlap one another from approximately 55°C to 85°C. It was not expected that the

stacking sequence would alter the density-specific heat values.

5.1.5.4 Comparison with Previously Published Results

The estimated thermal conductivity values perpendicular to the fiber axis using
[0°] oriented samples are compared to previously published results on comparable fiber-
epoxy systems in Table 5.4. At 20°C. the results in this study were within the 95%
confidence interval of the results presented by Loh (1989). and since the confidence
interval for the results by Ishikawa (1980) are not given. it can not be said with any
certainty whether or not these estimates are comparable. The results of Loh (1989) at
100°C were 12% higher than those found in this study.

5.2 Estimation of Kinetic Parameters Associated with the Curing of EPON
828/mPDA Epoxy

In this section. the results from the difi'erential scanning calorimetry (DSC) experi-
ments used to characterize the curing of an EPON 828/mPDA epoxy system are given.
The total heat of reaction was first found from the dynamic DSC experiments. and then
the degree of cure was calculated for each isothermal DSC experiment shown in Table
4.4; these results are presented and discussed in Section 5.2.1. The first half of the
cure was evaluated with the assumption that the reaction rate was limited by
autocatalyzation. and the second half of the cure was analyzed with the assumption
that the reaction rate was controlled by diffusion. Three different models were
evaluated for the first half of the cure: estimates for the kinetic parameters using these
models are presented and compared with previously published results on similar epoxy
systems in Section 5.2.2. In addition, estimates are given for the parameters associated

with the model shown in eq. (3.18) for the last half of the cure in Section 5.2.3. Finally.

126

Table 5.4. Comparison of Estimated Effective Thermal Conductivity
Perpendicular to the Fiber Axis. k. with Previously Published Results.

 

 

 

Composite Temp. k Estimation Reference
Material (°C) (W/m°C) Method
Carbon/ a 20 0.72 Infrared Ishikawa (1980)
3130 Epoxy Radiation Method
AS4 Carbon/ 20 0.8010.02 Gauss Loh (1989)
Epon 828 100 0.9310.02 Minimization
AS4 Carbon/ 20 0.7610.02 Gauss present study
Epon 828 100 0.8310.03 Minimization Exp. 1.1-1.6:
1.11-1.15

 

 

 

 

 

Similar in structure to Epon 828 epoxy, with similar thermal conductivity
(Ishikawa (1980).

 

127

the models and the associated parameters estimates selected for use in procedures for

the estimation of thermal properties during curing are presented in Section 5.2.4.

0- -m I. u -_ r . ._ .-._ . in. u .1. r. -- .

The total heat of reaction (eq. 3.9) was required to determine the degree of cure
(eq. 3.8) from the isothermal differential scanning calorimetry (DSC) experiments. The
total heat of reaction was first determined from the dynamic DSC experiments as
described in Section 4.2. In each of these experiments. a constant base-line was estab-
lished. and the magnitude of the baseline was subtracted from the experimental data.
The modified data was integrated over time using the DuPont Thermal Analysis System
9900 software to determine the total heat of reaction. The total heat of reaction values
calculated for the eight dynamic experiments shown in Table 4.4. along with the
average heat of reaction from all the experiments. are shown in Table 5.5.

The degree of cure was then determined for each isothermal experiment con-
ducted at 60°C. 70°C. 100°C. and 110°C (Table 4.4). The isothermal DSC data was
post processed as discussed in Section 4.2.2.3 to obtain heat of reaction rate values for
each isothermal experiment. These values were then transferred to a VAXstationII/GPX
microcomputer, where the cumulative heat of reaction was determined as a function of
time through numerical integration of the heat of reaction rates with time. and the de-
gree of cure was calculated by dividing the cumulative heat of reaction values by the
average total heat of reaction shown in Table 5.5.

These degree of cure values. which vary with time, for the various isothermal ex-
periments were then used to estimate the kinetic parameters in the kinetic models
shown in eqs. (3.4-6).

'qczn“ n. H o ;_01-..r.‘o 1'60’ ‘ "Dir-o

The degree of cure data less than 0.50 were assumed to follow an autocatalyzed

kinetic reaction without any influence from decreased molecular mobility. The use of

0.50 as a limit for the autocatalyzed controlled reactions is based on the observations

128

Table 5.5. Total Heat of Reaction Calculated from Dynamic Differential
Scanning Calorimetry Experiments (Exp. 2.1) using EPON 828/mPDA

Epoxy.

 

 

 

Repetition Heat of
No. Reaction

(J/kg)

1 0.459

2 0.444

3 0.468

4 0.452

5 0.432

6 0.451

7 0.439

8 0.447

Average 0.44910.009

 

 

 

 

129
by Sourour and Kama] (1976) as discussed in Section 2.2.3. and on similar observa-
tions in this study. The degree of cure data was first used to determine the kinetic
parameters (i.e. rate constants) at the different experimental temperatures associated
with the kinetic models shown in eqs. (3.4-3.6). and then the estimated parameters
were used to determine the activation energy constants and the pre-exponential factors

in eq. (3.7).

5.2.2.1 Estimation of Rate Constants and Exponents

The rate constants and exponents in each of the kinetic models shown in eqs. (3.4-
3.6) were first determined for each of the isothermal DSC experiments shown in Table
4.4. These experiments were conducted at 60°C. 70°C. 100°C. and 110°C. with three
repetitions at each temperature. In these analyses. all linear regression calculations
were performed using PLO'l‘itR (1989) software.

In eq. (3.4). the parameters to be estimated were the rate constants. c. and c; .
and the exponential. m. These parameters were estimated from the initial and maxi-
mum degree of cure rate data from each experiment. using the method given by Ryan
and Dutta (1979) and outlined in eqs. (3.10-3.13). The least squares method discussed
in Section 3.2.1 and shown in eq. (3.14). was also used to determine these parameters.
In this method. the parameters were estimated from all of the degree of cure data in
each experiment through the minimization of a least squares function. The estimated
values for c. . c. . and m. using the Ryan and Dutta method and the least squares
method are shown in Tables 5.6 and 5.7. respectively. In addition. 95% confidence in-
tervals are given for the rate constants estimated using the alternate approach.

The isothermal DSC data were then used to determine the rate constants. c. and
c. in eq. (3.5). using the method given by Sourour and Karma] (1976) and discussed in
Section 3.2.1. The estimated rate constants and their respective 95% confidence inter-
vals are shown in Table 5.8 for each isothermal DSC experiment shown in Table 4.4.

The rate constant. c. . and the exponential. n. from the model shown in eq. (3.6)

were also evaluated from the isothermal DSC data. These parameters were determined

Table 5.6. Rate Constants. c1

eq.

(3.4)

130

and Degree of Cure Values. 0:.
828/mPDA Epoxy. using the Ryan and Dutta (1979) Method.

and o. . and Exponent. m. Determined from
Less than 50% for EPON

 

 

 

 

 

 

 

Exper. Rep. Temp. Ryan and Dutta Method
No. No. (°C) (3'1) (3.1) m
1 60 .06E-5 .70E-4 0.809
2.2 2 60 .04E-5 .43E-4 0.817
3 60 .86E-5 .51E-4 0.777
1 70 .27E-5 .01E—4 0.797
2.3 2 70 .33E-5 .05E-4 0.815
3 70 .02E-5 .29E-4 0.848
1 100 .40E-4 .36E-3 0.757
2.4 2 100 .92E-4 .37E-3 0.749
3 100 .60E-4 .12E-3 0.727
1 110 .97E-4 .81E-3 0.793
2.5 2 110 .56E-4 .65E-3 0.705
3 110- .02E-4 .61E-3 0.668

 

 

 

 

 

 

a.

Experiment numbers and repetition numbers refer to those in Table 4.4

 

131

Table 5.7. Rate Constants. c. and c. . and Exponent. m. and Associated
95% Confidence Intervals. Determined from eq. (3.4) and Degree of Cure
Values. on. less than 50% for EPON 828/mPDA Epoxy. using a Least
Squares Method.

 

 

 

 

 

 

Exper. Rep. Temp. Alternate Method
No.a No.a (°C) c. (3'1) c. (s-l) m
1 60 (2.2210.12)E-5 (2.4610.05)E-4 0.832
2.2 2 60 (2.3110.08)E-5 (2.2710.02)E-4 0.834
3 60 (1.5510.08)E-5 (2.3010.03)E-4 0.781
1 70 (3.4610.17)E-5 (3.9210.06)E-4 0.816
2.3 2 70 (4.0610.22)E-5 (4.0210.08)E—4 0.847
3 70 (5.3310.40)E-5 (3.3610.16)E-4 0.880
1 100 (1.7610.20)E-4 (1.5910.07)E-3 0.765
2.4 2 100 (1.5910.36)E-4 (1.5710.12)E-3 0.691
3 100 (1.5810.22)E-4 (1.4510.07)E-3 0.731
1 110 (3.3810.19)E-4 (2.4910.08)E-3 0.809
2.5 2 110 (3.0910.18)E-4 (2.3910.09)E-3 0.713
3 110 (2.9810.17)E-4 (2.1810.07)E-3 0.665

 

 

 

 

 

 

 

 

a. Experiment and repetition numbers refer to those in Table 4.4.

132

Table 5.8. Rate Constants. cl and o. . and Associated 95% Confidence
Intervals Determined from eq. (3.5) and Degree of Cure Values. on, less than
50% for Epon 828/mDPA Epoxy. using the Sourour and Kamal (1976)
Method.

 

 

 

 

 

 

Experx Rep. Temp. Sourour and Kamal Method
No.a No.a (°C) c. (3-1) G; (3-1)
1 60 (1.2210.49)E-S (5.0410.19)E-4
2.2 2 60 (1.5210.34)E-5 (4.5710.13)E-4
3 60 (1.3510.21)E-S (4.4210.08)E-4
1 70 (2.1710.61)E-5 (7.9310.23)E-4
2.3 2 70 (2.1511.10)E-5 (8.3110.41)E-4
3 70 (3.6011.53)E-5 (7.1710.59)E-4
1 100 (1.5610.31)E-4 (3.2210.11)E-3
2.4 2 100 . (1.5810.61)E-4 (3.3410.23)E-3
3 100 (1.5110.41)E-4 (2.9610.15)E-3
1 110 (2.7010.37)E-4 (5.1610.15)E-3
2.5 2 110 (2.8310.44)E-4 (5.3010.18)E-3
3 110 (3.0810.35)E-4 (4.9710.14)E-3

 

 

 

 

 

 

 

a. Experiment and repetition numbers refer to those in Table 4.4.

133

using linear regression (PLOTitR. 1989). and they are shown. along with their respective

95% confidence intervals. in Table 5.9.

5.2.2.2 Estimation of the Activation Energy Constants and the Pre-Exponential Factors

Each of the rate constants shown in the kinetic models in eqs. (3.4-3.6) was as-
sumed to follow an Arrhenius relationship with temperature. The activation energ
constants and the pre-exponential factors associated with the Arrhenius relationship in
eq. (3.7) were estimated using linear regression (PLOTitR. 1989) from the rate constants.
c. and o. . shown in Tables 5.6-5.9. These rate constants are shown along with their
associated linear regression curves in Figures 5.10 and 5.1 1. respectively. Also. the
exponential values for m. in Tables 5.6 and 5.7. and n. in Table 5.9. indicated some

temperature dependence; this dependence was assumed to linear. Thus. for eq. (3.4).

m = m0 + rnI T (5.3)
and for eq. (3.6).
n = n, + n. T (5.4)

where rn0 . m. . no . and n. are estimated constants using linear regression. The es-
timated values for the activation energy constants. the pre-exponential factors. and the
exponential constants and their respective 95% confidence intervals are compared with
values obtained by Sourour and Kama] (1976). Kamal et al. (1973). Ryan and Dutta
(1979). and Prime (1970. 1973) for bisphenol-A-diglycidylether (BADGE-
DER332)/mPDA systems in Table 5.10. The DER332 BADGE resin has approximately
the same structure as the EPON 828 system used here.

In comparing the kinetic parameters estimated in this study with those estimated
in the other studies. the parameters characterizing the second rate constant. namely A.

and E. . are very similar for the models shown in eqs. (3.4) and (3.5). (It is difficult to

134

Table 5.9. Rate Constant. e.. and Exponential. n. and Associated 95%
Confidence Intervals Determined from eq. (3.6) and Degree of Cure Values.
01. less than 50% for EPON 828/mPDA Epoxy.

 

 

a.

 

 

 

 

Exper. Rep. Temp. c. n
No.a No.a (°C) (8‘1)
1 60 (1.8510.12)E-2 -1.2510.18
2.2 2 60 (1.8410.22)E-2 -1.1110.18
3 60 (1.4910.14)E-2 -1.4510.29
l 70 (3.0110.17)E-2 -1.2210.17
2.3 2 70 (3.2310.24)E-2 -l.1210.22
3 70 (3.3210.35)E-2 -0.8510.32
1 100 0.14110.023 -1.0010.46
2.4 2 100 0.16110.017 -0.8410.31
3 100 0.14510.016 -0.8310.32
1 110 0.23810.023 -0.8510.29
2.5 2 110 0.25410.017 -0.7910.21
3 110 0.25110.017 -0.7110.22

 

 

 

 

 

Experiment and repetition numbers refer to those in Table 4.4.

 

135

 

    

 

 

 

0.0
-2.°" .“‘|Hu‘.“\s.ts\|es
_4.0-4 H“|ttt|||t|18
-8.0-
”r?“
MO m. -3.0-1
5 c:
C
-10.0‘ -
‘--- Mean 1 cont. bands: sq (3.6)
_12 0- s 0.: sq (3.6)
‘ - - Mean & conf. bands: sq (3.6)
1 A c,: sq (3.5)
-140- — lsantconf.bands:sq(3.4);lsastsquarssrnsthod
' o e,: sq (3.4) using least squares method
4 — lean 8 cont. bands: sq (3.4); Ryan-Dutta method
0 cl: sq (3.4) M Ryan-Dutta method
-1600 ' I j v I fir 1 v I r
0.00031 0.00032 0.00033 0.00034 0.00035 0.00030 0.00037

(mcliggT/ld)

Figure 5. 10 Arrhenius Relationsz for the First Rate Constant. c. in
eq. (3.4). (3.5). and (3.6): Ln (c. )versus l/RI‘ (R = Gas Const.
(Ulmole'K): T. = Abs. Temp. (40).

 

 

 

 

 

—4.0
-5.0-«
—6.0-
’5
3.3, 47.04
5 c:
C}
-8.0-
- - lean & cont. bends: sq (3.5)
_9 o ‘ c.: “I (3-5)
’ 7 — leankeonf.bands:sq(3.4);lsestsquarssmethod
‘ o 12.: sq (3.4) using least squares method
— lean & cont. bands: sq (3.4); Ryan-Dutta method
_10 OJ 0 - 3.4 glen-Dutta method
. r ' I . I ' I '
0.00031 0.00032 0.00033 0.00034 0.00035 0.00033 0.00037

(mclrQET/kl)

5. 1 l Arrhenius Relationship for the Second Rate Constant. c. in
eq. (3.4). (3.5). and (3.6): Ln (Cr )versus 1/RT (R = Gas Const.
(kJ/mole°K): T. = Abs. Temp. (40).

 

 

 

136

 

.muonssc cofiumsvo on uuouou Hopoz

om

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Apogee: maoeemo
0cm upumnouom. mm.oH¢>.HIu.: AomHImHHV
AmpmacoanVoEwum m.aum.o I c «Hum mmlom anmc>a Am.m.
Acoemmoummu mumxfl.~«>.mvu.c
ummcflav mm.o«eb.alu.: AoHHIomv
xenon menu e_c+.e u e m.o«~.ma m.v«a.mm HmsumnuomH .m.m.
$th Hoe—ox Aomauowv
Eco usounom m.m o.mH me am HmEuwnuomH Am.mv
A.mwuowu newswav Aoaanomv
16506 when s.ofim.oa e.HHH.NH o.~He.om m.sfim.¢m HmeumnuomH im.m.
3pm: analog
cause new swam wo.H u E co>fio uoc m.HH we mm HmfiumnuomH A¢.m.
«mg: Search;
.38 no Hague H . e H.m 3.0H we we HagumeuomH xv.mv
Apognms mumxm.aam.mvnu.e
mmumsvm ammo: "3.923.002: .oHHIomv
xpsum menu 9.5+.e u s >.oHv.m o.m8«.m H.~Hm.m¢ m.maw.mm HassonuomH A¢.mv
Apogee: .meaa mumxm.flao.mvun_e
.muuso Ucm swam. madfimméflé Aoaauomv
16938 «any 9.5+.s u s m.oH~.oa H.Nfim.oa a.mae.om m.m«s.em HagumeuomH is.m.
Aponuozv mnemomcoo As-mvea As-mvea Aoaos\nxv Aoaoe\va .u...m&oov
mocmuwwmm amaucwcoaxm Au<vca A_<VCH «mm .mm .me Umo H0602
68%. 452888 20.5 .8 cs mm . .
1.8.. .08 an 3:328 28 esmfisom 5 3 8 82g 858 ”8398.65.81.33. .219 was

.<..mhzon

Am nxhw Aw

Bacocaxwéum awn—magma tum .63 QEmCOZBQm made

 

 

137
determine whether or not the parameters are statistically equivalent. since the con-
fidence intervals of the parameters estimated in the previous studies were not given.
and calculations of the confidence intervals for the activation energy constants and the
pre-exponential factors shown for this study did not include the confidence intervals
from the rate constants.) The parameters associated with the first rate constant differ

by a wider margin. This is to be expected. however. since the first rate constant repre-

 

sents the initial portion of the cure. which is difficult to measure using the DSC. and

 

therefore it is more susceptible to errors. This is evident by the larger confidence inter-

vals for E. and A. . compared with E. and A. . Fortunately. since the second rate

constant. c. . is approximately one order of magnitude greater than the first rate con-

__J

stant. c. . as shown in Tables 5.6-5.9. the first rate constant is only dominant during

the initial stages of the cure when the degree of cure rate is small.

52W

There have been relatively few studies to characterize the cure of an amine-epoxy
resin for diffusion controlled reactions (Section 2.2.3). The model presented by Hawley
et al. (1988) and shown in eq. (3.16) was first investigated to simulate the curing of the
EPON 828/mPDA resin for degrees of cure greater than 0.5. To check the validity of the

model. a natural logarithm was applied to eq. (3.16). and then the equation was rear-

 

ranged as follows.

dealt] _ _

ln[( l—a) — ln(c,) Do

A preliminary analysis indicated that the left hand side of the above equation was

highly nonlinear. and so this model was deemed to be inappropriate for this analysis.
The alternate model shown in eq. (3.18) was then used to characterize the curing

reaction for a greater than 0.50. This model is similar that shown in eq. (3.16) in that
the reaction rate was assumed to follow an exponential decay. however. this model is
inherently coupled to the kinetic model used for the degree of cure less than 0.50. in

which diffusion was assumed to be negligible through the 010 terms in the equation.

 

 

138

The rate constant. c. . and the diffusion constant. D. were evaluated from the
degree of cure rate at the time the reaction became diffusion controlled. which was as-
sumed to be at on = 0.50. and the degree of cure and associated degree of cure rate
values greater 01D. using the nonlinear parameter estimation scheme discussed in
Section 3.2.2. Once again is it noted that the selection of on was based on the results of
the studies discussed in Section 2.2.3 and on visual observation of plots of the degree of
cure rate data versus the degree of cure. The estimated parameters. c, and D. are
shown. along with their 95% confidence intervals. in Table 5.1 1 for each isothermal
DSC experiment. The rate constant was assumed to follow an Arrhenius relationship

with temperature (eq. 3.7) and the diffusion constant was assumed to vary linearly with

temperature as follows
D = D. + D. T (5.5)

The parameters. E: . A. . D0 . and DI . were estimated using linear regression (PLOTitR.

1989) from these relationships; the estimated parameters are also shown in Table 5.1 1.

'1’ inf l '11 " fruin h .t_010'l‘1‘1'ur-._-'100'1‘.‘

W

In order to estimate the thermal properties during the curing of the carbon/ epoxy
system discussed in Section 4.3. degree of cure rate values were required throughout
the cure cycle. as discussed in Section 3.2. The degree of cure rate was found from one
of the three kinetic models characterizing the autocatalyzed reaction discussed in
Section 3.2.1. and from the diffusion controlled models presented in Section 3.2.2. The
selection of the models was based on the confidence intervals of the estimated rate con-
stants. activation emery constants. and pre-exponential values.
I For the parameters estimated assuming an autocatalyzed controlled reaction rate
(Tables 5.6-5.8). the rate constants estimated using eq. (3.4) for the kinetic model had
the lowest confidence intervals. and from Table 5.10. the confidence intervals of the

activation emery constants and the pre-exponential factors estimated using eq. (3.4)

 

 

139

Table 5.1 1. Rate Constant. c, . and Diffusion Constant. D. and Associated
95% Confidence Intervals Determined from eq. (3.16) and Degree of Cure
Values. a. Greater than 50% for EPON 828/mPDA Epoxy.

 

 

 

 

 

 

 

 

 

 

 

Exper. Rep. Temp. c,b Db
No.3 No.“ (°C) (3“)
1 60 (2.3310.10)E-2 6.51i0.18
2.2 2 60 (4.47i0.17)E-2 6.5510.22
3 60 (4.9410.29)E-2 7.4210.33 x
1 70 (4.19i0.18)E-2 4.7610.16
2.3 2 70 (4.03io.25)E-2 4.8910.21
3 70 (5.57i0.57)E-2 5.71i0.46
l 100 0.265:0.006 1.87i0.07
2.4 2 100 0.276i0.013 1.6810.12
3 100 0.31110.013 1.5210.12
1 110 0.49610.009 0.94i0.05
2.5 2 110 0.47710.010 0.9610.05
3 110 0.493i0.011 0.97i0.06
Ea, ln(A,) Diffusion
(kJ/mole) 1n(s'1) Constants
56.9:8.5 17.112.9 D = D.+D,T
Du =13.4:t1.2
D, I -0.117i0.013

 

 

 

 

 

a. Experiment and repetitions numbers refer to those in Table 4.4.

 

 

140

were slightly higher than those found using eq. (3.5). Since eq. (3.4) resulted in the
lowest confidence intervals for the estimated rate constants. and since the confidence
intervals for the activation energy constants did not incorporate the variability of the
rate constants. eq. (3.4) was determined to be the best model for the EPON 828/mPDA
system studied here. The parameters estimated using eq. (3.4) with the least squares
method were used in the estimation procedure for the thermal properties during curing
because it incorporated all of the kinetic DSC data in the analysis. instead of relying on
just a few points.

The parameters estimated using the alternate difl'usion model (eq. 3.18) presented
in Section 3.2.2 were used in the estimation of the thermal properties during the second
half of the curing reaction. The degree of cure rates estimated using eq. (3.4) at “D =
0.50 were included in the procedure. The first model presented in Section 3.2.2 by
Hawley et al. (1988) was determined to be inappropriate. as discussed in Section 5.2.3.

The kinetic models used in the estimation of the thermal properties during curing

are summarized below.

Degree of cure less than 0.50 (autocatalyzed controlled reaction):

gtg=(c, +Czam)(l-a)n m+n=2

where c] =A| CXp(-E: /R(T+273.15)):
6; =A¢ exP(-Ei /R(T+273.15)):

and. rn=rno +m.T.

The estimated parameters are:

E. = 55.8 kJ/mole.
E. = 49.2 kJ/mole.
A. = 1.21x10‘ s".
A. = 1.21x10‘ s‘l.
m, = 0.98.

and. m1 = -0.0023.

 

 

141

Degree of cure greater than 0.50 (diffusion controlled reaction):

- (a -a)
gig _ [gig] _ D 1)
dt - dt asaD+ C’ (“D (1)6
where c, =A, exp(-Ea /R(T+273.15)).
“D = 0.5
and D = Do + D1 T.

The estimated parameters are:
E, = 56.9 kJ/mole.
A. = 2.75x1o' s".

Do

13.4.

and. D1 '0. 1 17.

These models and associated parameters were incorporated into the computer
programs CUREID and PROPID_CURE for the estimation of the thermal properties
during curing.

5.3 Estimation of the Thermal Properties of Composite Materials during Curing

The estimation of thermal properties of AS4/EPON 828 composite materials
during curing is the focal point of this section. First. an analysis of the estimation pro-
cedure through the use of simulated data is given in Section 5.3.1. The second section
(5.3.2) is devoted to the analysis of the results for the estimated properties from ex-
perimental data. The estimated thermal properties are presented and discussed. and
improvements for the experimental apparatus and design are proposed. It is em-
phasized that this is an introductory investigation into this problem. and that the
objective here is to establish the procedures and experimental design for the estimation

of thermal properties during curing. rather than to determine specific numerical values.

 

 

 

142

MW

The procedure for the estimation of thermal properties during the curing of carb-
on /epoxy composite materials described in Section 3.3 was first implemented and
analyzed using simulated temperature data. This simulated temperature data was gen-
erated through the use of the computer program CUREID. described in Section 3.3.1.
and the resulting temperatures were used as input for the program PROPID_CURE
(Section 3.3.2) for the estimation of the thermal properties. A description of the test
cases used in the analysis of the estimation procedure is first given. followed by the

results of the estimated thermal properties using the simulated data as input.

5.3.1.1 Description of Test Cases

The computer program. CUREl D. was used to generate temperature data as input
for the program PROPID_CURE. Several difi’erent test cases. involving different input
conditions for the program CUREID and added random errors. were used in this
analysis. The input conditions for CUREID were designed to simulate one of the heat
flux pulses in Exp. 3.2. which corresponds to the second experiment during curing

described in Section 4.3. The thermo-kinetic problem defined by these parameters can
be mathematically described as

%;[k(T.a)géE] + ping? = pcp('I‘.ot)'§';L 0 S x S L: t > 0

with the boundary conditions.

[0;O<tst.
«mg; = qm=i q.:t. <tst. x=o
0;t3 <tst,
T=TL(t) x=L; t>0

and the initial condition.

 

 

 

143

T=T.(x) Osst:
where.
§%=(c. +c.a"‘)(1-a)“ m+n=2

c. =A. exp(-E, /R(T+273.15));
c; =Aa arm-Fa lRi'I‘+273.15)):

m=rno +m.T.

with the initial condition.

and for a > O.

-D(a -a)

$195 - £195 D
(it - [dt]aaaD+ C3 (GD- (1)6

'c. =A, exp(-E. /R(T+273.15)).

D=Do +D1T.

with the initial condition.

D " toi=oz

where T is °C.

(150.5

a>0.5

The kinetic parameters presented in Section 5.4.2 were used in the analysis of the

kinetic equations shown above. The total heating time (t. - t, ) was set equal to 135

 

 

 

 

144

seconds and the input heat flux. q. . was set equal to 800 W/m’ ; these values are con-

 

sistent with the values used in the experimental procedures. A total time. t, . of 300
seconds was used in this analysis. Based on the measured thicknesses of the ex-
perimental composite samples. the hypothetical composite thickness. L. was taken to be
4.5 mm. All of these parameters mentioned thus far were identical in all of the case
tests studied. The temperature boundary condition at x = L. TL(t). and the initial condi-

tion. T. (x), were both taken to be constant and equal to one another in each test case

 

studied. Two difierent values were used for these parameters: one set of test cases was
evaluated at 75°C and a second set was evaluated at 125°C. The analysis also re-
quired the initial degree of cure; in all cases it was assumed that the initial degree of
cure was uniformly constant. For the test cases at 75°C. a value of 0.40 was used. and

for the test cases at 125°C. a initial degree of cure of 0.80 was used.

No difierent conditions were used for the thermal properties. in one instance the
properties were taken to be constant. and in the second instance. the properties were
taken to vary linearly with both temperature and degree of cure. For the first instance.

the constant properties were evaluated at 100°C using the equations shown in Table

5.2:

k(T) = 0.742 + 9.02x10'4T;
for T = 100°C. k = 0.83 W/m°C:

pcpfl‘) = 1.39 +4.5x10'3T;
forT = 100°C. pcp = 1.84 MJ/m’ °C.

and for the case were the properties were assumed to vary linearly with temperature

and degree of cure. the following hypothetical relationship was assumed:

kfl‘,a) = 0.9k. + k. T + 0.1k. a.

and.
pcp(T,ot) = 0.9(pcp). +(pcp).T+0.10(pcp.)a

 

145
(Note that when 0: equals 1.0. these equations reduce to those shown in Table 5.2.)

Using the values for k. . k. . (pop). and (pop). given in Table 5.2. one obtains

ma) = 0.6678 + 9.02x10'4r + 0.0742a. (W/m°C) (5.6)

and,
pcplT.a) = 1.251 + 4.5x10'3'r + 0.139a. (MJ/m’ °C) (5.7)

The program CUREID requires input values for k and pcp at two difl'erent tem-
peratures and two difi'erent degree of cure values for the case where the thermal
properties vary linearly with both temperature and degree of cure (see eq. 3.21).
Therefore. using eqs. (5.6) and (5.7) with temperature values of 75°C and 100°C. and
degree of cure values of 0.4 and 0.8. the input property values for CUREID were as fol-

lows:

For thermal conductivity. k, (W/m°C)

k(75.0.4) = 0.765: k(75.0.8) = 0.795:

k(100.0.4) = 0.810: and k(100.0.8) = 0.840.
For density-specific heat. pep. (MJ/m’ °C)

pcp(75.0.4) = 1.644: pcp (75.0.8) = 1.700:

pcp(100.0.4) = 1.869: pcp(100.0.8) = 1.925.

The temperatures were calculated using CUREID and the various input
parameters described in this section at the locations for x = 0.0 mm. x = 2.25 mm, x =
3.375. and x = 4.5 mm. Normally distributed random temperature errors (Abramowitz

and Stegun. 1970) were added to the results from CUREID prior to use in the

parameter estimation program PROP1D_CURE. '1\vo thermocouples were simulated at
each of the above locations by adding two different sets of normally distributed random

errors to the temperatures calculated using CUREID. Three different values for the

standard deviations of the temperature errors. 01.. were used: 0.1. = 0.0 (no errors): qr =

O.10°C: and. GT = 0.25°C.

146
The input parameters for the test cases used in this analysis are summarized in
Table 5.12. A total of eleven different test cases were studied. each with different com-
binations of initial temperature. thermal property. and standard deviation for
temperature errors values. The cases with the standard deviation. 0.1.. not equal to zero
were repeated six times using a different set of random errors with the same standard

deviation for each repetition. The temperature change associated with each case was

approximately 5°C. and the degree of cure change was approximately 5%.

5.3.1.2 Results from Test Cases
Each of the test cases shown in Table 5.12 was analyzed using PROPID_CURE for

the estimation of thermal conductivity and density-specific heat. In each case. the 95%
confidence intervals of the estimated parameters were calculated within the program
PROP1D_CURE assuming that the errors associated with the temperature measure-
ments are correlated and autoregressive (Beck. 1989). These conditions are typically
associated with actual temperature measurements. however. the errors in this case
were calculated as normally distributed and independent. The difi'erences between
these types of errors can be seen through the comparison of the residuals from actual
experiments (see Figure 5.1) and the residuals from the hypothetical cases. The
residuals from the hypothetical thermocouples at x = 0.. x = 1.125 mm. x = 2.25 mm.
and x = 3.375 for one test case with 0,1. = 0.25 are shown in Figure 5.12. These errors.
unlike those shown in Figure 5.1. are uncorrelated and unbiased. with a constant
variance throughout the time period. Despite these differences in the types of errors.
the use of hypothetical data in the parameter estimation procedure provides an excel-
lent means of testing and evaluating the process.

The estimated thermal conductivity and density-specific heat values for cases 1. 1.
2.1-2.6. and 3. 1-3.6. involving an initial temperature of 75°C and constant thermal
properties are given in Table 5.13. Similarly. results for cases 4.1. 5.1-5.6. and 6.1-6.6.

involving an initial temperature of 125°C and constant thermal properties are given in
Table 5. 14. The percent errors given in both of these tables represent the differences

between calculated properties and the input properties values to CUREID. Since the

147

Table 5.12. Test Cases for the Estimation of Thermal Conductivity. k. and
Density-Specific Heat. pcp. from Simulated Data for Composite Materials

 

 

 

 

 

 

 

 

 

 

 

 

during Curing.
Case T ' T a db oc k
L' ° T pcp
No. (°C) (°C) (W/m°C) (MJ/m’ °C)
1.1 '75 0.40 0.00 0.83 1.84
2rL4.6 75 0.40 0.10 0.83 1.84
3.1-3.6 75 0.40 0.25 0.83 1.84
4.1 125 0.80 0.00 0.83 1.84
5.1-5.6 125 0.80 0.10 0.83 1.84
6.1-6.6 125 0.80 0.25 0.83 1.84
7.1 75 0.40 0.00 linearc k linearc pcp
8.1-8.6 75 0.40 0.10 linearc k linearc pcp
9.1-9.6 75 0.40 0.25 linearc k linearc pcp
10.1 125 0.80 0.00 linearc k linearc pep
11.1-11.6 125 0.80 0.10 linearc k linearc pcp
12.1-12.6 125 0.80 0.25 linearc k linearc pcp
a . Temperature boundary condition and initial temperature (constant).
b. Degree of cure.

c. Standard deviation of temperature errors.

 

 

 

148

 

 

 

 

 

 

1.00
fi’ 0.50-
as omJ ‘ LL“ .ia.l. u I. I ulJn‘il. 1| 1 1“ d J
:93, ' 'l' ' 1 l l' H1!“ '71 1‘ ' “y ‘1 1 1
1.00
m ‘ '
0.50" ’
'50 0.00- - ' \l. ' 4' _ n .
(732" i' “ll" ‘1 " ""1 ( l:ll
flat." --50-I l
_100‘ — 1.125nnnfromheatedsurfaoe
1.00 .
I
tn , I ‘
0.50 1;! .
WI ' ' ' ' a ' I
38 a) lunch”. 1w“. 1 I“ ,55.("H."~ 5 {"1 h ’1 1'1...
”—00 I ‘ .I‘ '0‘.” m"-""
a?“ 50 ""i‘ "4"1":ii".":"."l' i . W" ‘ i i
-' i ' .

 

 

 

 

 

 

- 3. 375 mm from heated surface
I v
0.0 100.0 200.0 300,0

Time (sec)

 

Figure 5. l2 Residuals for a Simulated Test Case. using Input Temperature
Errors with a Standard Deviation of 0.25'C. at Four Difl‘erent Locations
(total thickness equal to 4.5 mm).

 

149

Table 5.13. Estimated Effective Thermal Conductivity. k. Perpendicular
to the Fibers and Density-Specific Heat. pc . and the Associated. 95%

Confidence

Intervals

for

a Simulated Composite

withan

Infihfl

Temperature of 75°C. an Initial Degree of Cure of 0.40. and Thermal

Properties: k: 0.83 W/m°C and pcp= 1.84 MJ/m’ °C.

 

Estimated

Thermal Conductivity

Estimated

Density-Specific Heat

 

 

 

 

 

(has a; RMSb k Errorc pcp Errorc
(°C) (°C) (W/m°C) (%) (MJ/m?°C) (%)
0.00 0.003 0.8299 0.01 1.840 0.00
0.10 0.100 0.821810.0063 0.99 1.821i0.034 1.03
O 10 0.101 0.8315r0.0022 0.18 1.84610.025 0.60
0.10 0.098 0.8268i0.0021 0.39 1.86510.019 1.35
0.10 0.100 0.8294i0.0021 0.12 l.827t0.039 0.71

0 10 0.103 0.829110.0025 0.11 1.82410.021 0.87
0.10 0.101 0.826710.0020 0.40 1.852i0.018 0.65
0.25 0.248 0.8312i0.0051 0.15 1.84310.045 0.08
0.25 0.253 0.8217t0.0054 1.00 1.940t0.049 5.44
(3.25 0.248 0.8389i0.0059 1.07 1.841i0.050 0.05
(3.25 0.252 0.823410.0056 0.79 1.882i0.049 2.28
().25 0.258 0.837310.0056 0.70 1.882i0.048 2.28
()u25 0.256 0.8294i0.0052 0.07 1.82010.045 1.09

 

 

 

 

 

 

a.
b.
c.

Standard deviation of temperature errors.

Root mean squared error between calculated and input temperatures in PROPlD_CURE.
% difference between input and estimated thermal properties.

 

 

 

 

 

 

150

Table 5.14. Estimated Effective Thermal Conductivity. k. Perpendicular
t0 the Fibers and Density-Specific Heat. pc . and the Associated 95%
Confidence Intervals for a Simulated C‘bmposite with an Initial
Temperature of 125°C. an Initial Degree of Cure of 0.80. and Thermal
Properties: k=0.83 W/m°C and pop: 1.84 MJ/m’ °C.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Estimated Estimated
Thermal Conductivity Density-Specific Heat
Case a; RMSb k l':‘.rrorc pcp Errorc
No. (°C) (°C) (W/m°C) (%) (MJ/m?°C) (%)
4.1 0.00 0.003 0.6296 0.05 1.639 0.05 F
5.1 0.10 0.100 0.6274io.0019 0.31 l.606i0.017 1.74
5.2 (L10 0 101 0.6266r0.0019 0.17 1.632r0.017 0.43
5.3 0.10 0.100 0.6297r0.0020 0.04 1.666i0.016 1.41
.4 0.10 0.100 0.6313i0.0016 0.16 1.647i0.016 0.36
.5 0.10 0.103 0.6321r0.0016 0.25 1.674r0.016 1.65
.6 0.10 0.101 0.6309i0.0019 0.11 1.616ro.017 1.30
.1 0.25 0.251 0.8107i0.0126 2.33 1.601i0.073 2.12
.2 0.25 0.261 0.626110.0043 0.47 1.795i0.039 2.45
43 0.25 0.249 0.6336i0.0046 0.43 1.666i0.043 1.41
(1.25 0.250 0.6369r0.0049 1.07 1.660ro.044 2.17
(1.25 0.246 0.6373ro.0044 0.66 1.633r0.039 0.36
(>n25 0.255 0.835710.0049 0.69 1.696r0.044 3.04
a. Standard deviation of temperature errors.
b. Root. mean squared error between calculated and input temperatures in 9140910501113.
c. 4 difference between input and estimated thermal properties.

 

 

151

two programs (CUREID and PROPID_CURE) utilize similar numerical methods. the er-

rors associated with the case of 0,1. = 0.0 were very small (<0.05°/o). as expected. and

 

these errors are assumed to be numerical round-off errors. In all cases. the root mean
squared errors (RMS) are close in magnitude to the standard deviation of the input tem-

perature errors. or. as expected. The estimated thermal conductivity values associated

with initial temperatures equal to 75°C and 125°C. and (31, equal to 010°C all had er-

 

rors less than 1% of the input thermal properties used in CUREID. The errors

 

associated with the estimated thermal conductivity values with 0,1. equal to 025°C were
all less than 2.5%. The errors associated with the estimated density—specific heat were
typically higher: errors with CT = 010°C were less than 2%. while the errors in the es-
timated density-specific heat values with 0.1. = 0.25°C were less than 2.5%. with the
exception of one case for which the error was over 5%. It should be noted that in many
cases. the confidence intervals calculated using PROP1D_CURE of estimated thermal
properties with 0.1. > 0.0°C did not include the themial properties determined with or“ =
00°C. as expected.
The results for the cases in which the properties were taken to be linear functions
of temperature and degree of cure are given in Table 5.15 for Cases 7.1. 8.1-8.6. and
9. 1-9.6. for which the initial temperatures were all equal to 75°C. and in Table 5.16 for
Cases 10.1. 11.1-11.6. and 12.1-12.6. in which the initial temperatures are all were
equal to 125°C. In these tables. the percent errors given were calculated from the dif-
ferences between the estimated properties with (3,1. = 0.0 and the estimated properties
with 0.1. a: 0.0. Exact property values were not used in the error analysis in these cases
since the properties were assumed to be linear with temperature and degree of cure.
and these values varied slightly over each run. However. if the estimated parameters
are associated with the average surface temperature over each run and the average de-
gree of cure. one can compare these estimated values with using the same temperature
and degree of cure values in eqs. (5.6) and (5.7). The average surface temperatures and

degree of cure values were calculated from the cases with or = 0.0. For the case with an

initial temperature of 75°C and initial degree of cure of 0.40.

 

 

152

Table 5.15. Estimated Effective Thermal Conductivity. k. Perpendicular to the
Fibers and Density-Specific Heat. pc . and the associated 95% Confidence
Intervals for a Simulated Composite with an Initial Temperature of 75°C. an
Initial D gree of Cure of 0.40. and Thermal Properties: 1: (W/m°C) = 0.6678 +
9.031410" 4» 0.074201. and pcp (MJ/m’ °C) = 1.251 + 0.0045T + 0.13901.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Estimated Estimated
Thermal Conductivity Density-Specific Heat
Case 0: RMSb k Erroré pcp Errorc
No. (°C) (°C) (W/m°C) (%) (MJ/m’ °C) (%)
7.1 0.00 0.003 0.767810.0003 - 1.653i0.002 - k
8.1 0.10 0.102 0.768010.0019 0.03 1.654i0.016 0.06
8.2 0.10 0.103 0.7663t0.0019 0.20 1.62910.015 1.45
8.3 0.10 0.100 0.7664:0.0018 0.18 1.668i0.015 0.91
8.4 0.10 0.100 0.770010.0019 0.29 1.653:0.015 0.00
8.5 0.10 0.104 0.7697:0.0018 0.25 l.650¢0.018 0.18
8.6 0.10 0.098 0.768310.0017 0.07 1.657i0.016 0.24
9.1 0.25 0.253 0.7646i0.0050 0.42 1.61010.042 2.60
9.2 0.25 0.252 0.7652t0.0043 0.34 1.69110.038 2.30
9.3 0.25 0.255 0.773210.0052 0.70 1.706t0.044 3.21
9.44 0.25 0.255 0.767810.0047 0.00 1.66410.041 0.67
9.55 0.25 0.262 0.767210.0051 0.08 l.608:0.043 2.72
9..6 0.25 0.251 0.7658i0.0047 0.26 1.64910.040 0.24
a. Standard deviation of temperature errors.
b. Root mean squared error between ca1cu1ated and input temperatures in PROP1D_CURE.
(r. a (difference between test cases with OT > 0 and GT = 0.

 

 

153

Table 5.16. Estimated Effective Thermal Conductivity. k. Perpendicular to the
Fibers and Density-Specific Heat. pc . and the Associated 95% Confidence
Intervals for a Simulated Composite with an Initial Temperature of 125°C. an
Initial Digree of Cure of 0.80. and Thermal Properties: k (W/m°C) = 0.6678 +

 

 

 

 

 

 

 

 

9.03x10 + 0.074201 and and pcp (MJ/rn’ °C) = 1.251 + 0.0045T + 0.13901.
Estimated Estimated
Thermal Conductivity Density-Specific Heat
Case 0: RMSb k Errorc pcp Errorc
No. (°C) (°C) (W/m°C) (%) (MJ/m?°C) (%)
10.1 0.00 0.004 0.844310.0000 — 1.93010.000 -
11.1 0.10 0.101 0.844610.0009 0.04 .91910.017 0.57
11.2 0.10 0.104 0.8442:0.0019 0.01 .921:0.0l7 0.47
11.3 0.10 0.100 0.845210.0019 1.65 .94810.018 0.93
11.4 0.10 0.105 0.845110.0021 0.10 .93310.019 0.16
11.5 0.10 0.105 0.846910.0021 0.31 .933:0.019 0.16
11.6 0.10 0.099 0.843410.0019 0.11 .938:0.017 0.42
12.1 0.25 0.252 0.845110.0051 0.10 .937i0.046 0.36
12.2 0.25 0.255 0.8479i0.0054 0.43 .92610.049 0.21
12.3 0.25 0.250 0.856810.0048 1.48 .96210.004 1.66
:12.4 0.25 0.254 0.8407i0.0055 0.43 .928:0.051 0.10
12.55 0.25 0.248 0.8436:0.0045 0.08 .92610.041 0.21
122.6 0.25 0.259 0.845910.0052 0.08 .97210.047 2.18

 

 

 

 

 

 

 

a.
b.
c.

Standard deviation of temperature errors.
Root mean squared error between calculated and input temperatures in PROP1D_CURE.

% difference between test cases with OT

>0ando =
'1‘

0.

 

 

 

154

‘E: 768°C

and. 0: = 0.425;

for the case with an initial temperature of 125°C and initial degree of cure of 0.80.

T = 126.8°C
and. (.1 = 0.856.

Substituting the above values into eqs. (5.6) and (5.7). the thermal conductivity and

density-specific heat values are:

k(76.8.0.425) = 0.7686 W/m°C (5.88)
pcp(76.8.0.425) = 1.656 MJ/m’ °C (5.8b)
k(126.8.0.856) = 0.8457 W/m°C (5.8c)
pcp(76.8.0.425) = 1.940 MJ/m’ °C (5.8d)

The the parameters estimates resulting from six repetitions in each of the test
cases in Tables 5.15 and 5.16 were averaged and compared with the values shown in
eqs. (5.8a-d). The average parameter estimates. along with the percent error. based on
the difference between these averaged parameter values and those shown in eqs. (5.8a-
d) are shown in Table 5.17. In all cases the percent errors are less than 1%. and due to
the uncertainty of the temperature and degree of cure values. the estimates from the
cases with the lowest variance in the ’measured‘ temperature values do not necessarily

have the lowest percent error.

L141 1 o h E irn. _o 0fTh‘u -. ' H‘ !“ 011-110'131‘14 0.. ._
This section involves the analysis of the temperature data from the experiments
discussed in Section 4.3 for the estimation of thermal properties during the curing of

AS4/EPON 828-mPDA composite samples. The results from this estimation procedure

 

 

 

 

155

Table 5.17. Comparison of Averaged Estimated Thermal Conductivity.

It, and Density-Specific Heat. p7. Values with Values Calculated Using
eqs. (5.6) and (5.7). p

 

 

 

 

 

 

 

 

 

 

 

Case 0: TL, T.b ; Errorc 3;; Errorc

No. (°C) (°C) (W/m°C) (%) (MJ/m?°C) (%)
7.1 0.00 75 0.7678 0.10 1.653 0.16
8.1-8.6 0.10 75 0.7681 0.06 1.652 0.25
9.1-9.6 0.25 75 0.7673 0.17 1.655 0.08
10.1 0.00 125 0.8443 0.17 1.930 0.52
11.1-11.6 0.15 125 0.8449 0.10 1.932 0.41
12.1-12.6 0.25 125 0.8467 0.11 1.942 0.10

a. Standard deviation of temperature errors.

b. Temperature boundary condition and initial temperature (constant).
c. % difference between estimated parameters and parameters calculated using eqs.
(5.6) and 5.7).

 

156
are first presented in Section 5.3.2.1. followed by analysis of the parameter estimates
with regards to errors in Section 5.3.2.2. In the last sub-section. suggestions for im-

provements in the experimental design are presented.

5.3.2.1 Results from Experimental Data

The thermal properties. effective thermal conductivity perpendicular to the fiber
axis and effective density—specific heat. were estimated using the temperature measure-
ments from the three experiments discussed in Section 4.3. The first two of these
experiments were conducted during the curing process. and the third experiment was
conducted after curing. The third experiment was used as a basis for comparison with
the results previously presented for cured composite samples in Section 5.1. The tem-
perature measurements from the first experiment (Exp. 3.1). along with the associated
heat flux measurements are shown in Figure 5.13. and temperature measurements
from six of the twelve thermocouples in the second experiment (Exp. 3.2) are shown in
Figure 5.14. The break in the temperature data at approximately 140 minutes was due
to an unexpected abort of the data acquisition program. The calculated temperatures
resulting from a simulated experiment. using similar experimental conditions as those
used in Exp. 3.2 and thermal properties evaluated at 100°C from the results shown in
Table 5.1. are shown in Figure 5.15. Exp. 3.2 and 3.3 were conducted using the new
Ectron amplifiers and the computer controlled power supply. while Exp. 3.1 was con-
ducted using the Data Translation amplifiers and the manually controlled power
supply.

In estimating the thermal properties as functions of temperature and degree of
cure. the series of heat flux pulses associated with each experiment were evaluated in-
dependently. There were a total of 15 heat flux pulses associated with Exp. 3.1. 16
pulses were associated with Exp. 3.2. and 18 pulses were associated with Exp. 3.3. The
average magnitude of the heat flux pulses in Exp. 3.1 was approximately 2000 W/m’ .
and the duration time of each of the pulses was 40 seconds. The average magnitudes of
the heat flux pulses in Exps. 3.2 and 3.3 were approximately 800 W/m’ and 1500

W/m2 . respectively. and the duration of each pulse was 135 seconds in both cases.

 

157

 

.: .n .5. war—:0 mar—:6 Eugen—Km 838383. 3:29:29

Ema... 05 E 83:085..on 56m 80.... 03E use: pea—an? 98 35805302 83809th 9 .m 833m

$355ch oEE.

 

 

 

 

 

 

 

0mm OWN om“ cm" 0.0 . 0
ram
new.
row

.3}!le in}. . J{\\
.\. . .. , r
\v.“\\ om
.\\\ ..
t -cos
\x
\i.) \\\ ooeuezn .033: do Eofion 89: 85 ad. I
{.3 \\\ \ A8353 costs» peace: an .I now“
(J .IIJ. we 30.: course peace: we - i .
ooetzm cocoon no d3 Bob 85 one i:
3;
_ _ _ _ _ _ _ ............... z 1 . , - _ l o
g E _ _ _ _ _ .se

 

 

 

L

9

m

.d

9

.4

Q

11

n

l

8

\w

70

m
/M%
mm.
7. 1.
(

158

.85. dam. goo wag—p Eon—tog 9.53.8988. «aim—nab.
vacuum 05 E seafloooawofi. now 88.. 5E 30: boa—9‘ and 5.888832 056.89th 3 .m 333..—

AmBSEEV 053.

can 9.4m om“ 0w; ow o

 

 

 

.
II
0‘
‘

 

003.25 noose: do Eofion 50¢ 55 0N4. - - roofi
008.25 beacon do Eofion 50.: SE 9am I
ooouezm voodoo: do Eofion be I
1.....I....ll .. . 003.26 peace: so .| Iowa
ooanSm cocoon do d3 Eot EE on; - I
mocceam concor— eo 33 50.: EE mmé

(30) SJHTQladUIGLL

 

 

 

 

 

 

 

......-....- ...... u. . ‘ Jim.

 

 

(JH/M)
TQGH

 

159

 

 

 

.062 so 2.8.580 6250 such Bronson 82895 3855.
means an. announced assess? ocooow 2: do c8585 2: ape essence accountant... n; .n 85m:

Amobsfifiv oEEL

 

 

 

 

 

 

 

 

 

 

oom Gem om“ ONH cm 0
b b 1— h — h b b — -
tom
L
foe m
row 7%
. J
m.
low H
J
. 3
[God \/
o
. 3
(
..\. 005.25 peace: be In ION~
d d d G < d d d < 003.26 noose: 50¢ 55 don - i .
ooeesam cocoon 80.: 85 man .
, on:
\I/
............................ IO MYHQH
/Tl..e
. m m e
Z 1..
.l.\
com

 

 

160
Also. the steady state temperatures evident in Figures 5.13 and 5.14 between the heat
flux intervals were removed by determining the average temperature from the all of the
thermocouple measurements prior to the onset of each heat flux. and subtracting the
difference between each thermocouple measurement and the overall average tempera-
ture from each thermocouple measurement. Each analysis for the estimation of
thermal properties included temperatures from ten seconds prior to the onset of each
heat flux interval to at least 150 seconds after the end of each heat flux interval. over

which time the thermal properties were assumed constant.

In the experimental design. the heater was placed in the center of the laminate.
separating the laminate into two sections. top and bottom. and in using this design. it
was assumed that the heat flow was symmetric on either side of the heater. In the es-
timation of thermal properties. the top and bottom sections were first analyzed
separately. assuming half of the measured heat flux was applied to the top section and

the other half was applied to the bottom section.

The top and bottom sections associated with the eighteen heat flux pulses in the
experiment after curing (Exp. 3.3) were first analyzed for the estimation of the thermal
properties. efi'ective thermal conductivity perpendicular to the fiber axis and effective
density-specific heat. using PROP1D_CURE. The estimated thermal conductivity and
density-specific heat values for the top and bottom sections. along with the temperature
range over each heat flux pulse and the root mean squared error between the calculated
and measured temperatures. are shown in Tables 5.18 and 5.19. respectively. The tem-
perature data from the second experiment during curing (Exp. 3.2) was analyzed in a
similar manner. and parameter estimates for the top and bottom sections are given in

Tables 5.20 and 5.21. respectively. These tables also include the average degree of cure
values at the beginning and end of each analysis interval. In both experiments. during
and after curing. the estimated thermal conductivity and density-specific heat values for
the bottom section were higher than those estimated for the top section.

The difl'erences in the parameter estimates between the top and bottom sections
suggested that. even though the steady state temperatures were subtracted from the

data. there was uneven heat flow within the top and bottom sections of the sample. To

161

Table 5.18. Estimation of Thermal Conductivity. k. Perpendicular to
the Fiber Direction and Density-Specific Heat. pc . of Cured
AS4/EPON 828-mPDA Composites using Temperature fiata from the
Top Composite Section Only in Exp. 3.3.

 

 

 

Pulse Temperature k pcp RMSa
No. Range (°C) (W/m°C) (MJ/m?°C) (°C)
1 51-57 0.493 0.655 0.147
2 54-60 0.504 0.906 0.142
3 57-63 0.460 0.903 0.166
4 60-65 0.506 0.919 0.153
5 62-66 0.499 0.951 0.153
6 63-69 0.516 0.923 0.143
7 65-70 0.506 0.970 0.146
6 66-71 0.546 0.993 0.149
9 67-71 0.573 0.963 0.219
10 76-64 0.532 1.053 0.160
11 9 61-66 0.499 0.960 0.220
12 85-91 0.518 1 02l 0.149
13 90-96 0.499 1.012 0.167
14 94-100 0.519 1.051 0.164
15 99-105 0.506 1.055 0.211
16 104-110 0.516 1.063 0.160
17 109-114 0.515 1.075 0.167
16 112-117 0.501 1.120 0.162

 

 

 

 

 

 

 

a. Root mean squared error determined from the differences between cal-
culated and experimental temperatures.

162

Table 5.19. Estimation of Thermal Conductivity. k. Perpendicular to
the Fiber Direction and Density-Specific Heat. pc . of Cured
AS4/EPON 828-mPDA Composites using Temperature [fata from the
Bottom Composite Section Only in Exp. 3.3.

 

 

 

Pulse Temperature k pcp RMSa
No. Range (°C) (W/m°C) (MJ/m’ °C) (°C)
1 51-57 1.762 . 1.535 0.136
2 54-60 1.655 1.664 0.139
3 57-63 1.677 1.695 0.127
4 60-65 1.862 1.705 0.151
5 62-68 1.589 2.030 0.176
6 63-69 1.796 1.792 0.138
7 65-70 1.564 1.977 0.144
8 66-71 1.912 1.755 0.155
9 67-71 1.751 1.950 0.171
10 78-84 1.533 1.917 0.160
11 81-86 1.533 1.681 0.163
12 85-91 1.643 1.784 0.156
13 90-96 1.476 1.682 0.137
14 94-100 1.526 1.606 0.144
15 99-105 1.304 1.475 0.157
16 104-110 1.143 1.412 0.175
17 109-114 1.205 1.320 0.167
18 112-117 1.084 ' 1.455 0.178

 

 

 

 

 

 

 

a. Root mean squared error determined from the differences between cal-
culated and experimental temperatures.

Table 5.20.

163

Estimation of Thermal Conductivity. k. Perpendicular to
the Fiber Direction and Density-Specific Heat. pc . of AS4/EPON 828-
mPDA Composites during Curing using Temperatu‘i’e Data from the Top
Composite Section Only in Exp. 3.2.

 

 

 

 

 

 

 

 

 

Pulse Temperature Degree k pcp RMSa
No. Range (°C) of Cure (W/m°C) (MJ/m3°C) (°C)
1 71-73 0 11-0 15 0.761 1.409 0.161
2 70-72 0.19-0.23 0.806 0.686 0.364
3 70-72 0.27-0.31 0.837 0.702 0.345
4 70-72 0.35-0.39 0.781 0.708 0.287
5 70-73 0.43-0.47 0.870 0.784 0.300
6 70-73 0.51-0.55 0.794 0.720 0.256
7 71-74 0.58-0.62 0.737 0.853 0.195
8 92-96 0.82-0.84 0.590 1.041 0.131
9 97-100 0.85-0.87 0.646 1.122 0.140
10 102-105 0.88-0.90 0.573 1.140 0.116
11 107-110 0.91-0.93 0.564 1.040 0.148
12 111-114 0.94-0.96 0.567 1.119 0.139
13 114-117 0.96-0.98 0.674 1.045 0.114
14 116-118 0.99-1.00 0.661 1.157 0.152
15 116-119 1.00 0.578 1:087 0.119
16 117-120 1.00 0.583 1.166 0.120

 

a. Root mean squared error determined from
and experimental temperatures.

the differences between calculated

 

a.

Table 5.2 1.

Bottom Composite Section Only in Exp. 3.2.

164

Estimation of Thermal Conductivity. k. Perpendicular to
the Fiber Direction and Density-Specific Heat. pc . of AS4/EPON 828-
mPDA Composites during Curing using Temperfiture Data from the

 

 

 

 

 

 

 

 

 

Pulse Temperature Degree k pcp RMSa
No. Range (°C) of Cure (W/m°C) (MJ/m’ °C) (°C)
1 71-73 0.11—0.15 5.665 5.168 0.124
2 70-72 0.19-0.23 1.279 2.700 0.104
3 70-72 0.27-0.31 1.533 2.793 0.112
4 70-72 0.35-0.39 1.762 1.515 0.162
5 70-73 0.43-0.47 1.808 2.018 0.116
6 70-73 0.51-0.55 1.786 1.717 0.139
7 71-74 0.58-0.62 1.762 1.762 0.145
8 92-96 0.82-0.84 3.662 2.024 0.143
9 97-100 0.85-0.87 3.205 1.842 0.138
10 102-105 0.88-0.90 3.152 1.590 0.145
11 107-110 0.91-0.93 4.419 1.921 0.143
12 111-114 0.94-0.96 3.651 1.561 0.155
13 114-117 0.96-0.98 3.543 1.721 0.168
14 116-118 0.99-1.00 2.860 1.840 0.147
15 116-119 1.00 2.501 1.453 0.133
16 117-120 1.00 2.845 1.583 0.136

 

and experimental temperatures.

Root mean squared error determined from the differences between calculated

 

165

eliminate the need for the assumption of symmetrical heat flow. the temperature dis-

tributions in the top and bottom sections were averaged. The basis for this averaging

can be expressed mathematically as follows:

From eqs. (3.19a-d) and the assumption of constant thermal properties over each

analysis interval. the governing equations for the top section can be expressed as.

kg;[§;r] + thgf‘ = pcpz—fr 0<x<LT; t>0
with the boundary conditions.
BTT
-ka—£ =qT(t) x=0: t>0
T = Twit) x=LT: t>0
- and the initial condition.
T=T.(x) OsstT: t=0

(5.9a)

(5.9b)

(5.90)

(5.9d)

where TT(x.t) represents the temperatures associated with the top section. q. (t) repre-

sents the heat flux boundary condition due to the heater. and LT is the thickness of the

top section.

Likewise. for the bottom section.

3T 61‘
—B 119. - _B .
kg;[ax] + ptht - ”pat O<x<LB' t>0
with the boundary conditions.
8T

3 _ _. .
4:5; — qB(t) x-0. t>0

(5.10a)

(5. 10b)

166

T = TLB(t) x = 1.3: t > 0 (5.10c)
and the initial condition.
T = T. (x) 0 s x 5 LB: t = 0 (5.10d)

where TB(x.t). qB(t). and LB are the temperatures. heat fluxes and thickness associated
with the bottom section. By assuming that bottom thicknesses of the top and bottom
sections are similar. and substituting the average of these thickness. L = (LT + LB)/2. for
LT and LB. eqs. (5%) and (5.10a); (5.9b) and (5.10b): (5.9c) and (5.10c): and. (5.9d) and

(5. 10d) can be added together and averaged as follows:

11] 11s _ 3:: .
kg;[ax + thdt _ pcpat 0<x<L. t>0 (5.11a)
with the boundary conditions.
«3% = q(t) x = 0; t > 0 (5.11b)
T = TL(t) x = L; t > 0 (5.11c)

and the initial condition.
T=T.(x) Osst; t=0 (5.11d)
where. T = (TT «1» TB)/2.

T = (q.r + qu/Z.
and, T = ('1‘LT + TLB)/2

167
Therefore. the temperature data from both top and bottom sections of the composite
can be used by using the average heat flux (which was used when the sections were
analyzed separately) for the first boundary condition and the average temperature for
the second boundary.

The heat flux intervals in the experiment during curing and the two experiments
after curing were then analyzed using temperature measurements from both top and
bottom sections. The results for the estimated thermal properties for some of the heat
flux pulses in Exp. 3.3. the experiment after curing; Exp. 3.2. the second experiment
during curing; and. Exp. 3.1. the first experiment during curing. are shown in Tables
5.22. 5.23. and 5.24. respectively. The 95% confidence intervals shown for these values
were determined in the program PROP1D_CURE (Beck. 1989). .

The results from the experiment after curing were first compared with the pre-
vious estimates of the thermal properties shown in Table 5.1. All of the estimated
thermal conductivity values shown in Table 5.22 were approximately 90% higher than
the values at corresponding temperature ranges in Table 5.1. and the density-specific
heat values in Table 5.22 were approximately 150% higher than the density-specific
heat estimates at corresponding temperature ranges in Table 5.1. In addition. the
parameter estimates for both thermal conductivity and density-specific heat indicated
no significant temperature dependence. since all of the confidence intervals for each
parameter were over lapping. The temperature rises for each heat flux pulse were also
compared with the temperature rises for the heat flux pulses resulting from the simu-
lated experiments discussed in Section 5.3.1. The heat fluxes in the simulated
experiments were 800 W/m’ . and the resulting temperature rises were approximately
5°C. using thermal properties based on the values shown in Table 5.1. However. the
temperature rises shown in Table 5.22 for Exp. 3.3 were only 5°C to 6°C for a heat flux
of approximately 1500 W/m’ .

The results from the second experiment during curing (Exp. 3.2) in Table 5.23
were then compared to the results from Exp. 3.3 shown in Table 5.22. As was found for

Exp. 3.3. both the estimated thermal conductivity and density-specific heat values in

Table 5.22.
the Fiber

168

Direction and Density-Specific

Heat.

PC-

Estirnation of Thermal Conductivity. k. Perpendicular to

of Cured
AS4/EPON 828-mPDA Composites using Temperature Data from Both
Top and Bottom Composite Sections in Exp. 3.3.

 

 

Pulse Temperature k pcp RMSa
No. Range (°C) (W/m°C) (MJ/m?°C) (°C)
2 54-60 1.44i0.10 3.86:0.46 0.330
5 54-60 1.54:0.17 4.25i0.68 0.397
8 66-71 1.47:0.06 3.83iO.31 0.282
11 81-86 1.5110.14 4.33i0.60 0.340
14 94-100 1.52i0.16 4.4710.66 0.359
17 109-114 1.5610.19 4.75:0.79 0.416

 

 

 

 

 

 

 

a. Root mean squared error determined from the differences between cal-
culated and experimental temperatures.

Table 5.23.

Estimation of Thermal Conductivity. k. Perpendicular to
the Fiber Direction and Density-Specific Heat. pc . of AS4/EPON 828-
mPDA Composites during Curing using Tempera‘ture Data from both
Top and Bottom Composite Sections in Exp. 3.2.

 

 

 

 

 

 

Pulse Temperature Degree k pcp RMSa
No. Range (°C) of Cure (W/m°C) (MJ/m’ °C) (°C)
2 70-72 0.19-0.23 1.67i0.09 3.9010.56 0.258

4 70—72 0.35-0.39 1.73:0.10 3.8010.61 0.223

6 70-73 0.51-0.55 1.7710.08 3.73:0.48 0.205

9 97-100 0.85-0.87 1.6510.10 4.60:0.57 0.216
12 111-114 0.94-0.96 1.6210.19 5.24i0.96 0.267
15 116-119 1.00 1.6010.12 4.6210.70 0.256

 

 

Root mean squared error determined from differences between
experimental temperatures.

 

 

calculated and

 

Table 5.24.

169

Estimation of Thermal Conductivity. k. Perpendicular to
the Fiber Direction and Density-Specific Heat. pc . of AS4/EPON 828-
mPDA Composites during Curing using Tempera’fure Data from Both
Top and Bottom Composite Sections in Exp. 3.1.

 

 

 

 

 

 

 

 

Pulse Temperature Degree k pcp RMSa
No. Range (°C) of Cure (W/m°C) (MJ/m’ °C) (°C)
2 73-81 0.34-0.38 1.2010.11 .6010.45 0.356

4 74-81 0.49-0.54 1.2410.20 .95i0.91 0.415

6 74-81 0.65-0.68 1.24:0.13 .9Sio.60 0.351

9 117-123 0.98-0.99 1.52t0.37 .50i1.16 0.441
11 121-129 0.99 1.13:0.10 .7lio.50 0.275
14 123-131 1.00 1.60i0.23 .40iO.77 0.168

 

Root mean squared error determined from differences between
experimental temperatures.

calculated and

 

170

Table 5.23 are statistically independent of temperature. The estimated thermal conduc-
tivity values from Exp. 3.3 were approximately 10% higher than those estimated for
Exp. 3.2. and the estimated density-specific heat values were statistically equivalent in
both experiments. Also. the temperature rise during this experiment was expected to be
on the order of 5°C to 6°C. as discussed in Section 4.3; however. the observed tempera-
ture rise was only 2°C to 3°C. This is consistent with the results shown for Exp. 3.3.
for which the temperature rise was much less than anticipated. '

Finally. the results from the two experiments during curing (Exp. 3.1 and Exp.
3.2) were compared. Both the thermal conductivity and density-specific heat estimates
shown in Table 5.23 for Exp. 3.2 were significantly higher than the values shown in
Table 5.24 for Exp. 3.1.

Due to the discrepancies between the results from the experiment after curing
(Exp. 3.3) and those shown in Table 5.1. the difference in the required heat flux for a
given temperature rise between the simulated experiments discussed in Section 5.3.1
and Exp. 3.3. and the differences between the results for the two curing experiments
(Exps. 3.1 and 3.2). the parameter estimates from these three experiments were taken
to be unreasonable. and additional efforts were concentrated on finding the reasons for
the discrepancies listed above with the goal of providing improvements for the ex-

perimental design.

5.3.2.2 Investigation of Possible Errors in the Estimation of Thermal Properties during
Curing

The residuals. as discussed previously. are useful in gaining insight into possible
errors in the experimental design and estimation procedure. The residuals for three of
the five thermocouples at the heated surface during the fifth heat flux pulse for Exp.
3.3. conducted after curing. are shown in Figure 5.16. These values are typical of the
residuals for all the heat flux pulses in Exp. 3.3. The temperature rise at the heated
surface in Exp. 3.3 was approximately 6°C. and the magnitude of the maximmn

residuals of the three thermocouples shown in this figure were approximately 3.0°C.

 

Residuals Residuals
(°C) (°C)

Residuals
(°C)

171

 

 

 

 

 

 

1.0

 

0.0
— 1.0 -
—2.0 -

-3.0 -

 

¢

 

1..

Heat flux I

 

 

1.0 -
0.0
-1.0-
-26.?

-3.0 -

 

A

II
I

_. TC”

 

i

 

0.0

200.0 300.0

Time (see)

Figure 5. 16 Residuals Associated with Three of the Five Thermocouples
r0041. TC#2. and TCii3) Located at the Heated Surface for the Fifth Heat Flint

Pulse in Exp. 3.3.

172
0.9°C. and 08°C. and the magnitude of the maximum residuals for the two ther-
mocouples at the heated surface not shown in this figure were 2.8°C and 1.5°C. These
residuals were from 12% to 50% of the maximum temperature rise. which suggests sig-
nificant errors could be associated with these measurements. The residuals are also
very biased. suggesting that the heat flux distribution across the heater was extremely
nonuniform: this is especially evident for the first thermocouple.

To further investigate other possible sources of error. the temperatures measure-
ments at the end of the heat flux pulses shown in Tables 5.22. 5.23. and 5.24 were
plotted as functions of location with respect to the heated surface in Figures 5.17. 5.18.
and 5.19. respectively. The 95% confidence bands shown in each figure were found
from the linear regression of temperatures with respect to location using PIiO'l‘itR (1989).
In all of these figures. the temperature differences across the composite samples were
within the error limits of the thermocouples. This is evident by comparing the con-
fidence bands near the heated surface with the confidence bands of the corresponding
regions farthest from the heated surface. In each of the heat flux pulses shown in each
figure. the confidence intervals overlapped one another. signifying that the tempera-
tures were statistically equivalent. This also indicated that the temperatures at the
boundary away from the heated surface increased. which is illustrated by comparing
the measured temperatures farthest from the heated surface in Figures 5.13 and 5.14
with the calculated temperatures farthest from the heat flux boundary condition in the
simulated experiment shown in Figure 5.15. The experimental temperatures farthest
from the heated surface were shown to increase significantly during the heat flux inter-
vals. while ideally they would remain constant. as shown by the calculated
temperatures farthest from the heated surface in Figure 5.15. This suggests that the
mechanical press and oven apparatus used in these experiments were inappropriate
clue to lack of a means to control the temperature at the boundaries of the composite
away from the heated surface.

Another point of question is the degree of temperature rise resulting from the ap-
plied heat flux. As noted previously, the temperature rises in both Exp. 3.2 and Exp.

3.3 were much less than expected. One possible explanation for this is that the thin

173

. House:
05 .8 550832 05 g 8583 “adage—F .8 0:050:2— 3 30:08:80 08:0. ad .3 B moo—.5
5E «so: 52 use .53 .5" a .5m .5m .38 05.3 9.5 05 «e 030885302 Edam—09:89 Sn 3:9..—

 

 

 

 

 

 

 

 

AEEV
3350008539 co 003.804
06 on... own ohm or; 0.0
$13 been .250 a. zoo: 1| . om
2 used w
2.: econ .38 a coo: .l a0.................................................o..c.... .............................
30min 6 ................ -..-.........:53”””Hnnmomnnnfiufififififiwm row
23 corn .38 a one: -i do .............................. ah. - - - - o .
: ..aa . E... Hum Hm - .a- . H 6mm
3v pcon .ccoodceoz -1 «newHHHHHHHHHHH.1.HHIMWHWIHWMIHWIH£WMWNMMHHHHHm 10b mm
m 025n— 4 iiiii q . w
3v 0:09 .250 on coo: - - d
n and... o .8 m
Amy each .300 a. duo: llllllllllllll o B
m 025d 0 cm H H H H H H H H H H H H HlHWIMIHlHlHlHlHIHlHIHlHMmIHlHlHlHl lllllllllllll (m m.
iii 1 J
om a
)
111111111 0
HH HH H HH H Hmwmmmwllmmmmmmswmmmmmmflflw - cos (0
((1)) lo:
“II M N m m
in

 

 

 

Mean 8: cont. band (2)
Mean 8c conf. band (4)
Pulse 6

-— — Mean & conf. band (6)

Pulse 4
Pulse 9
- Mean & conf. band (9)

Pulse 2

O
D
A
0

Mean 8: cont. band (12)

Pulse 15
— Mean 8c conf. band (15)

Pulse 12

V

X

 

 

 

 

 

 

 

Ill

:8:
III
III
III
m
III
III
III
m
m
m
m
m
I?
H?
m
m
m
m
m
m

III
III
III
III
III
III
III
|||

%

. r:
if};

0

n '
I'll:
ct“:
H '
(0:
tilt.
is"

5.0

3.0

2.0

1.0

 

 

130

1201

l ' I
O O
C) no

110-
100-«

(30) aaneJadtueJJ

 

0.0

ples

(mm)

Location of Thermocou

Figure 5.18 Temperature Measuremems at the End of the 2nd. 4th. 6th. 9th. 12th. and 15th Heat Flux
Pulses In up. 3.2 (uncured composites; second experiment) as Functions of Thermocouple Location from
the Midsection of the Heater.

175

:5 2:5
:3 scan
:3 EB
6v 2:5
3 2:5

A8 33

£3on 05 .«0 83883 05

Bob .8333 fiancee—Eva. .3 32655 no 985.898 3.5 ”933358 6898:. a .m .5 5 mom—«S
x3... .8: £3 93 .5“ u .fim .So ..3 .25 23o ecu 2: 3 888858»: 8382.83 a3 95mm

.250 on new:

«A mom—3m

.300 on new:

: omgm

.300 a. semi

m 025m

.230 a. com:

© Mom—3m

.250 a. Geo:

v 023nm

.300 a. Geo: .

N 02:;

'.
'<: ololul
. I

0.0

 

 

 

 

 

AEEV
meadooofismze Ho Cofimooq
o a Qm o m w o“ 0.0
_ L _ _

Huh“ ........ Toe
u I! lo .I l I III!“ “I‘llolv.ll.lhll. W .0 .I-la h .I .l.l. P. lo H.l.l-l. ........................ f
@uwwwwuwnmwarcwmwunwn----Muhwuhwuhuuhwuh
llullldlllllllHlllluuulnh aluluhudhhluhflfhahh. .un......u..u.u.u.ul.:nn....n. .... Q 100

0|IJJflhllJflvlJvadhlldalflhllllllll.I..".Ilvlll [m
[cm
r
TooH
no:
fiuflHHHHHHHHHHHHHHHHHH uuuuuuuuuuu AS
fl». .. Ml: uflnxuwlhh n u HUI. u u HUI. n w WW” .w
IIIIIIIIIIIIIHJIMHIIH lllllllllll D
llllllffllw nomfi

 

 

(3°) adnqeaadmal,

176
sheets of aluminum foil placed on either side of the heater extended past the heater on
all sides. acting as a fin around the heater. Due to the low thermal conductivity of the
composite compared to that of the heater. it is possible that some of the heat was dis-
sipated through the fin area. reducing the effective heat flux. If the heat flux is
calculated using the foil surface area instead of the heater surface area. the resulting
heat flux is 65% less than the heat flux calculated from the heater surface area. The
cases shown in Table 5.22 for the cured composite were re-evaluated using the adjusted
heat flux. The resulting parameter estimates are shown in Table 5.25. Comparing
these values to those shown in Table 5.1. the confidence intervals of the density-specific
heat values overlapped one another at corresponding temperature levels. The con-
fidence intervals of the thermal conductivity values did not all overlapping. and the
results from Exp. 3.3 were approximately 5 - 8% higher than those shown in Table 5.1.
However. due to the uncertainties discussed in the previous paragraphs. these es-

timates were still regarded as unreliable.

5.3.2.3 Improvements in the Experimental Design for the Estimation of Thermal
Properties in Composite Materials during Curing
From the investigation of possible sources for error presented in the previous sec-
tion, several suggestions for improvement in the experimental design are presented.
1. Due to the difficulties of maintaining the proper temperature boundary condi—
tions. it is recommended that the samples be cured in an autoclave or a hydraulic
press with heated platens. While the autoclave is generally more effective. the
hydraulic press is practically preferred. because the thermocouple wires and the
heater connections would not interfere with its operation. while special ports for
these wires would be required to use the autoclave. This would also reduce the
heating times between room temperature and 75°C. and from 75°C and 125°C.
Ideally. the composite would be heated at approximately 25°C/minute; however.
as shown in Figures 5.13 and 5.14, the samples were heated at a rate less than

1 °C/minute.

177

Table 5.25. Estimation of Thermal Conductivity. k. Perpendicular to
the Fiber Direction and Density-Specific Heat. pc . of Cured
AS4/EPON 828-mPDA Composites using Temperature Data from both
Top and Bottom Composite Sections in Exp. (3.3). with the Applied
Heat Flux Calculated from the Measured Power Input and the Surface
Area of the Aluminum Foil Protection Layer.

 

 

Pulse Temperature k pcp RMSa
No. Range (°C) (W/m°C) . (MJ/m’ °C) (°C)
2 54-60 0.86:0.02 1.61:0.07 0.144
5 54-60 0.84i0.04 1.71:0.17 0.205
8 66-71 0.9010.02 1.64:0.07 0.133
11 81-86 0.84:0.04 1.89:0.15 0.195
14 94-100 0.89:0.03 1.82:0.11 0.152
17 109-114 0.88:0.05 1.81:0.18 0.192

 

 

 

 

 

 

 

a. Root mean squared error determined from the differences between cal-
culated and experimental temperatures.

178
2. In order to increase the temperature difference across the samples. it is recom-
mended that sample thickness be doubled; this would result in a total thickness
of eighty-eight plies. which would require the fabrication of two prepregs for each
experiment.
3. Because of the high variations between the thermocouples at the heated sur-
face. it is recommended that a heater with a more uniform heat distribution
across the surface be used.
4. In addition. it is recommended that the thermocouples closest to the heated
surface be placed one ply away from the heated surface. instead of directly next to
the heater.
5. Finally. due to the possible dissipation of heat due to the increased surface

area of the foil. the foil should only be applied to the surface area of the heater.

Chapter 6

Summary and Conclusions

The focus of this study was on the estimation of thermal properties for cured
carbon/ epoxy composites and on the estimation of both thermal and kinetic properties
for these composites during curing. Although much attention has been devoted to the
determination of the mechanical properties of these materials. relatively little effort has
been given to the determination of thermal properties. especially during the curing
process.

The first overall objective was related to the estimation of thermal properties of
cured carbon/ epoxy composites as functions of temperature. Experimental set-ups
were designed using AS4/EPON 828 composite samples with the fibers oriented in two
different directions: [0°]24 and [0°. 130°. 160°. 90°],usym). These experiments were con-
ducted at different initial temperatures ranging from 25°C to 125°C. and the thermal
response of the composite to an applied heat flux was measured using thermocouples
embedded between composite samples. The thermal conductivity perpendicular to the
fiber direction. 1:. and the density-specific heat. pcp. were estimated using an estab-
lished parameter estimation program. PROPID (Beck. 1989). which utilizes a Gauss
minimization procedure. From this analysis, both it and pcp were found as functions of
temperature. This analysis differs from previous studies in that the experiments were
transient. resulting in the simultaneous estimation of both it and pcp. and the

properties were estimated as a function of temperature.

179

EH

180

The estimation of the kinetic properties associated with the curing of the EPON
828/mPDA epoxy matrix was the second overall objective of this investigation.
Isothermal experiments were performed at four different temperatures using differential
scanning calorimetry. and the heat of reaction rate was recorded as a function of time.
Rate constants were estimated from this data using three different kinetic models for
the first half of the curing process. and two different models for the second half of the
curing process in which diffusion was assumed to be significant. The kinetic
parameters. including activation energy constants and pre-exponential factors. were
determined from the eStimated rate constants. assuming an Arrhenius relationship with
temperature. The most appropriate model was then selected for the first half of the
cure. based on the confidence intervals of the estimated kinetic parameters. and a new
model was proposed for the diffusion-controlled. second half of the cure cycle.

The last overall objective was related to the estimation of thermal properties as
functions of temperature and degree of cure. A new and potentially powerful estimation
procedure was proposed for the simultaneous estimation of k and pcp throughout the
curing process. This procedure was once again based on the minimization of a least
squares function. In this case. the parameter estimation program. PROPID (Beck.
1989). was modified to account for the heat of reaction of the epoxy during curing. This
required the solution of an additional diiIerential equation for the kinetic reaction rate
in the computations. The estimation procedure was tested and verified using simulated
data from the one dimensional curing program. CUREID. with added normally dis-
tributed independent errors. Experiments using AS4/EPON 828 composites. conducted
both during and after the curing process. were used in conjunction with the modified
parameter estimation program. PROP1D_CURE. for the estimation of thermal properties.
Unrealistic estimates of the thermal properties from the experimental data led to the
conclusion that the experimental design and equipment used in this study were insuffi-
cient for the estimation of the thermal properties. Proposed improvements for the
experimental design included curing the samples in a hydraulic press with heated

platens and doubling the sample thickness.

18 l
The following conclusions were drawn from this study. (All references to thermal

conductivity are perpendicular to the fiber axis.)

1. Both the thermal conductivity and the density-specific heat of cured
AS4/EPON 828 composite materials were found to increase with temperature from
25°C to 145°C.

2. The estimates of the thermal conductivity of AS4/EPON 828 composite
samples with an orientation of [0°] were significantly higher than the thermal conduc-
tivities estimated for the same material with an orientation of [0°. t30°. 160°. 90°].
based on a comparison of the estimated linear regression curves given in Table 5.2. For
example. at 100°C. the thermal conductivity estimated for the [0°] samples was 7%
higher than the thermal conductivity estimated for the [0° . i30° . i60° . 90°] samples.

3. The estimates for thermal conductivity with an orientation of [0°] were within
5% of previously published values (Table 5.4) for the same or similar materials at 25°C.

4. The thermal history above the glass transition temperature of the epoxy had a
significant effect both on the thermal conductivity and density-specific heat of
AS4/EPON 828 composite materials; heating a sample for two hours at 150°C in-
creased the thermal conductivity 15% and the density-specific heat 5%. again based on
a comparison of the estimated linear regression curves given in Table 5.2 at 100°C.

5. For the EPON 828/mPDA epoxy system used in this study and degree of cure
values less than 50%. the kinetic model used by Ryan and Dutta (1979) was shown to
be the most appropriate model of the three kinetic models investigated based on the
confidence intervals of the estimated parameters and assuming an autocatalyzed
second order reaction.

6. A new model for the degree of cure rate was presented for degree of cure
greater than 50%. In this model. the rate of reaction is assumed to be diffusion-
controlled and to follow an exponential decay with the degree of cure.

7. The procedure for the estimation of the thermal properties during the curing
process was verified using the program PROPID_CURE with simulated data plus nor-

mally distributed independent errors of 0.25°C (5% of the maximum temperature rise).

182
The estimated parameters were found to be within 5% of the input parameters used to
generate the simulated temperature data.

8. The experimental design and equipment used for the estimation of thermal
properties during curing. which included a unheated mechanical press. were not ade-
quate to provide accurate estimates of these parameters. It is recommended that a
hydraulic press with heated platens be used in place of the mechanical press and that
thicker composite samples be used in the experiments.

9. In the experiments conducted during curing. the thermocouples located at the
heated surface had the highest residuals: to reduce these errors it was recommended
that these thermocouples be placed at least one ply layer away from the surface of the

heater.

APPENDIX A

APPENDIX A

ONE DMNSIONAL CURING PROGRAM. CUREID

A.1 Summary of Program

The one dimensional curing program. CUREID. discussed in Chapter 3. is
presented in this appendix. An outline of the program is given in Table A.l. and a list-

ing of the program. written in Fortran 77 for a VaxstationII/GPX microcomputer is
given in Table A l.

183

184

Table A. 1 Description of the One Dimensional Curing Program. CUREID.

W
PROGRAM CURE 1D

SUBROUTINE PROPER

SUBROUTINE INPUTI

SUBROUTINE INPUT2

SUBROUTINE SOLN

SUBROUTINE COEFF

SUBROU'UNE HEAT

SUBROUTINE BCF IND
SUBROUTINE PFIND

SUBROUTINE OUTPUT

Description

 

Main program; contains program menu.

Allows interactive input of thermal
properties. Writes data to a file.

Allows interactive input of ambient condi-
tions and product geometry. Writes data to
a file.

Allows interactive input of kinetic properties.
Writes data to a file.

Computes temperature distribution and
quality retention as a function of tempera-
ture. Calls output subroutine.

Determines matrix coefficients used in finite
difference algorithm.

Determines heat generation term from
kinetic equations.

Interpolates boundary conditions.

Interpolates thermal property values required
for the finite difference solution.

Writes input data and resulting temperature
and degree of cure values to an output file.

 

185

A2 Program Listing for CUREIDJ‘OR

PROGRAM CUREID

ct*ittii****fi*****fifiiii*t*tittti*tttitititititfitttitttt*tti‘ki’itiit

Ciii***t****i****litt*iii***t**t**it***tttttttltttititt*iitiifitit*

C

One Dimensional
Curing Program for Composite Materials
by
Elaine Scott, Ph.D.
1989
Copyright (c) 1989 Michigan State University

All rights reserved.

ctiii*iiii‘t'k'k*i’tiittii*ii****fi**********ti’it*iittiiii'ktiittiiiitti

00

0 0000 00000

00

Note: this program was adapted from program 'FREEZElD.FOR', by
E.P. Scott, 1987.

This program calculates temperature and extent of cure as a
function of time and location in composite materials during
curing. It is assumed that one dimensional heat transfer and a
second order autocatalyzed chemical reaction occurs with an
Arrhenius relationship with temperature.

Input parameters include the effective composite density,
thermal conductivity, and specific heat. The kinetic properties,
including rate constants and activation energy constants are
required to determine the extent of cure.

Boundary conditions are assumed to be in the form of a known
heat flux or temperature as a function of time. The initial
condition must be a known function of position.

ct'k‘k‘ki‘i’ii‘k'kttiitti’iit‘kii’i’i’ii*iittt***ittt*t**i**‘ktiitt‘kitiitiitit"

parameter(maxp=100,maxm=101)
integer model

character title'20,ttlfil*4,filyn1*1.fi1yn2*1,filyn‘l,
£tt1prp*4,ttlkin*4,fildat*12,inpdat*12,kindat*12

logical itmode

common/mod/model,/itm/itmode,/ttl/tit1e,tt1fi1,ttlprp,ttlkin,
s/datfil/fildat,inpdat,kindat

Set ITMODE = .FALSE. if running batch.
ITMODE a .TRUE.

IF(ITMODE)THEN

write(5,1000)

186

1000 format(’1’,72("’),/,’0’,t21,’0ne Dimensional Curing Program’,
&/,’0’,t35,’by’,/,’O’,t27,'Elaine Scott, Ph.D.',
&/,’0’,t14,’Copyright (c) 1987 Michigan State’,

5’ University’,/,' ’,t26,’A11 rights reserved.’,/,'O',72('*’))
WRITE(5,100)
100 FORMAT(’0’,’Program Menu:’,/

&,/,’ ’,’ 1. Temperature only (no chemical reactions)’,

8/,’ ’,’ 2. Temperature & extent of cure: exact kinetic prop.’.
&/.' ’,’ 3. Temp. & extent of cure: kinetic prop. with variance’,
&//,’ ’,’Se1ection?’)

ENDIF

READ(5,10)mode1
10 FORMAT(I1)
IF(ITMODE) write(5,200)
200 format(’ ’,/,’ ’,’Title: ’)
READ(S,20)TITLE .
IF(ITMODE)then
write(5,300)
300 format(’ ’,/,' ’,'Key word for input boundary conditions ’,/
5 ’ ’,2x,’and geometry data file; 4 Characters: ’)
READ(5,20)TTLFIL
endif
IP(ITMODE) write(5,320)
320 format(’ ’,/,’ ’,'Key word for thermal property data ’,
& ’file; 4 Char.: ’)
READ(S,20)TTLprp
IF(ITMODE.and.model.ne.1)then
write(5,340)
340 format(' ’,/,’ ’,’Key word for kinetic property data ’,
& 'file; 4 Char.: ’)
READ(5,20)TTLkin
endif
20 FORMAT(A)
if(itmode) write(5,400)
400 format(’ ’,/,’ ’,’Are thermal properties approximations',/,’ ',2x,
s'with temperature stored on file? (y/n) ’)
read(5,20)filyn1
if(itmode)then
write(5,500)
500 format(’ ’,/,’ ’,’Are input initial and boundary conditions’,/,’ ’
& ,2x,’and geometrical dimensions stored on file? (y/n) ’)
read(5,20)filyn2
endif
if(model.ne.1) then
if(itmode) write(5,600)
600 format(’ ’,/,’ ’,’Are the kinetic properties stored on file? ’,
&’(y/n) ’)
read(5,20)filyn3
endif
if(filynl.eq.'n'.or.filynl.eq.’N')then
call proper
endif
if(filyn2.eq.’n’.or.filyn2.eq.'N')then
call inputl
endif
if(model.ne.1)then
if(filyn3.eq.’n’.or.filyn3.eq.’N')then
call input2
endif
endif
call soln
end

00000 0 0 0000

00

SUBROUTINE PROPER

This subroutine provides the input for the property functions.
Input values include effective product thermal conductivities and the
density-specific heat products at given temperatures and/or extent of
cure values.

Output includes a printout of thermal conductivity and density-
specific heat products as functions of temperature.

The variables used in this subroutine are:

Constants-

Input Variables-
Ntemp - number of temperatures at which preperties are given
Kp(I) - Effective thermal conductivity (W/mK) at Ith temp.
DCp(I) - Density-specific heat product (kJ/m3K) at Ith temp.

Misc. Variables-
Tc = Temperature for printout.
Yn = Character- Y or N

integer ntemp,prpscr

double precision dcp(10,10),kp(10,10),tdc(10),tk(10),aldc(10),
aalk(10)

character yn*1,title*20,ttlfil‘4,ttlprp*4,tt1kin*4,fildat*12,
&inpdat*12,kindat*12

logical itmode
SAVE

common /ITM/ITMODE,
&/ttl/tit1e,ttlfil,tt1prp,ttlkin,/datfil/fi1dat,inpdat,kindat,
&/mod/mode1

IF(.NOT.ITMODE)THEN
READ(10,*)NTK,NTDC
IF(MODEL.NE.1)THEN

READ(10,*)NALK,NALDC
ELSE
NALK - 1
NALDC = 1
ENDIF
READ(10,*)(TK(I),I a 1,NTK)
IF(MODEL.NE.1)THEN
READ(10,*)(ALK(I),I = l,NALK)
ELSE
ALKll) = 1.000
BNDIF
DO I - 1,NTK
READ(10,*)(KP(I,J),J = l,NALK)
ENDDO
READ(10,*)(TDC(I),I = l,NTDC)
IF(MODEL.NE.1)THEN

1!

1883

READ(10,*)(ALDC(I),I = l,NALDC)
ELSE
ALDC(1) - 1.0D0
ENDIF
DO I = 1,NTDC
READ(10,*)(DCP(I,J),J = 1,NALDC)
ENDDO
GO TO 20
ENDIF
5 write(5,2000)
2000 format(’l',72(’-’),/,’O’,t27,’Thermal Properties',/,’O’,72('-’))
write(5,600)’Enter number of temperatures for thermal ’,
s’conductivity, k, values:'
600 format(’ ’,/,’ ’,A,A)
READ(5,*)NTK
IF(MODEL.NE.1)THEN
write(5,600)’Enter number of extent of cure values for k:’
605 format(' ’,/,’ ’,A)
READ(5,*)NALK
ELSE
NALK - l
ENDIF
WRITE(S,610)’Enter temperatures (C) for k (',NTK,’) :’
610 FORMAT(1X,/,1X,A,I2,A)
READ(5,*)(TK(I),I - 1,NTKl
IP(MODEL.NE.1)THEN
WRITE(5.610)’Enter extent of cures for k (',NALK,’) :’
READ(5,*)(ALK(I),I = 1,NALK)
ELSE
ALK(1) - 1.0D0
ENDIF
WRITE(S,620)’Enter k (W/mC) values: ’
620 FORMAT(1X,/,1X,A,/)
DO I = 1,NTK
IF(MODEL.NE.1)THEN
WRITE($,630)’Enter k at ’.TK(I),’C for ’,NALK,

& ’ extent of cure value(s):’
630 FORMAT(1X,A,F6.2,A,I2,A)
ELSE
WRITE(S,640)’Enter k at ’,TK(I),’C :’
640 FORMAT(1X,A,F6.2.A)
ENDIF
READ(5,*)(KP(I,J).J - 1,NALK)
ENDDO

c End of thermal conductivity input

'WRITE(S,900)
900 FORMAT(’ ’,/,' '72(’-’),/,’ ’,’Are these values correct? (y/n) ')
read(5,200)yn
200 FORMAT(A)
if(yn.ne.’y’.and.YN.NE.’Y’)goto S

c Start density-specific heat input

6 write(5,600)’Enter number of temperatures for density-',
8’specific heat, d-Cp, valuesz’
READ(5,*)NTDC
IF(MODEL.NE.1)THEN

write(5,605)’Enter number of extent of cure values for d-Cp'
READ(5,*)NALDC

ELSE
NALDC = 1

189

ENDIF
WRITE(5,610)’Enter temperatures (C) for d-Cp (’,NTDC,’) :’
READ(5,*)(TDC(I),I = 1,NTDC)
IF(MODEL.NE.1)THEN
WRITE(S,610)’Enter extent of cures for d-Cp (’,NALDC,') :’
READ(5,*)(ALDC(I),I = l,NALDC)
ELSE
ALDC(1) = 1.0D0
ENDIF
WRITE(5,620)’Enter d-Cp values: ’
DO I B 1,NTDC
IF(MODEL.NE.1)THEN
WRITE(5,630)’Enter d-Cp at ’,TDC(I),’C for ’,NALDC,

& ' extent of cure value(s)’
ELSE
WRITE(S,640)’Enter d-Cp at ’,TDC(I),'C ’
ENDIF
READ(5,*)(DCP(I,J),J I l,NALDC)
ENDDO

WRITE(5,900)
read(5,200)yn
if(yn.ne.'y’.and.YN.NE.’Y’)goto 6
20 CONTINUE

IF(ITMODE)THEN
write(6,908)
908 format(’ ’,72(’-’),//’ ’,’END OF PROPERTY DATA INPUT’)

ENDIP
C
C Convert C temperatures to K temperatures:
C

DO I = 1,NTK

TK(I) = TK(I)+273.15D0

ENDDO

DO I = 1,NTDC

TDC(I)= TDC(I)+273.ISDO

ENDDO
C
C Write thermal property data to file
C

WRITE(FILDAT,1000)TTLPRP,’PRP.DAT’
1000 FORMAT(' ',A,A)
OPEN(UNIT=12,NAME=FILDAT(1:12),TYPE=’NEW',CARRIAGECONTROL=’LIST')
WRITE(12,*)NTK,NTDC
WRITE(12,*)NALK,NALDC
WRITE(12,*)(TK(I),I = 1,NTK)
WRITE(12,*)(ALK(I),I = 1,NALK)
DO I a 1,NTK
WRITE(12,*)(KP(I,J),J = 1,NALK)
ENDDO
WRITE(12,*)(TDC(I),I = 1,NTDC)
WRITE(12,*)(ALDC(I),I = l,NALDC)
DO I = l,NTDC
WRITE(12,*)(DCP(I,J),J = 1,NALDC)
ENDDO

return
END

 

0000

00

190

subroutine inputl

This subroutine provides the input for the boundary condi-
tions assuming a known heat flux or temperature at the boundary.
The boundary conditions may be constant, a linear function of
time,or a combination of the two.

Input varibles include initial temperature, type of boundary
condition, and time and temperature and/or heat flux at each

c boundary.

00000

20

parameter(maxp=100)
integer Isym,ISTEP,Ishape,m,nbc1,nbc2,ibcl,ibc2

double precision TI,TQl(maxp),T02(maxp),time1(maxp),
stime2(maxp),H,L,DZ,DT,PDT

character yn*1,title*20,tt1fil*4,ttlprp*4,ttlkin*4,fildat*12,
sinpdat*12,kindat*12

logical itmode

common /ttl/title,ttlfi1,ttlprp,tt1kin,/mod/mode1,
&/itm/itmode,/datfi1/fildat,inpdat,kindat

save

IBCl and IBCZ indicate type of boundary condition at each surface.
IBCl, IBCZ - 1 indicates temperature boundary condition, and
IBCl, IBCZ - 2 indicates heat flux boundary condition.
NTBCl and NTBCZ indicates the number of temperature and/or heat flux
values given for each boundary.

if(itmode)go to 3
read*,ti,ibc1.nbcl,ibc2,nbc2,ttime,DT,PDT
do i = 1,nbc1
read*,time1(i),TQl(i)
enddo
do i = 1,nbc2
read*,time2(i).T02(i)
enddo
read Ishape
if(Ishape.lt.3)then
read*,l
h = 1.0d0
else
read*,1
h = 0.0d0
endif
go to 500

write(5,l)
1 format(’l’,72(’-’),/,'0’,27x,’Boundary Conditions’,/,’0’,72(’-’))

write(6,10)’Initia1 product temperature (C): ’

format(' ’,/,’ ’,A)

read*,ti

write(6,20)

format(’ ’,/,’ ’,’Are the boundary conditions symmetrical? ’,
&’(y/n) ’,/,’ ',’(Enter ”y” for solid cylindrical & spherical ',
s’geometries) ’)

read(5,2)yn

if(yn.eq.'y’.or.yn.eq.'Y')Isym = l

191

write(6,30)
30 format(’ ’,/,’ ’,'Enter type of boundary condition for IBC1:',
&/,5x,’1 - temperature boundary,’,/,5x,’2 = heat flux boundary’)
read*,Ibcl
if(Ibcl.eq.1)then
write(6,15)’Enter number of temperatures given at IBCl ’,
&’(include values at time = 0, and at time = tmax):’
15 format(/.lx,A./,1X,A)
read*,nbc1
else
write(6,15)'Enter number of heat fluxes given at IBCl',
&’(inc1ude values at time = 0, and at time = tmax):’
read*,nbc1
endif
if(Isym.ne.1)then
write(6,32)
32 format(/,1x,’Enter type of boundary condition for IBC2:’,
&/,5x,’l 8 temperature boundary,’,/,5x,’2 - heat flux boundary’)
read*,Ibc2
if(Ibc2.eq.1)then
write(6,15)’Enter number of temperatures given at IBCZ ’,
5 ’(include values at time - 0. and at time - tmax):’
read*,nbc2
else
write(6,15)’Enter number of heat fluxes given at IBCZ ’,
& ’(include values at time = 0, and at time = tmax):’
read*,nbc2
endif
endif
write(6,10)’Enter total time (s):'
read*,ttime
write(6,lO)'Are these values correct? (y/n) ’
read(5,2)yn
2 format(a)
if(yn.ne.’y’.and.yn.ne.’Y’)goto 5
write(6,106)

 

c input boundary conditions: time and temperature and/or heat flux

100 if(ibc1.eq.1)then
write(6,10)’Enter time (s) and temperatures (C) for IBClz’
do i=1,nbcl
write(6,120)i
120 format(6x,12,’: ’)
read*,time1(i),TQl(i)
enddo
else
write(6,10)’Enter time (s) and heat flux (W/m2) for IBClz’
do i=1,nbc1
write(6,120)i
read*,timel(i),TQl(i)
enddo
endif
if(ibc2.eq.l)then
write(6,10)’Enter time (s) and temperatures (C) for IBC2:'
do i=l,nbc2
write(6,120)i
read*,time2(i),T02(i)
enddo
else
write(6,10)’Enter time (s) and heat flux (W/mZ) for IBC2:'
do i=1,nbc2
write(6,120li

192

read',time2(i),T02(i)
enddo
endif

write(6,10)’Are these values correct? (y/n) ’
read(5,2)yn
if(yn.ne.’y'.and.yn.ne.'Y’)goto 100
write(6,106)

106 format(' ’/.' ’,72('-'))

c input geometry and size

140 write(6,150)
150 format(’O’,'Enter product geometry: ’,/,’ ',5x,’1 = slab’,/,’ ’,Sx
+,’2 - cylinder',/,’ ’.5x,’3 = sphere’)
read*,Ishape
if(Ishape.gt.1.and.Isym.eq.0)then
print*,’Boundary conditions must be symmetrical for cylinder and’,
8’ sphere; try again!!’
go to 3
endif
if(Ishape.eq.1)then
write(6,160)

160 format(’ ’,/,’ ’,’Enter dimensions for slab:’,/,’ ’,5x,
G’thickness in direction of heat transfer (m) - ’)
read*,l
h - 1.0d0
else

if(Ishape.eq.2)then
write(6,180)
180 format(’ ’./,’ ’,'Enter dimensions for cylinder:',/,
+’ ’,5x,’radius (m)= ')
read *,1
h = 1.0d0
else
write(6,200)
200 format(’ ’,/,’ ’,’Enter dimensions for sphere (m):’,/,
+’ ’,5x,’radius (m)= ')
read *,1
h=0.0d0
endif
endif
500 ti - ti+273.150d0
if(Isym.eq.1.and.Ishape.eq.l)L a L*0.50d0
do i = 1,nbc1
if(ibcl.eq.1)TQl(i) - TQl(i)+273.150d0
if(Isym.eq.1.or.Ishape.ne.1)then

if(Ishape.eq.1)then
T02(i) 8 0.0d0
ibc2 a 2

else
T02(i) = TQl(i)
TQl(i) = 0.0d0

endif

endif
enddo

do i a 1,nbc1
if(ibc2.eq.l)TQZ(i) = T02(i)+273.150d0
enddo
write(6,220)
220 format(’ ’,/’ ’,’Enter total number of spatial increments:',/
&’ ’,’(Must be a multiple of four.)’)
read*,m

C

0000000000000000

193

ISTEP = m/4
write(6,230)
230 format(lx,/,1x,’Enter time step for finite difference ’,
s'calculations:’)
read',dt
write(6,240)
240 format(lx,/,lx,’Enter time step for print out:’,/
8’ ’,'(Must be a multiple of the time step.)’)
read*,pdt

Check input values

write(6,10)’Are these values correct? (y/n) ’
read(5,2)yn
if(yn.ne.’y’.and.yn.ne.’Y')goto 140
write(6,106)
write(6,580)
580 format(' ',//,' ','END OF BOUNDARY CONDITIONS AND GEOMETRY INPUT’)

Write data to file.

590 write(inpdat,600)ttlfil,’inp.dat’

600 format(’ ’,a,a)
open(unit=12,name-inpdat(1:12),typea’new’,carriagecontrol=’list')
write(12,700)ibc1,ibc2,nbc1,nbc2,Isym,ti,ttime

700 format(’ ’,5(iZ,2x),2(2x,E11.5))
do i a 1,nbc1

write(12,800)T01(i),timel(i)

800 format(’ ’,2x,E11.5,2x,f18.2)
enddo
do i = 1,nbc2

write(12,800)T02(i),time2(i)
enddo
write(12,900)ISHAPE,L,H,M,ISTEP,DT,PDT

900 format(' ’,il,2(2x,ElO.4),2x,i3,2x,i2,2x,ElZ.5)
close(unit=12)
return
end

subroutine inputZ

Kinetic properties to determine extent of cure in a composite
during curing are entered in this subroutine. A file titled
’TTLFILkin.dat’ containing the kinetic properties is created. This
file is reopened in the solution subroutine, so that it may be reused
again in subsequent runs.

The extent of cure is determined assuming an autocatalyzed, second
order reaction exists for extent of cure < 0.5, and a diffusion control-
led reaction for extent of cure > 0.5. Nine parameters are required:

a) for extent of cure < 0.5; the activation energy constants, EAl and
EA2, the pre-exponential factors, A1 and A2, and the exponent, MEXP;
for extent of cure > 0.5; the activation energy constant, 8A3, the pre-
exponential factor, A3, and the diffusion constant, D3; and for all
extent of cure; the density, DENS, and the total heat of reaction, HT.
In addition the variances associated with Al, A2, 8A1, and EAZ are
included in model 3.

194

integer model

double precision LNA1,LNA2,EA1,EA2,M0,M1,DENS,HT,VA1,VA2,VEA1,
1 VEAZ,LNA3,EA3,DO,Dl,D2,Al,A2,A3

character title*20,tt1fil*4,ttlprp*4,ttlkin*4,fildat*12,
&inpdat*12,kindat*12

logical itmode

common /tt1/tit1e,ttlfi1,tt1prp,ttlkin,
&/mod/model,/itm/itmode,
&/datfil/fildat,inpdat,kindat

Read batch file data (if itmode = .false.)
if(itmode)go to 1

Read in values for extent of cure < 0.5
read',LNA1,LNA2,EA1,EA2,M0,M1

Read in values for extent of cure < 0.5

read*,LNA3,EA3,DO,Dl,D2,DENS,HT

if(mode1.eq.3)then
read*,VA1,VA2,VEA1,VEA2

endif

go to 30

Read interactive input

1 write(5,2l
2 format(’l’,72(’-'),/,'0’,t40,’Kinetic Data’,/,’0',
E72('-’))
10 write(5,100)
100 format(’ ’,/,’ ’,’Enter parameters for extent of cure < 0.5 :’,/,
a ' ’,4x,’Pre-exponentia1 factor, LN(A1) LN(1/s) : ')
read*,LNA1
write(5,110)’Pre-exponential factor, LN(A2) LN(1/s) : '
110 format(Sx,A)
read*,LNA2
write(5,200)’Activation Energy Constant, EAl, (kJ/mole) : '
200 format(’ ',/.5x,A)

read*,EAl

write(5,110)’Activation Energy Constant, EA2, (kJ/mole) : ’
read*,EA2

write(5,200)'Exponent, M = M0 + M1*T(C): '

WRITE(S,110)’ M0 = '

read*,M0

WRITE(5,110)’ M1 = ’

read*,M1

write(5,210)
210 format(’ ’,/,' ’,’Enter parameters for extent of cure > 0.5 :’,/,
8 ' ’,4x,’Pre-exponential factor, LN(A3) LN(1/s) : ’)

read*,LNA3

write(5,110)’Activation Energy Constant, EA3, (kJ/mole) : ’
read*,EA3

write(5,200)’Diffusion coefficient, D = D0 + Dl*T + DZ*T*T (T=C):’
WRITE(S,110)' D0 = '

read*,DO

WRITE(5,110)’ D1 = ’

read*,Dl

195

WRITE(5,110)’ D? a '

read*,DZ

write(5,200)’Density (kg/m3) : ’

read*,DENS

write(5,200)’Total heat of reaction, Ht, (J/kg) : ’
read*,HT

if(model.eq.3)then
write(5,200)’$tandard deviation of LN(A1) LN(l/s) :’
read*,VA1
write(5,110)’Standard deviation of LN(A2) LN(1/s) :’
read*,VA2
write(5,200)’$tandard deviation of EAl (kJ/mole) :'
read*,VEA1
write(5,110)’$tandard deviation of EA2 (kJ/mole) :’
read*,VEA2
endif
write(5,550)
550 format(’ ’,/,’ ’,’Are these values correct? (y/n) ')
read(5,20)yn
20 format(a)
if(yn.eq.’n'.or.yn.eq.'N’)go to 10

WRITE(6,560)
560 FORMAT(' ’//,' ’,’END OF KINETIC DATA INPUT’)
30 A1 - EXP(LNA1)
A2 ' EXP(LNA2)
A3 - EXP(LNA3)
EAl - EA1*1000.0d0
EAZ - EA2*1000.0d0
EA3 - EA3*1000.0d0
if(model.eq.3)then
veal - (vea1*1000.0d0)**2.0d0
vea2 - (vea2*1000.0d0)**2.0d0
vAl s (vAl)**2.0d0
vAZ - (vA2)**2.0d0
endif

c Write kinetic data to file.

write(kindat,600)ttlkin,’kin.dat’
600 format(’ ’,a,a)

open(unit=12,name=kindat(1:12),type=’new',carriagecontrol=’list')

write(12,*)Al,A2,EAl

write(12,*)EA2,M0,Ml

write(12,*)A3,EA3,DO

write(12,*)Dl,D2,DENS

write(12,*)HT

if(model.eq.3)then

write(lZ,800)VA1,VA2,VEA1,VEA2

800 format(’ ’,4(2x,e11.5))

endif

close(unit=12)

return

end

C *******************iit****I********ti*‘kt*ttt‘kttit’tt‘k‘kttitiii‘ki*‘kitittt

SUBROUTINE soln
parameter(maxm=101,maxp=100,tol=0.10d0,r=8.3140d0)

IMPLICIT DOUBLE PRECISION(A-H,P-Z)

196

double precision BCTQ(2,2),TAVG,cc(maxm),dd(maxm),a(maxm),
1 b(maxm),c(maxm),d(maxm),t(maxm,2),pi,pr,AL(MAXM),ALAVG,
1 ABC(5),AL_5(MAXM),DAL_5(MAXM),TC(12),SIGMA,ALPHA

character fildat*12,inpdat'lZ,kindat'12,0UTFIL2*12,0UTFIL3*12
logical itmode
C Declare variables in common statements

C Common block /ttl/
character tit1e*20,ttlfi1'4,ttlprp'4,ttlkin*4

C Common block /kin/
DOUBLE PRECISION Al,A2,EAl,EA2,MO,Ml,DENS,HT,VA1,VA2,VEA1,
l VEA2,A3,EA3,DO,D1,D2

C Common block /prop/
INTEGER NTDC,NALDC,NTK,NALK
DOUBLE PRECISION dcp(10,10),tdc(10),aldc(10),kp(10,10),tk(10).
salk(10)

C Common block lbound/
INTEGER IBCl, IBCZ , NBCl, NBCZ
DOUBLE PRECISION TI,TQl(MAXP),TQZ(MAXP),TIME1(MAXP),TIMEZ(MAXP)

C Common block /geom/
INTEGER ISHAPE,ISYM,M,MP1,ISTEP
DOUBLE PRECISION H,L,DZ

C Common block /tim/
DOUBLE PRECISION DT,TTIME,PDT

c Common blocks.
common/bound/ibcl,ibc2,nbcl,nbc2,ti,TQl,TQZ,time1,timeZ,
s/ttl/title,ttlfi1,ttlprp,tt1kin,
S/geom/ISHAPE,ISYM,M,MP1,ISTEP,H,L,DZ,
s/mod/model,/itm/itmode,
s/datfil/fildat,inpdat,kindat,
s/prop/dcp,tdc,aldc,kp,tk,alk,ntdc,naldc,ntk,nalk,
G/KIN/Al,A2,EA1,EA2,M0,M1,DENS,HT,VA1,VA2,VEA1,VEA2,A3,EA3,D0,Dl,DZ
S/TIM/TTIME,DT,PDT

save
c Read in boundary and initial conditions

write(inpdat,600)ttlfil,'inp.dat’
600 format(’ ’,a,a)
open(unit-12,name-inpdat(1:12),type=’old',carriagecontrol=’list’)

read(12,*)ibc1,ibc2,nbcl,nbc2,Isym,ti,ttime
do i - 1,nbc1
read(lZ,*)TQl(i),time1(i)
enddo
do i - 1,nbc2
read(12,*)T02(i),time2(i)
enddo

C Input geometry and dimensions

read(12,*)Ishape,L,h,m,ISTEP,dt,PDT
close(unit=12)

197'

c Read in constant property assumptions

WRITE(FILDAT,310)TTLprp,’PRP.DAT’
310 Foanart' ’,A,A)
OPEN(UNIT-12,NAME=FILDAT(1:12),TYPE=’OLD’,CARRIAGECONTROL=’LIST’)
READ(12,*)NTK,NTDC
READ(12,*)NALK,NALDC
READ(12,*)(TK(I),I = 1,NTK)
READ(12,*)(ALK(I),I = 1,NALK)
DO I - 1,NTK
READ(12,*)(KP(I,J),J = 1,NALK)
ENDDO
READ(12,*)(TDC(I),I = 1,NTDC)
READ(12,*)(ALDC(I),I = l,NALDC)
DO I - 1,NTDC
READ(12,*)(DCP(I,J),J = 1,NALDC)
ENDDO
CLOSE(UNIT=12)

 

c Read in kinetic data

if(model.ne.1)then
write(kindat,600)tt1kin,’kin.dat’

open(unit-12,name-kindat(1:12),type-’old’,carriagecontrols'list’)
READ (12, *)A1,A2,EA1
READ(12,*)EA2,M0,M1
READ(12,*)A3,EA3,DO
READ(12,‘)Dl,DZ,DENS
READ(12,*)HT
if(model.eq.3)READ(12,*)VA1,VA2,VEA1,VEA2
close(unit-12)

endif

C Read standard deviation for errors added to TC for PROPlD

WRITE(*,110)’Enter standard deviation (C) for TC errorsz’
READ (*, *) SIGMA

WRITE(*,110)’Enter seed for random numbers:’
READ(*,*)IAR

c Read in initial extent of cure values

IF(MODEL.NE.1)THEN
WRITE(*,110)’Enter initial extent of cure (alpha) values:’
110 FORMAT(' ’,5X,A)
WRITE(*,120)’Alpha (constant) = ’
120 FORMAT(’ ’,10X,A)
READ(*.*)ALPHA
DO I a l,M+1
AL(I) ' ALPHA
ENDDO
ENDIF

if(itmode)write(6,1)
1 format(’ ','PROGRAM IS RUNNINGI’)
ntime = int(ttime/dt)

pi - dacos(-l.0d0)
dz = L/m
mp1 = m+1

C Initialize temperature

198

DO I - 1,mp1
DO K a 1,2
t(I,k)=ti
enddo
enddo

tavg = ti
time=0.0DO
BCTQ(1,1) TQl(1)
BCTQ(1,2) T02(1)
BCTQ(2,1) = T01(1)
BCTQ(2,2) = T02(1)

ll

Ncount-O

NPR - PDT/DT
nprint - 0
IP(MODEL.eq.1)THEN

 

HEADTQal
else
headtq=2
endif
call output(nprint,headtq,t,tavg,time,a1,alavg,al_5,dal_5,val)
c
c Write temperatures to file for PROPlDMA.FOR
c

write(outfilZ,1000)ttlfil,’OUT.TEM'
1000 format(’ ’,a,a)
open(unit-13.name-outfi12(1:12),typea’new',carriagecontrol-’list’)
DO I - 1,5
ABC(I) - T((I-1)*ISTEP+1,2)
ENDDO
WRITE(13,1010)TIME,BCTQ(2,1),t(l,2),t(1,2).t(3,2),t(3,2),
1 t(5,2),t(5,2),t(7,2),t(7,2),t(9,2),t(9,2)
1010 FORMAT(1X,F8.2,1X,F10.2,12(1X,F7.2))

write(outfil3,1000)tt1fi1,’OUT.ALP’
open(unit=l4,name=outfil3(1:12),type=’new',carriagecontrol='list’)
DO I - 1,5
ABC(I) = AL((I-l)*ISTEP+1)
ENDDO
WRITE(14,1020)TIME,(ABC(I),I=1,5)
1020 FORMATllX,P8.2,5(1X,F7.4))
i=1
c finite difference solution
do 160 III=1,ntime
time - III*dt

c Find boundary conditions for each time step

CALL BCPIND(TIME,BCTQ)

 

c thomas algorithm
c find coefficients for thomas algorithm
call coeff(III,BCTQ,T,A,B,C,D,AL,ALAVG,AL_5,DAL_5)

cc(l)-c(1)/b(1)

199

dd(l)=d(1)/b(1)

do k=2,mpl
kk-k-l
cc(k)-c(k)/(b(k)-a(k)*cc(kk))
ddlk)-(d(k)-a(k)*dd(kk))/(b(k)-a(k)*cc(kk))

enddo

t(mp1,2)=dd(mp1)

tavg - t(mp1,2)

do k=2,mp1
kk-m-k+2
t(kk,2)=dd(kk)-cc(kk)*t(kk+1,2)
tavg - tavg+t(kk,2)

enddo

tavg = tavg/mp1

 

c find quality distribution and adjust time step

c find mass average quality

 

Ncount=Ncount+l
c printout

IF(NCOUNT.EQ.NPR)THEN
nprint - 1
call output(nprint,headtq,t,tavg,time,al,a1avg,al_5,dal_5,val)
DO I - 1,5

ABC(I) = T((I-l)*ISTEP+1,2)
ENDDO
tc(1)=t(1,2)
tc(2)=TC(1)
tc(3)=t(3,2)
tc(4)-TC(3)
tc(5)=t(5,2)
tc(6)=TC(5)
tc(7)=t(7,2)
tc(8)=TC(7)
tc(9)-t(9,2)
tc(10)=TC(9)

CALL RANDNU(tc,lO,iii,sigma,IAR)
WRITE(13,1010)TIME,BCTQ(2,1).(TC(I),I = 1,10)
DO I = 1,5

ABC(I) = AL((I-1)*ISTEP+1)
ENDDO
WRITE(14,1020)TIME,(ABC(I),I=1,5)
NCOUNT - 0

ENDIF

 

c initial t for next time step

do 100 i=1,mpl
100 t(i,1)=t(i,2)

c end of finite difference calculations

c *‘ki’i‘k‘kfi‘ki’fi*i*************iitttt*t‘kt*i’ii’it‘kt‘kii’itkititttt‘k

BCTQ(1,1) 8 BCTQ(2,1)
BCTQ(1,2) = BCTQ(2,2)

200

160 continue
NPRINT-Z
call output(nprint,headtq,t,tavg,time,a1,alavg,a1_5,da1_5,val)
close(13)
CLOSE(14)
return
end

C itiiiittitiIt‘ktitiittttititti'ttittittit!iiiitt*tit**t*iltitt**fit**iiit

subroutine coeff(III,BCTQ,T,A,B,C,D,AL,ALAVG,AL_S,DAL_5)
parameter(maxm=101,maxp=100,r=8.3140d0)

IMPLICIT DOUBLE PRECISION(A-H,P-Z)

double precision beta,nu,omega,gama,BCTQ(2,2),aar,ar(maxm),
Garl(maxm),area,avgl,avg2,da,db,dc,ddd(MAXM),ck(maxm),csd(maxm,2),
&a(maxm),b(maxm),c(maxm),d(maxm),t(maxm,2),AL(MAXM),
&DAL(MAXM),ALAVG,pi,dzz,HDEN,AL_S(MAXM),DAL_5(MAXM)

C Declare variables in common statements

C Common block /kin/

DOUBLE PRECISION A1,A2,EA1,EA2,M0,M1,DENS,HT,VA1,VA2,VEA1,
l VEA2,A3,EA3,DO,Dl,DZ

C Common block lprop/
INTEGER NTDC,NALDC,NTK,NALK
DOUBLE PRECISION dcp(10,10),tdc(10),aldc(10),kp(10,10),tk(10),
salk(10)

C Common block /bound/

INTEGER IBC1,IBC2,NBC1,NBC2
DOUBLE PRECISION TI,TQl(MAXP),TQZ(MAXP),TIME1(MAXP),TIME2(MAXP)

C Common block lgeom/

INTEGER ISHAPE,ISYM,M,MP1,ISTEP
DOUBLE PRECISION H,L,DZ

c Common block /tim/
DOUBLE PRECISION TTIME,DT,PDT
common/bound/ibcl,ibc2,nbcl,nbc2,ti,TQl,T02,time1,time2,
S/geom/ISHAPE,ISYM,M,MP1,ISTEP,H,L,DZ,
a/prop/dcp,tdc,aldc,kp,tk,alk,ntdc,naldc,ntk,nalk,
&/KIN/A1,A2,EA1,EA2,M0,M1,DENS,HT,VA1,VA2,VEA1,VEA2,A3,EA3,DO,D1,DZ
s/MOD/MODEL,
t/TIM/TTIME,DT,PDT
pi = dacos(-1.0d0)

c Weighting functions for Crank-Nicolson finite difference method:

c weighting coefficients for d2T/d22:
c for time t:

 

201
beta=0.50d0
c for time t+1:

nu-O.50d0

c weighting coefficients for dT/dt:

c for time t:
omega--1.0d0

c for time t+1:
gama=1.0d0
dzz=1.0d0/dz

if(Ishape.eq.2)then
aar=2.0d0*pi*h
else
if(Ishape.eq.3)then
aar=4.0d0*pi
endif
endif

c Call Subroutine HEAT to determine heat of reaction

IF(MODEL.NE.1)CALL HEAT(T,AL,DAL,DT,MP1,ALAVG,AL_5,DAL_5)

do 10 i=1,mp1
c slab

if(Ishape.eq.1)then
ar(i)-h
ar1(i)=h

else

c cylinder
if(Ishape.eq.2)then
ar(i)=aar*(i-l)*dz

ar1(i)=ar(i)+aar'dz/2.0d0
else

c sphere
ar(i)=aar*((i-l)*dz)**2.0d0
arl(i)=aar*((i-l)*dz+dz/2.0d0)**2.0d0
endif
endif

C Find heat generation term from dalpha/dt:

HDEN - HT*0ENS*DZ
10 continue

CALL PFIND(T,AL,M,CK,CSD,DT,DZ)

C iii**************************ttfl'tit**********************t

c lst boundary point

AVGl 8 (AR(1)+AR1(1))*0.50d0
a(1)=0.0d0
IP(IBC1.EQ.1)THEN

202

Bil) 8 1.000

C(l) = 0.000

0(1) - BCTQ(2,1)
ENDIF

IF(IBC1.EQ.2)THEN
c(l)-nu*dzz'CK(l)*arl(l)
dc=-beta*dzz*CK(l)*ar1(1)

b(1)--gama*CSD(1,1)*avgl-c(1)
db=omega*CSD(1,l)*avg1-dc
000(l)=(-beta'BCTQ(1,1)-nu*BCTQ(2,l)-HOEN*DAL(1)*0.50d0)*ar(1)
d(1)=db*t(l,1)+dc't(2,1)+ddd(l)

ENDIF

C tiiitititi’t*iii’i’iiitt*ititttttitittiiitiiiititt*******t***

C

c interior points

20

do 20 i-2,m

AVGl - (AR(I)+AR1(I))*0.50d0
AVGZ - (AR(I)+AR1(I-1))*0.50d0
alI) - nu*dzz*CK(I-1)*arl(i-l)
da - -beta*dzz*CK(I-l)*arl(i-1)

c(I) = nu'dzz*CK(I)*ar1(i)
dc a -beta'dzz*CK(I)*arl(i)

b(I) = -gama*(CSD(I,1)*avg2+CSD(I,2)*avg1)-a(i)-c(i)
db = omega*(CSD(I,1)*avg2+CSD(I,2)*avg1)-da-dc
DDD(I) 3 -HDEN*DAL(I)*ar(I)

d(I) - da*t(i-1,1)+db*t(i,l)+dc*t(i+l,l)+000(I)
continue

C t!*I‘I‘kitttfi'*t‘ktiti’ititiifi‘kii‘ktt‘ltt‘ki’*ttittittttttttiiitittt

c 2nd boundary point

AVGZ = (AR(mp1)+AR1(M))*0.50d0
c(mpl)=0.0d0

IF(IBC2.EQ.1)THEN
A(MP1) = 0.000
B(MP1) - 1.000
0(MP1) - BCTQ(2,2)

ENDIP

IF(IBC2.EQ.2)THEN
a(mp1)=nu*dzz*CK(M)*arl(m)
da--beta*dzz*CK(M)*ar1(m)

b(mp1)=-gama*CSD(mp1,2)*avg2-a(mpl)
db=omega*CSD(mpl,2)*ang-da
ddd(MPl)=(-beta*BCTQ(1,2)-nu*BCTQ(2,2)-HDEN*DAL(MP1)*0.500)

1 *ar(mp1)
d(mp1)=da*t(m,1)+db*t(mpl,1)+ddd(MPl)

ENDIF

return

end

C *****‘kifl’i‘k'k‘k‘ki’ttt‘kfittti‘k*ti‘k'k'kt'l’ti**************t***************‘k‘kti‘tt

SUBROUTINE HEAT(T,AL,DAL,DT,MP1,ALAVG,AL_S,DAL_5)

PARAMETER(MAXM=101,MAXP=100,R=8.31400)

0000000000

00

0

100

2CX3

IMPLICIT DOUBLE PRECISION(A-H,P-Z)

DOUBLE PRECISION AL(MAXM),AL_5(MAXM),ALAVG,C1,C2,C3,D3,DAL(MAXM).
1 DAL_5(MAXM),DEXP,DT,MEXP,NEXP,SUM,T(MAXM,2),TT

Declare variables in common block.

DOUBLE PRECISION A1,A2,EA1,EA2,M0,M1,DENS,HT,VA1,VA2,VEA1,
1 VEA2,A3,EA3,D0,D1,D2

COMMON /KIN/A1,A2,EA1,EA2,M0,M1,DENS,HT,VA1,VA2,VEA1,VEA2,A3,EA3,
1 D0,D1,D2

Extent of cure is found assuming an autocatalyzed second order reaction

Use Ryan and Dutta method for a < 0.5:
da/dt = (cl+c2*a**m)(1-a)**(2-m);
where, cl - Alexp(-El/T*R), c2 = A2exp(-E2/T*R), m = m0+m1*(T-273.15)

Use E. Scott’s method for a > 0.5:
da/dt - (da/dt)(a-.5) + c3*(0.5-a)exp(-D*(0.5-a))
where, c3 - A3exp(-E3/T*R), D = d0+dl*(T-273.15)

SUM B 0.0D0
DO 100 J = 1,MP1

For extent of cure < 0.5:

IF(AL(J).LE.0.50dO)THEN
MEXP = M0+M1*(T(J,1)-273.1500)
NEXP = 2.000-MEXP
c1 - Al‘EXP(-EA1/(T(J,l)*R))
c2 - A2*EXP(-EA2/(T(J,l)*R))
DAL(J) = (C1+C2*(AL(J)**MEXP))*(l.000-AL(J))**NEXP
AL_S(J) = AL(J)
0AL_S(J) = DAL(J)
ELSE

For extent of cure > 0.5:

AL_5(J) = 0.5000
MEXP = M0+Ml*(T(J,l)-273.1500)
NEXP = 2.000-MEXP
C1 = Al*EXP(-EAl/(T(J,l)*R))
C2 = A2*EXP(-EA2/(T(J,l)*R))
DAL_5(J) = (C1+C2*(AL_5(J)**MEXP))*(1.000—AL_5(J))**NEXP
'TT - 150.000-(T(J,l)-273.1500)
DEXP - 00+01*TT+02*TT*TT
IP(0EXP.LT.0.5000)DEXP = 0.500
DIFAL a AL_5(J)-AL(J)
C3 = A3*EXP(-EA3/(T(J,l)*R))
DAL(J) = DAL_5(J)+DIFAL*C3*EXP(-DEXP*DIFAL)/(HT*lD-3)
IF(DAL(J).LT.0.000)DAL(J)=0.0DO
ENDIF
AL(J) = AL(J)+DT*DAL(J)
IF(AL(J).GT.1.0DO)THEN
AL(J) 1.000
DALlJ) = 0.000
ENDIF
SUM = SUM+AL(J)
continue
ALAVG = SUM/MP1

 

204

RETURN
END

C itiiiiti*****iifltitttitttttittiittttti*til’ttttt‘ltit"ittfiitttittttititi

SUBROUTINE BCPIND(TIME,BCTQ)
PARAMETER(MAXP=100)

IMPLICIT DOUBLE PRECISION(A-H,P-Z)
DOUBLE PRECISION TIME,BCTQ(2,2)

C Common block /bound/
INTEGER IBC1,IBC2,NBC1,NBC2
DOUBLE PRECISION TI,TQl(MAXP),TQZ(MAXP),TIME1(MAXP),TIME2(MAXP)

common/bound/ibcl,ibc2,nbcl,nbc2,ti,TQl,TQZ,time1,time2
C First boundary condition

00 J - 2,NBC1
JMl - J-l
IF(TIME.LE.TIME1(J))THEN
BCTQ(2,1) - TQl(JM1)+(TIME-TIME1(JMl))*(TQl(J)-TQI(JM1))/
5 (TIME1(J)-TIMEI(JM1))
GOTO 10
ENDIF
BCTQ(2,1) = TQl(NBC1-l)+(TIME-TIME1(NBC1-l))*(TQl(NBC1)-
& TQl(NBC1-l))/(TIMEl(NBCl)-TIME1(NBC1-l))
ENDDO
10 CONTINUE

C Second boundary condition

00 J - 2,NBC2
JM1 = J-l
IF(TIME.LE.TIME2(J))THEN
BCTQ(2,2) = TQZ(JM1)+(TIME-TIME2(JMl))‘(T02(J)-T02(JM1))/
& (TIME2(J)-TIME2(JM1))
GOTO 20
ENDIP
BCTQ(2,2) = T02(NBC2-1)+(TIME-TIME2(NBC2-l))*(T02(NBC2)-
& TQZ(NBC2-1))/(TIME2(NBC2)-TIME2(NBCZ-l))
ENDDO
20 CONTINUE

RETURN
END

C *iitt‘kt‘ki’ii’iiitittti‘tt*t'kii’ii*‘li’i’l‘ktitittit********t***t*ittittt******

SUBROUTINE PFIND(T,AL,M,CK,CSPD,DT,DZ)
PARAMETER (MAXm=101)
IMPLICIT DOUBLE PRECISION(A-H,P-Z)

double precision TAVGK,TAVGDC(2),ALAVGK,ALAVGDC(2),CK(MAXM),
&CSPO(MAXM,2),t(maxm,2),AL(maxm),dt,dz

205

C Declare variables in common statements
C Common block /prop/

INTEGER NTDC,NALDC,NTK,NALK
DOUBLE PRECISION dcp(10,10),tdc(10),a1dc(10),kp(10,10),tk(lO),
&a1k(10),cc

COMMON /prop/dcp,tdc,aldc,kp,tk,alk,ntdc,naldc,ntk,nalk
C Common block /MOD/

INTEGER MODEL

COMMON /MOD/MODEL

MP1 = M+l
cc = 1.0d0*dz/2.0d0

C Look up table for thermal conductivity

DO 100 I I 1,MP1
IF(I.LE.M)THEN
TAVGK - (T(I,1)+T(I+1,l))*0.5000
TAVGDC(1) - 0.7500*T(I,1)+0.2500*T(I+1,1)
IF(MODEL.NE.1)THEN
ALAVGK - (AL(I)+AL(I+1))*0.5000
ALAVGDC(1) = 0.7500*AL(I)+0.2500*AL(I+1)
ENDIF
ENDIF
IP(I.GT.1)THEN
TAVGDC(2) = 0.7SDO*T(I,1)+0.2500*T(I-1,1)
IF(MODEL.NE.1)THEN
ALAVGDC(2) = 0.7500*AL(I)+0.2500'AL(I-l)
ENDIF
ENDIP

C First find thermal conductivity
C For MODEL 1, thermal conductivity is a function of temperature only

IP(MODEL.EQ.1)THEN

IP(TAVGK.LE.TK(1))THEN
CK(I)=KP(1,1)+(TAVGK-TK(1))‘(KP(2,1)-KP(1,1))/(TK(2)-TK(1))

ENDIP

IF(TAVGK.GT.TK(1).AND.TAVGK.LT.TK(NTK))THEN
00 J - 1,NTK
IF(J.NE.NTK)THEN
IF(TAVGK.LE.TK(J+1))THEN
CK(I)=KP(J,1)+(TAVGK-TK(J))*(KP(J+1,l)-KP(J,1))/
& (TK(J+1)—TK(J))
ENDIF
ENDIF
ENDDO
ENDIF

IF(TAVGK.GE.TK(NTK))THEN
CK(I)=KP(NTK,l)+(TAVGK-TK(NTK))*(KP(NTK,l)-KP(NTK-l,l))/
& (TK(NTK)-TK(NTK-1))
ENDIF

ELSE

206

C For MODEL’s 2 and 3, find thermal conductivity as a function of
C temperature and extent of cure

IF(NTK.EQ.1.AND.NALK.EQ.1)THEN
CK(I)-KP(1,1)
GO TO 105
ENDIP
IF(TAVGK.LE.TK(1))THEN
IP(ALAVGK.LE.ALK(1))THEN
CK(I)=KP(1,1)+(TAVGK-TK(1))*(KP(2,l)-KP(1,1))/

8 (TK(2)-TK(1))+(ALAVGK-ALK(1))*(KP(1,2)-KP(1,1))/
& (ALK(2)*ALK(1))
ENDIF

IF(ALAVGK.GT.ALK(1).AND.ALAVGK.LT.ALK(NALK))THEN
DO N = 1,NALK
IF(N.NE.NALK)THEN
IF(ALAVGK.LE.ALK(N+1))THEN
CK(I)=KP(1,N)+(TAVGK-TK(1))‘(KP(2,N)-KP(1,N))/

s (TK(2)-TK(1))+(ALAVGK-ALK(N))‘(KP(1,N+1)-KP(1,N))/
& (ALK(N+l)-ALK(N))
ENDIF
ENDIF
ENDDO
ENDIP

IF(ALAVGK.GE.ALK(NALK))THEN
CK(I)-KP(l,NALK)+(TAVGK-TK(1))*(KP(2,NALK)-KP(1,NALK))/

8 (TK(2)-TK(1))+(ALAVGK-ALK(NALK))*(KP(l,NALK)-KP(1,NALK-l))/
& (ALK(NALK)-ALK(NALK-l))
ENDIF
ENDIP

IP(TAVGK.GT.TK(1).AND.TAVGK.LT.TK(NTK))THEN
DO J a 1,NTK
IP(J.NE.NTK)THEN
IF(TAVGK.LE.TK(J+1))THEN
IF(ALAVGK.LE.ALK(1))THEN
CK(I)-KP(J,1)+(TAVGK-TK(J))*(KP(J+1,1)-KP(J,1))/

8 (TK(J+1)-TK(J))+(ALAVGK-ALK(1))*(KP(J,2)-KP(J,1))/
& (ALK(2)-ALK(1))
ENDIF

IF(ALAVGK.GT.ALK(1).AND.ALAVGK.LT.ALK(NALK))THEN
00 N - 1,NALK
IF(N.NE.NALK)THEN
IF(ALAVGK.LE.ALK(N+1))THEN
CK(I)=KP(J,N)+(TAVGK-TK(J))*(KP(J+1,N)-KP(J,N))/

& (TK(J+1)-TK(J))+(ALAVGK-ALK(N))*(KP(J,N+1)
5 -KP(J,N))/(ALK(N+1)-ALK(N))
ENDIP
ENDIP
ENDDO
ENDIF

IF(ALAVGK.GE.ALK(NALK))THEN
CK(I)=KP(J,NALK)+(TAVGK-TK(J))*(KP(J+1,NALK)

& -KP(J,NALK))/(TK(J+1)-TK(J))+(ALAVGK-ALK(NALK))
& *(KP(J,NALK)'KP(J,NALK-1))/(ALK(NALK)-ALK(NALK-l))
ENDIF
ENDIP
ENDIP
ENDDO
ENDIF

IF(TAVGK.GE.TK(NTK))THEN
IF(ALAVGK.LE.ALK(1))THEN

207

CK(I)-KP(NTK,1)+(TAVGK-TK(NTK))*(KP(NTK,1)-KP(NTK—1,l))/

G (TK(NTK)-TK(NTK-1))+(ALAVGK-ALK(1))*(KP(NTK,2)-KP(NTK,1))/
E (ALK(2)-ALK(1))
ENDIF

IF(ALAVGK.GT.ALK(1).AND.ALAVGK.LT.ALK(NALK))THEN
DO N I 1,NALK
IF(N.NE.NALK)THEN
IF(ALAVGK.LE.ALK(N+1))THEN
CK(I)'KP(NTK,N)+(TAVGK-TK(NTK))*(KP(NTK,N)

& -KP(NTK-1,N))/(TK(NTK)-TK(NTK-l))+(ALAVGK-ALK(N))
s *(KP(NTK,N+1)-KP(NTK,N))/(ALK(N+1)-ALK(N))
ENDIF
ENDIF
ENDDO
ENDIF

IP(ALAVGK.GE.ALK(NALK))THEN
CK(I)-KP(NTK,NALK)+(TAVGK-TK(NTK))*(KP(NTK,NALK)

8 -KP(NTK-1,NALK))/(TK(NTK)-TK(NTK-l))+(ALAVGK-ALK(NALK))
*(KP(NTK,NALK)-KP(NTK,NALK-1))/(ALK(NALK)-ALK(NALK-l))
ENDIP
ENDIP
ENDIF

C Next find density—specific heat, CSPD(l) and CSPD(2):
105 00 50 NC - 1,2
C For MODEL 1, density-specific heat is a function of temperature only

IP(MODEL.EQ.1)THEN
IF(TAVGDC(NC).LE.TDC(1))THEN
CSPD(I,NC)=DCP(1,l)+(TAVGDC(NC)-TDC(1))*(DCP(2,l)-DCP(1,1))/
& (TOC(2)-TOC(l))
ENDIP

IF(TAVGDC(NC).GT.TOC(1).ANO.TAVGDC(NC).LT.TOC(NTDC))THEN
00 J - 1,NTDC
if(j.ne.ntdc)then
IF(TAVGDC(NC).LE.TDC(J+1))THEN
CSPD(I,NC)-0CP(J,1)+(TAVGDC(NC)-TDC(J))*(DCP(J+1,1)
& -0CP(J,1))/(TOC(J+l)-TDC(J))
ENDIF
endif
ENDDO
ENDIF

IF(TAVGDC(NC).GE.TDC(NTDC))THEN
CSPD(I,NC)-DCP(NTDC,l)+(TAVGDC(NC)-TDC(NTDC))‘(DCP(NTDC,1)
& -DCP(NTDC-1,l))/(TDC(NTDC)-TDC(NTDC-1))
ENDIF

ELSE

C For MODEL’s 2 and 3, find specific heat as a function of temperature
C and extent of cure

IF(NTDC.EQ.1.AND.NALDC.EQ.l)THEN
CSPD(I,NC)=DCP(1,1)
GO TO 50
ENDIF
IP(TAVGDC(NC).LE.TDC(1))THEN

 

208

IP(ALAVGDC(NC).LE.ALDC(1))THEN
CSPD(I,NC)-DCP(1,l)+(TAVGDC(NC)-TDC(1))‘(DCP(2,1)-DCP(1,1))/

& (TDC(2)-TDC(1))+(ALAVGDC(NC)-ALDC(1))*(DCP(1,2)-DCP(1,1))/
(ALDC(2)-ALDC(1))
ENDIP

IF(ALAVGDC(NC).GT.ALDC(1).AND.ALAVGDC(NC).LT.ALDC(NALDC))THEN
00 N - 1,NALDC
IF(N.NE.NALDC)THEN
IF(ALAVGDC(NC).LE.ALDC(N+1))THEN
CSPD(I,NC)=DCP(1,N)+(TAVGDC(NC)-TDC(1))*(DCP(2,N)~
DCP(1,N))/(TDC(2)-TDC(1))+(ALAVGDC(NC)*ALDC(N))*
(DCP(1,N+1)-DCP(1,N))/(ALDC(N+1)-ALDC(N))
ENDIP
ENDIP
ENDDO
ENDIP
IF(ALAVGDC(NC).GE.ALDC(NALDC))THEN
CSPD(I,NC)=DCP(1,NALDC)+(TAVGDC(NC)-TDC(1))‘(DCP(2,NALDC)-
DCP(1,NALDC))/(TDC(2)-TDC(1))+(ALAVGDC(NC)-ALDC(NALDC))*
(DCP(1,NALDC)-DCP(l,NALDC-1))/(ALDC(NALDC)-ALDC(NALDC-l))
ENDIP
ENDIP

IF(TAVGDC(NC).GT.TDC(1).AND.TAVGDC(NC).LT.TDC(NTDC))THEN
00 J = 1,NTDC
IF(J.NE.NTDC)THEN
IP(TAVGDC(NC).LE.TDC(J+1))THEN
IP(ALAVGDC(NC).LE.ALDC(1))THEN
CSPD(I,NC)-DCP(J,l)+(TAVGDC(NC)-TDC(J))*(DCP(J+1,1)-
DCP(J,1))/(TDC(J+l)-TDC(J))+(ALAVGDC(NC)-ALDC(1))*
(DCP(J,2)-DCP(J,1))/(ALDC(2)-ALDC(1))
ENDIP
IP(ALAVGDC(NC).GT.ALDC(1).AND.ALAVGDC(NC).LT.ALDC(NALDC))THEN
DO N = 1,NALDC
IF(N.NE.NALDC)THEN
IP(ALAVGDC(NC).LE.ALDC(N+1))THEN
CSPD(I,NC)'DCP(J,N)+(TAVGDC(NC)-TDC(J))*(DCP(J+1,N)-
DCP(J,N))/(TDC(J+l)-TDC(J))+(ALAVGDC(NC)-ALDC(N))*
(DCP(J,N+l)-DCP(J,N))/(ALDC(N+1)-ALDC(N))
ENDIF
ENDIP
ENDDO
ENDIF
IF(ALAVGDC(NC).GE.ALDC(NALDC))THEN
CSPD(I,NC)=DCP(J,NALDC)+(TAVGDC(NC)-TDC(J))*(DCP(J+1,

& NALDC)-DCP(J,NALDC))/(TDC(J+l)-TDC(J))+(ALAVGDC(NC)-
& . ALDC(NALDC))*(DCP(J,NALDC)-DCP(J,NALDC-1))/(ALDC(NALDC)-
8 ALDC(NALDC-1))
ENDIF
ENDIF
ENDIF
ENDDO
ENDIF

IF(TAVGDC(NC).GE.TDC(NTDC))THEN
IP(ALAVGDC(NC).LE.ALDC(1))THEN
CSPD(I,NC)=DCP(NTDC,l)+(TAVGDC(NC)-TDC(NTDC))*(DCP(NTDC,1)-

& DCP(NTDC-1,1))/(TDC(NTDC)-TDC(NTDC-l))+(ALAVGDC(NC)-ALDC(1))
& *(DCP(NTDC,2)-DCP(NTDC,1))/(ALDC(2)-ALDC(1))
ENDIF

IF(ALAVGDC(NC).GT.ALDC(1).AND.ALAVGDC(NC).LT.ALDC(NALDC))THEN
00 N = 1,NALDC
IF(N.NE.NALDC)THEN

209

IF(ALAVGDC(NC).LE.ALDC(N+1))THEN
CSPD(I,NC)=DCP(NTDC,N)+(TAVGDC(NC)-TDC(NTDC))*

8 (DCP(NTDC,N)-DCP(NTDC-1,N))/(TDC(NTDC)-TDC(NTDC-l))+
& (ALAVGDC(NC)-ALDC(N))*(DCP(NTDC,N+l)-DCP(NTDC,N))/
(ALDC(N+l)-ALDC(N))
ENDIF
ENDIP
ENDDO
ENDIF

IF(ALAVGDC(NC).GE.ALDC(NALDC))THEN
CSPD(I,NC)-DCP(NTDC,NALDC)+(TAVGDC(NC)-TDC(NTDC))*
8 (DCP(NTDC,NALDC)-DCP(NTDC-1,NALDC))/(TDC(NTDC)-TDC(NTDC-1))+
E (ALAVGDClNC)-ALDC(NALDC))*(DCP(NTDC,NALDC)-DCP(NTDC,NALDC-l)
& )/(ALDC(NALDC)-ALDC(NALDC-l))
ENDIF
ENDIP
ENDIF
50 CONTINUE
100 CONTINUE

do i - 1,mp1
DO KK - 1,2
CSPD(I,KK) = CSPD(I,KK)*cc/dt
ENDDO
enddo
RETURN
END

C tiiititittiitiiititittiiiiiiiittt‘ktfittiitiit*‘ktt‘kiiititiiiiittiii*iiii

subroutine output(nprint,headtq,t,tavg,time,al,alavg,al_5,dal_5,
l val)

parameter(maxp=100,maxm=101)
integer model

double precision a1(maxM),abcd,alavg,val,tavg,tavg1,abc(5),time,
1 T(MAXM,2),AL_S(MAXM),DAL_5(MAXM)

character outfil*12,hh11*29,hh22*21,prunit*6
C Declare variables in common statements

C Common block /tt1/
character title*20,ttlfi1*4,ttlprp*4,ttlkin*4

C Common block /kin/

DOUBLE PRECISION A1,A2,EA1,EA2,MO,M1,DENS,HT,VA1,VA2,VEA1,
1 VEA2,A3,EA3,DO,D1,D2

C Common block /prop/
INTEGER NTDC,NALDC,NTK,NALK
DOUBLE PRECISION dcp(10,10),tdc(10),aldc(10),kp(10,10),tk(10),
&a1k(10)

C Common block /bound/
INTEGER IBC1,IBC2,NBC1,NBC2
DOUBLE PRECISION TI,TQl(MAXP),TQZ(MAXP),TIMEI(MAXP),TIME2(MAXP)

C Common block /geom/
INTEGER ISHAPE,ISYM,M,MP1,ISTEP

 

C

C
C
C

7

5

210
DOUBLE PRECISION H,L,DZ

Common block /tim/
DOUBLE PRECISION DT,TTIME,PDT

common/ttl/title,ttlfil,tt1prp,ttlkin,
G/mod/model,
&/KIN/Al,A2,EA1,EA2,MO,M1,DENS,HT,VA1,VA2,VEA1,VEA2,A3,EA3,00,01,02
s/prop/dcp,tdc,a1dc,kp,tk,a1k,ntdc,naldc,ntk,nalk,
s/bound/ibcl,ibc2.nbcl,nbc2,ti,TQl,T02.timel,timeZ,
S/geom/ISHAPE,ISYM,M,MP1,ISTEP,H,L,DZ,
&/TIM/TTIME,DT,PDT

NPRINT = 0 if printing input parameters and headings
NPRINT = 1 if printing temperature and/or extent of cure
NPRINT = 2 if printing end line

IF(NPRINT.EQ.0)THEN
GO TO 1100
ELSE
if(nprint.eq.1)then
go to 1200
else
go to 1300
endif
endif

1100 write(outfil,1000)ttlfi1,’out.dat’

1000 format(’ ',a,a)
open(unit-12,name-outfil(1:12),types’new’,carriagecontrola'list’)
write(12,1)title
format(’ ’,3x,’Title: ’,a20,/3x,’ ----- ’,//,l4x,'Input Para’,

+’meters’,/,14x,16(’—’)/)
if(model.ne.1)then
write(12,3)’Kinetic Parameters; Extent of Cure < 0.5:’

format(’ ’/,’ ’,A,/)

write(12,5)’Pre-exponential factor, Ln(Al),(Ln(1/sec)).’,LOG(A1)
write(12,5)’ Ln(A2),(Ln(1/sec)).’,LOG(A2)
abcd-ea1/1000.0d0

write(12,6)’Activation energy const., Eal, (kJ/mole)...’,abcd

format(3x,A,f8.2)

abcd=ea2/1000.0d0

write(12,6)’ Ea2, (kJ/mole)...’,abcd
write(12,4)’Exponent, M0, (dimensionless) .............. ’,MO
write(12,4)’ M1, (dimensionless) .............. ’,Ml

format(3x,A,e10.4)

write(12,3)’Extent of Cure > 0.5:’
write(12,5)’Pre-exponential factor, Ln(A3),(Ln(1/sec)).’,LOG(A3)
abcdsea3/1000.0d0

write(12,6)’Activation energy const., Ea3, (kJ/mole)...’,abcd
write(12,4)’Diffusion constant, 00, (dimensionless)....’,00
write(12,4)' 01, (dimensionless)....’,01
write(12,4)’ 02, (dimensionless)....’,02
WRITE(12,7)'0ensity (kg/m3)............................',DENS
FORMAT(/,3X,A,F10.l)

WRITE(12,7)’Tota1 heat of reaction (J/kg) .............. ’,HT
if(model.eq.3)then

abcd=vAl**0.50d0

write(12,5)'$t. dev. of rate constant, Al, (l/sec) ..... ’,abcd
FORMAT(3X,A,F8.4)

abcd=vA2**0.50d0

write(12,5)’St. dev. of rate constant, A2, (1/sec) ..... ’,abcd
abcd=vea1**0.50d0/1000.0dO

write(12,5)’$t. dev. of Eal (kJ/mole) .................. ’,abcd

211

abcd-vea1*'0.50d0/1000.0d0

write(12,5)’St. dev. of Ea2 (kJ/mole) .................. ’,abcd
endif
endif
write(12,10)’Thermal Properties’
10 format(' ',/,’ ’,A,/)
write(12,11)’Therma1 conductivity table:’
11 format(3x,A)

if(model.eq.1)then
do i - 1,ntk
write(12,12)tk(i),’C’,kp(i,1),’W/mC’
12 format(6x,f6.2,lx,a,2x,f6.2,1x,a)
enddo
else
do i - 1,ntk
write(12,13)tk(i),’C’,(alk(n),kp(i,n),n = 1,nalk)
l3 format(5x,f6.2,a,2(3x,F6.3,3X,f6.2,’W/mC’),4(/,12x,
& 2(3x,F6.3,3X,f6.2,'W/mC’)))
enddo
endif
write(12,14)
14 format(’ './)
write(12,ll)’Density-Specific Heat table:'
if(model.eq.l)then
do i - 1,ntdc
write(12,15)tdc(i),’C’,dcp(i,1),’m/52’
15 format(6x,f6.2,1x,a,2x,ElO.4,1x,a)
enddo
else
do i - 1,ntdc
write(12,16)tdc(i),'C’,(ALDC(n),DCP(I,n),n = 1,naldc)
l6 format(5x,f6.2,a,2(3x,f6.3,3X,E10.4,’m/sZ’),4(/,12x,
& 2(3x,f6.3,3X,ElO.4,’m/sZ’)))
enddo
endif
write(12,l4)
abcd=ti~273.150d0
write(12,17)abcd
17 format(' ’,/,’ ’,’Initial Condition:’,/,’ ’,2x,’Product temp.’
+,’ (C) at time=0 ........... ’,f6.2)

c product geometry

if(Ishape.eq.1)then
if(Isym.eq.1)l = l*2.0d0
write(12,18)1
18 format( ’ ',/,’ ’,’Slab Geometry:’,/,3x,’thickness (m)’,
+26('.’),f10.6)
else
if(Ishape.eq.2)then
write(12,20)l
20 format(’ ’,/,’ ',’Cylindrica1 Geometry:’,/,3x,'radius (m)',
+29(’.’),f10.6)
else
write(12,22)1
22 format(’ ',/,’ ’,’Spherica1 Ceometry:’,/,3x,’radius (m)’,
+29(’.’),f10.6)
endif
endif

c boundary conditions

212

if(Ibc1.eq.1)then
write(12,23)’Temperature Boundary Condition (IBC1)’
format(’ ',/,3x,a,/)
else
write(12,23)’Heat Flux Boundary Condition (IBC1)’
endif
do i = 1,nbc1
if(Ibc1.eq.1)then

write(12,24)’Temperature (C) at ’,time1(i),’ sec ..... ',th(i)
format(6x,a,f8.l,a,f9.2)
else
write(12,24)’Heat Flux (W/m2) at ’,timel(i),’ sec ..... ’,TQl(i)
endif

enddo

if(Ibc2.eq.1)then
write(12,23)’Temperature Boundary Condition (IBC2)’

else
write(12,23)’Heat Flux Boundary Condition (IBC2)’

endif

do i - 1,nbc2
if(Ibc2.eq.1)then
write(12,24)’Temperature (C) at ’,time2(i),’ sec ..... ’,

GTQZ(i)-273.1500
else
write(12,24l’Heat Flux (W/mZ) at ’,time2(i),’ sec ..... ’,T02(i)
endif

enddo

 

Temperature history

write(12,100)title
100 format(’ ’,//,' ’,’Title= ’,a20,/)
if(Isym.eq.1)then
write(12,110)
110 format(’ ’,’Note: Distribution is symmetrical;’/,6x,’results’,
+’ are shown for half-thickness only.’/)
endif

hh22=’DISTRIBUTION HISTORY’
if(headtq.eq.1)then

hh11=’ TEMPERATURE (C) ’
else

hhll-‘TEMPERATURE (C) 8 EXTENT OF CURE'
endif

write(12,120)hhll,hh22
120 format(’ ’,/,’ ',17x,a,/,23x,a,/,19x,27(’-’),/)
if(model.eq.1)then
write(12,130)
130 format(’ ’,32x,’position (m)’,/' ',7x,'time’,5x,':’,40x,
S’lAvg Temp’)
else
write(12,135)
135 format(’ ’,32x,’position (m)’,/’ ',7x,’time’,5x,’:',40x,
s’lAvg Temp ’)
endif
do i - 1,5
abc(i)=(i-1)*ISTEP*dz
enddo
if(model.eq.1)then
write(12,137)abc(1),abc(2),abc(3),abc(4),abc(5)
137 format(’ ’,16x,’:’,5(f8.4))
else

213

if(model.eq.2)then
write(12,140)abc(l),abc(2),abc(3),abc(4),abc(5)
140 format(’ ’,16x,':’,5(f8.4),'|Ext Cure’)
else
write(12,145)abc(1),abc(2),abc(3),abc(4),abc(5)
145 format(’ ',16x,’:',5(f8.4),'|Ext Curel St0(%)’)
endif
endif
if(model.ne.3)then
write(12,150)
150 format(' ’,67(’='))
else
write(12,155)
155 format(' ',72(’='))
endif

c Printout time heading

1200 tavgl - tavg-273.150d0
do i - 1,5
abc(i)-t((i-l)*ISTEP+1,2)-273.150d0
enddo
prunit - ' sec:’
write(12,l90)time,prunit,abc(1),abc(2),abc(3),abc(4),abc(5),
stavgl
190 format(’ ’,f10.2,1x,a,5(f7.2,1x),'l’,f7.2,’C’)

 

if(headtq.eq.2)then

C Printout extent of cure values

do i a 1,5
abc(i)=al((i-l)*ISTEP+1)
enddo
if(model.eq.2)then
write(12,210)abc(1),abc(2),abc(3),abc(4),abc(5),alavg
210 format(17x,’:',5(f7.3,1x),'|’,lx,f7.3)
else
write(12,215)abc(l),abc(2),abc(3),abc(4),abc(5),alavg,
& (va1)**0.50d0
215 format(17x,’:’,5(f7.3,1x),’l’,f7.3,lx,’|’,f6.3)
endif
endif
return

c Printout end line

1300 if(model.ne.3)then
write(12,300)
300 format(’ ’,67(’-'))
else
write(12,305)
305 format(’ ',72(’-’))
endif
close(unit=12)
return
end

c program RANDNU

214

SUBROUTINE RANDNU(TCC,ntc,iii,sigma,IAR)

C
C ...... SUBROUTINE FOR GENERATING PSEUDO-RANDOM NUMBERS FROM NORMAL
C ...... DISTRIBUTION WITH ZERO MEAN AND UNITY VARIANCE.
C
DOUBLE PRECISION U(SOO),XP(500),TCC(12),SIGMA
DATA A0,Al,Bl,BZ /2.30753,0.2706l,0.99229,0.04481/
C
IAR=IAR+iii
C
C ...... GENERATION OF A SEQUENCE OF PSEUDO-RANDOM NUMBERS OF UNIFORM
C ...... DISTRIBUTION ON THE INTERVAL (0,1)).
C
00 10 I=1,Ntc
10 U(I)=RAN(IAR)
C
C .............................................
C
C ...... GENERATION OF PSEUDO-RANDOM NUMBERS FROM NORMAL DISTRIBUTION,
C ...... WITH ZERO MEAN AND UNITY VARIANCE,USING INVERSE METHOD(EQ.26.2.23;
C ...... ABRAMOWITZ AND STEGUN).
C
00 20 J=1,Ntc
C
IP ( U(J) .GT. 0.5 ) THEN
TEMPU=1.0 - U(J)
SIGN=-l.
ELSE
TEMPU= U(J)
SIGN=1.
ENDIF
C
T=SORT(ALOG(1.0/TEMPU**2))
XP(J)=T-(A0+A1*T)/(1.+Bl*T+BZ*T**2)
XP(J)=SIGN*XP(J)*sigma
TCC(j) - TCC(j)+xp(j)
20 continue
C
RETURN
END
c \ ___
c@@@@@@@@@@@@@@@@@@@@@@@@@@@ ____\ /___\ @@@@@@@@@@@@@@@@@@@@@@@@@@@@@
c .......................... / \|\___/ .............................
c!!!!!!!!!!!!!!!!!!!!!!!!! / / \ !!!!!!!!!!!!!!!!!!!!!!!!!!!!
c//////////////////////// I l @ -< I \\\\\\\\\\\\\\\\\\\\\\\\\\\
c\\\\\\\\\\\\\\\\\\\\\\\\ I l \_/ l ///////////////////////////
cl!lllllllllllllllllllllll \ \ / ll!!!llllllllllllllllillllll
c .......................... \ \ / .......................... EPS

 

c@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@

 

APPENDIX B

APPENDIX 3

DATA ACQUISITION PROGRAM, DATA_DA_AD

 

8.1 Summary of Program

The data acquisition program. DATA_DA_AD. discussed in Chapter 4, is presented
in this appendix. An outline of the program is given in Table BL and a listing of the
program. written in Fortran 77 for a VaxstationII/GPX microcomputer is given in Table
B. 1.

215

216

Table B. 1 Description of the Data Acquisition Program. DATA_DA_AD.

Subroutine Title

PROGRAM DATA

SUBROUTINE AUTO_ZERO

SUBROUTINE PROCESS_DATA

SUBROUTINE SETUP_AMP
SUBROUTINE SETUP_DATA
SUBROUTINE SETUP_POWER
SUBROUTINE TRANS_ADC

SUBROUTINE TRANS_DAC

DCSCI’lDtiOI‘l

Main program: contains data acquisition
control software.

Calculates zeros (0°C) of amplifiers.

Averages and converts A/D data to tempera-
ture and heat flux.

Sets gains. weights. and zeros for amplifiers.
Initial set-up routine for data acquisition.
Data input for D/A power supply.

Converts binary data into volts.

Converts volts into binary format.

 

217

3.2 Program Lining for DATA_DA_ADJ‘OR

PROGRAM DATA

0

0

This program was written by Elaine Scott, 7/15/89, with the invaluble
help of Mike McPherson of the MSU Case Center. It is based
partly on Example 4-16, pages 4-217-219, VAXlab/LabStar Program-
mers Guide.

The purpose of this program is to read data from the A/D converters
on the AXVll-Ca and AXVll-Cb boards, while writing to the D/A on
the AXVll-Cb board.

The program is designed to read thermocouple measurements (mV) and
write voltages to a power supply to activate a heater. The
output voltages of the power supply are also read. The thermo-
couple measurements are converted to degrees centigrade (Type E
TC), and are written to a file along with the power supply out-
put. The file is compatible to the format required by the
parameter estimation program, PROPlD.

 

Other features of this program include data averaging over sampling
interval and automatic ’zeroing' of input temperature data.

000000000000000000000

c .......................................................................
c
IMPLICIT NONE
c
c Six external functions are used. Page numbers in descriptions refer
c to the Vaxlab/LabStar Programmers guide.
c
EXTERNAL LIOSATTACH lReturns the device ID for the specified
device. See pages 4-48 to 4-51.
EXTERNAL LIOSDETACH lDetaches the specified device, returns
c any associated storage to the system,
c and closes and deallocates associated
c VMS devices. See page 4-54.
EXTERNAL LIOSREAD lFills a buffer with data from the spe-
c cified device. See pages 4-58 to 4-59.
EXTERNAL LIO$SET_I lSets up a device according to a parame~
c ter code and any number of interger
c values. See pages 4-60 to 4-61.
EXTERNAL LIO$SET_R lSets up a device according to a parame—
c ' ter code and any number of real values.
c See pages 4-62 to 4-63.
EXTERNAL LIOSWRITE !Empties a buffer through the specified
c device. See pages 4-68 to 4-69.
c
c Include library containing external functions
c
INCLUDE ’SYS$LIBRARY:LIOSET.FOR’ !External function symbol
c definitions
c
c Define external functions
c

INTEGER*4 LIOSATTACH, LIO$DETACH, LIOSREAD, LIO$SET_I,
1 LIO$SET_R, LIOSWRITE

 

2113

c Declare integer variables
c

C
C
C

INTEGER AXA_ID !AXVll-Ca device ID variable
INTEGER AXB_ID !AXVll-Cb device I0 variable
INTEGER A0_LENGTH_AXA !no. of data pts read (AXA)
INTEGER AD_LENGTH_AXB !no. of data pts read (AXB)
INTEGER*2 BUFF_AD_AXA(8) !8 word buffer for A/D
INTEGER*2 BUFFwAD_AXB(8) !8 word buffer for A/D
INTEGER'Z BUFF_0A_AXB(500) 3500 word buffer for D/A '
INTEGER CAL_ZERO i=1 for calculating zeros
INTEGER CLK_ID !KWVll-C clock I0 variable
INTEGER I,J,K,DUMMY !Dummy variables

INTEGER NAVG !Total no. of data averaged
INTEGER NCHAN,NCHANA,NCHANB !No. channels (total, board)
INTEGER NDATAP !Total no. data points
INTEGER NSAMPL !No. data avg./samp. interval
INTEGER NSTEPS !No. time steps for zeroing
INTEGER*4 NVOLT !No. voltages to be converted
INTEGER POWER !Indicates use of D/A

INTEGER STATUS !Status returned by LIOS

Declare real variables

REAL'4 GAIN(16) !Gains for amplifiers

REAL HEAT_AREA !Area of heated surf. (m'm)
REAL RATE !Clock rate (Hertz)

REAL RESIST !Resistance of heater (ohms)
REAL SRATE !Interval rate

REAL TIMES(100) !Times(s) for power supply
REAL VOLTS(100) !Volts(v) for power supply
REAL*4 WEIGH(16) !Weights for amplifiers
REAL*4 ZERO(16) !Zeros for amplifiers

 

Declare character variables

0

CHARACTER*15 AMP(16) !Ectron amplifier number
CHARACTER BINARYFIL*20 !Binary data file
CHARACTER TEMPPIL*20 !Temperature data file
CHARACTER YN*2 !Yes/No (Y/N)

Common statements

0

COMMON /AMPLIPIER/ GAIN,WEIGH,ZERO,NAVG,NSTEPS,AMP
COMMON /FILES/BINARYFIL,TEMPFIL
COMMON /POWER/ BUFF_DA_AXB,NVOLT,POWER,HEAT_AREA,TIMES,VOLTS,
l RESIST
COMMON /SETUP/NCHAN,NCHANA,NCHANB,NDATAP,NSAMPL,RATE,SRATE
c
cit****************t***************t*************ttikitiiitiii’ttttiitttii
c
c Call SETUP_DATA to begin set up for data
c
CALL SETUP_DATA

0

Call SETUP_POWER to begin set up for input to 0/A for power supply

WRITE(*,10)’DO you want to input voltage values for D/A?’
l ,’ (N a no; F = enter data file; K = use keyboard)’
10 FORMAT(’ ',/,’ ’,5X,A,/,’ ’,5X,A)
READ (*, 1) YN
l FORMAT(A)
IF(YN.NE.'N'.AND.YN.NE.’n’)THEN
CALL SETUP_POWER(YN)

00

00000

20

C
C
C

219

ELSE
POWER = 0
ENDIF

Call SETUP_AMP to change gains, weights and/or zeros of amplifiers
CALL SETUP_AMP(NCHAN,CAL_ZERO)

Name output file for binary and temperature measurements. Binary
data is stored during the experiment in case of unintended
program interuption.

WRITE(*,30)’Two data files (binary and temp.) are created:’
FORMAT(' ’,/,’ ’,72("’),/,’ ',5X,A)

WRITE(*,20)’Fi1e name for binary data file:'
READ(*,1)BINARYPIL

WRITE(*,20)’File name for temperature data file:'

FORMATU ',/,' ',5X,A)

READ(*,1)TEMPFIL

End subroutine set up calls

cittittttttttttttitiiit*iittttttt*ttiittt‘lt*iiittitttitiItitiittitiitti‘k

C
C
C

000000

00

0000

C

Begin hardware set up for data aquisition:
WRITE(*,20)’Begin hardware set up:'
First, attach devices:

Attach the KWVll-C clock
Gets a device ID for the KWV and tells LIO to use QIO I/O

STATUS = LIO$ATTACH(CLK_ID, ’KZAO’, LIO$K_QIO) !Attach to KWV
IF(.NOT.STATUS) CALL LIBSSIGNAL(%VAL(STATUS))

Attach AXVll-Ca device for A/D
Gets a device ID for the AXVa and tells LIO to use 010 I/O

STATUS = LIO$ATTACH(AXA_ID, 'AXAO', LIO$K_QIO) !Attach to AXVa
IF(.NOT.STATUS) CALL LIBSSIGNAL(§VAL(STATUS))

Attach AXVll-Cb device for A/D and D/A
Gets a device ID for the AXVb and tells LIO to use QIO I/O

IF(NCHANB.GT.0.0R.POWER.EQ.1)THEN
STATUS - LIO$ATTACH(AXB_ID, ’AXBO', LIO$K_QIO) !Attach to AXVb
IF(.NOT.STATUS) CALL LIB$SIGNAL(%VAL(STATUS))
ENDIP

cttti’ititttii‘k‘kttti*ti’tiitit*t‘k‘k************tittttttittii’tititt*‘kt‘kii‘k‘ti

Set up the clock
rate - Hertz rate (a real number)
function - repeat count
trigger - immediate (this only sets up the start condition, the
clock will be started by the LIOSREAD from the A/D)

STATUS = LIO$SET_R(CLK_ID, LIO$K_CLK_RATE, 1, RATE)

22K)

IF(.NOT.STATUS) CALL LIBSSIGNAL(%VAL(STATUS))
STATUS - LIO$SET_I(CLK_ID, LIO$K_FUNCTION, 1, LIO$K_REP_COUNT)
IP(.NOT.STATUS) CALL LIB$SIGNAL(%VAL(STATUS))
STATUS - LIO$SET_I(CLK_ID, LIO$K_TRIG, l, LIO$K_IMMEDIATE)
IF(.NOT.STATUS) CALL LIBSSIGNAL(%VAL(STATUS))
C

Cti*iii*************t******t*i**iit****ti****‘k*tit'kttifi‘ktl‘i‘k‘kttt‘ktttrtit

C

 

c Set up AXVa and AXVb:
c
c AXVa:
c Sychronous interface (LIOSREAD):
c
STATUS - LIO$SET_I(AXA_ID, LIO$K_SYNCH, 0)
IP(.NOT.STATUS) CALL LIBSSIGNAL(%VAL(STATUS))
c
c Set up A/D:
c use A/D channels 0-7, depending on value of NCHANA
c
STATUS - LIO$SET_I(AXA_ID, LIO$K_AD_CHAN, NCHANA, 0, 1, 2, 3,
1 4, S, 6, 7)
IP(.NOT.STATUS) CALL LIBSSIGNAL(§VAL(STATUS))

0

use a gain of one
STATUS = LIO$SET_I(AXA_ID, LIO$K_A0_GAIN, NCHANA, 1, 1, 1, l,
1 1, 1, 1, 1)
IE(.NOT.STATUS) CALL LIBSSIGNAL(%VAL(STATUS))

Sweep through all channels on each clock tic:

0

STATUS - LIO$SET_I(AXA_ID, LIO$K_TRIG, 1, LIO$K_CLK_SWEEP)
IP(.NOT.STATUS) CALL LIBSSIGNAL(%VAL(STATUS))

c End AXVa set up
IF(NCHANB.GT.0.0R.POWER.EQ.1)THEN

c AXVb:
c Sychronous interface (LIOSREAD):

0

STATUS a LIO$SET_I(AXB_ID, LIO$K_SYNCH, 0)
IF(.NOT.STATUS) CALL LIBSSIGNAL(%VAL(STATUS))

Set up A/D:

Note, channel 4 is reserved for reading output of power supply,
and channels 0-3 are for thermocouple output. 010 I/O requires
channels to be in consecutive, ascending order; therefore, the
fifth channel is designated for the power supply, instead of
the last, and all five channels must be read regardless whether
or not they are all used.

WARNING: If additional amplifiers are installed, the channel
designation must be changedll

000000000000

STATUS - LIO$SET_I(AXB_ID, LIO$K_AD_CHAN, 5. o, 1, 2, 3, 4)
IF(.NOT.STATUS) CALL LIBSSIGNAL(%VAL(STATUS))

00

use a gain of one

STATUS . LIO$SET_I(AXB_I0, LIO$K_A0_GAIN, 5, 1. 1, 1. 1, 1)
IP(.NOT.STATUS) CALL LIB$SIGNAL(§VAL(STATUS))

221

c Set up D/A:

c use 0/A channel X
c
STATUS - LIO$SET_I(AXB_ID, LIO$K_DA_CHAN, l, 1)
IP(.NOT.STATUS) CALL LIBSSIGNAL(%VAL(STATUS))
c
c Use immediate start burst mode for AXVll-Cb. This starts immediately
c on the LIOSREAD call for the AXVll-Ca board:
c
STATUS - LIO$SET_I(AXB_ID, LIO$K_TRIG, 1, LIO$K_IMM_BURST)
IF(.NOT.STATUS) CALL LIBSSIGNAL(%VAL(STATUS))
ENDIF
C
c End AXVb set up
c

ctritttit’itt‘tt*ititiiiiittitttttifitit*ttt*******t******************i***
c

c Initialize the power supply and open output file before starting the
c clock:

c

IF(POWER.EQ.1)THEN
STATUS - LIOSWRITE(AXB_ID,BUFF_DA_AXB(1),2,LIO$K_OUTPUT)
IF(.NOT.STATUS) CALL LIBSSIGNAL(‘VAL(STATUS))
ENDIF
OPEN(UNIT=10,NAME=BINARYFIL,TYPE='NEW’,ERR=99)
c Now, initialize the experiment:
WRITE(*,40)’Type "1 <CR>" to start data collection’

40 FORMAT(’ './,' ',72('*')./.' ’.15X.A./,' '.72('*')./)
READ *,DUMMY

c Start the clock:

STATUS = LIO$SET_I(CLK_ID, LIO$K_START,0)
IF(.NOT.STATUS) CALL LIBSSIGNAL(§VAL(STATUS))

c
c Start loop:
c
K a 2
DO I - 1,NDATAP
c
C Write D/A values to power supply using AXVb
c
IF(POWER.EQ.1)THEN
IF(K.LE.NVOLT-1)THEN
IF(TIMES(K).GE.I/RATE.AND.TIMES(K).LT.(I+1)/RATE)THEN
STATUS - LIOSWRITE(AXB_ID,BUFF_DA_AXB(K),2,LIO$K_OUTPUT)
IF(.NOT.STATUS) CALL LIBSSIGNAL(%VAL(STATUS))
K - K+1
ENDIE
ENDIF
ENDIF

c
c Read A/D values from thermocouples (channels 0-7 on AXVa and 0-3 on
c AXVb) and power supply (channel 4 on AXVb)
c

STATUS = LIO$READ(AXA_ID, BUFP_AD_AXA(1),

as:

1222

l 2*NCHANA, AD_LENGTH_AXA, LIO$K_INPUT)
IF(.NOT.STATUS) CALL LIBSSIGNAL(%VAL(STATUS))
c
c Read all data for channel B
c
c WARNING: If additional amplifiers are installed, the number of
c channels read must be changed!!
c
IF(NCHANB.GT.0.0R.POWER.EQ.1)THEN
STATUS - LIO$READ(AXE_ID, BUFF_AD_AXB(1),
l 2*5, AD_LENGTH_AXB, LIO$K_INPUT)
IF(.NOT.STATUS) CALL LIBSSIGNAL(%VAL(STATUS))
ENDIF
c
c Write data to file: The data from the AXVll-Ca are written on
c the first line, and the data from the AXVll-Cb, including
c output from the D/A, are written on the next line.
C
WRITE(*,50)I,(BUFF_AD_AXA(J), J - l,NCHANA),
1 (BUFF_AD_AXB(J), J - 1,NCHANB),BUFF_AD_AXB(5)
WRITE(10,60)(BUFF_AD_AXA(J), J - 1,NCHANA),
1 (BUFF_AD_AXB(J), J - 1,NCHANB),BUPF_AD_AXB(5)
50 FORMAT(’ ’,I6,2X,8(IS,3X)./,9X,8(IS,3X)l
60 PORMAT(’ ’,8(IS,3X),/,’ ',8(IS,3X))
ENDDO
C
c End loop
c
c .......................................................................
c
c Begin shutdown procedure
C
c Close binary data file
c
CLOSE(UNIT=10)
c
c Terminate input voltage to power supply at the end of the run:
c
IP(POWER.EQ.1)THEN
STATUS = LIOSWRITE(AXB_ID,BUFP_DA_AXB(NVOLT),2,LIO$K_OUTPUT)
IF(.NOT.STATUS) CALL LIBSSIGNAL(%VAL(STATUS))
ENDIF
c
c Detach devices:
c
STATUS = LIO$DETACH(AXA_ID, )
IP(.NOT.STATUS) CALL LIB$SIGNAL(%VAL(STATUS))
IF(NCHANB.GT.0.0R.POWER.EQ.1)THEN
STATUS - LIO$DETACH(AXB_ID, )
IF(.NOT.STATUS) CALL LIBSSIGNAL(%VAL(STATUS))
ENDIF
c
c Stop and detach the clock:
c
STATUS - LIO$SET_I(CLK_ID,LIO$K_STOP,0)
IP(.NOT.STATUS) CALL LIBSSIGNAL(%VAL(STATUS))
STATUS a LIO$DETACH(CLK_ID, )
IF(.NOT.STATUS) CALL LIBSSIGNAL(%VAL(STATUS))
c
c Data collection is complete
c

WRITE(*,30)'Data collection is completed, now process data:’

223

cititiiiittiititttiittitii'kiit"ittflittii*fl**i***ttittitttti*itittt‘tttttt

0000

00

70

99
300

C
C
C
C

Start data analysis:

First calculate zeros for amplifiers if needed.

IF(CAL_ZERO.EO.1)THEN
CALL AUTO_ZERO(POWER)
ENDIF

Now, average binary data over sampling interval, and convert

binary data into millivolts and then to degrees centigrade.

CALL PROCESS_DATA(POWER,HEAT_AREA,RESIST)

All done now.... stop and end

WRITE(*,70)’Program is completed.’

FORMAT(//,’ ’,72(’*’)./,’ ’,5X,A,/)

GO TO 300

STOP ’Error during opening master data file’
STOP

END

Finill

ctit*i’iititii*i‘iitttiiii*itfitiiit*iii‘kiitl‘iitt*tt‘fii‘iitifl'**********t****i

 

C

c End main program.

c

c Begin subroutines:

c There are six subroutines in this program; they are listed in

c alphabetical order:

c Title Arguments Brief Description

c

c

c AUTO_ZERO POWER Calculates zeros(0C) of amplifiers
c PROCESS_DATA POWER,HEAT_AREA Avgerages & converts A/D data

c SETUP_AMP NCHAN,CAL_ZERO Sets gain, weight, zero for ampl.
c SETUP_DATA (none) Initial set up for data aquisition
c SETUP_POWER YN Data input for D/A

c TRANS_DAC VOLTS,BUFF_DA_AXB Converts volts to binary format

c

ctitt*itti‘t'ktt*ttttt****t*t*i**ti*i*****iit*‘lit‘kflttt‘kiiitiiiti*tttiitiit

ctit*‘k‘k'k*I’****t‘kt'k'k'k'l’******fii'k'k'ktttfi't'ki‘kt'tt‘kttt*‘k******ttt’ti’ktttt‘ktil’tit

C

0000

00

SUBROUTINE AUTO_ZERO(POWER)

This subroutine calculates the zeros (C) for the amplifiers, based

on initial isothermal data.

IMPLICIT NONE

Declare integer values

INTEGER I,J,K !Dummy

INTEGER NAVG !Total no. of data averaged

INTEGER NCHAN,NCHANA,NCHANB !Total number of channels
INTEGER NDATAP !Total no. data points

INTEGER NSAMPL !No. data avg./samp. interval
INTEGER NSTEPS !No. time steps for zeroing

INTEGER NZERO !No. of zeros changed

 

 

224

INTEGER NWEIGH !No. of weights changed
INTEGER POWER !Indicates use of D/A
c
c Declare real values

c
REAL'4 AVGZERO(100,16) !Binary data to be averaged
REAL*4 CONV(16) !Inter. value for zeros
REAL'4 GAIN(16) !Gains for amplifiers
REAL RATE !Clock rate (Hertz)

REAL SRATE !Interval rate

REAL SUMI(16) !Sum of binary data
REAL SUMAVG !Average binary data
REAL SUMWGH !Average weights

REAL‘4 WEIGH(16) !Weights for amplifiers
REAL'4 ZERO(16) !Zeros for amplifiers

c

c Declare character values

c
CHARACTER'lS AMP(16) !Ectron amplifier number
CHARACTER BINARYFIL*20 !Binary data file
CHARACTER TEMPEIL'ZO !Temperature data file
CHARACTER YN‘Z !Yes/No (Y/N)

c

c Common statement

c
COMMON /AMPLIFIER/ GAIN,WEIGH,ZERO,NAVG,NSTEPS,AMP
COMMON /FILES/BINARYFIL,TEMPFIL
COMMON /SETUP/NCHAN,NCHANA,NCHANB,NDATAP,NSAMPL,RATE,SRATE

c

cit*t*********fl**ttttfltiit!***********t************ti*************tti‘kt‘k

c
c The zero's are calculated internally by averaging the first NSTEPS
c data points for each thermocouple prior to the onset of applied heat
c flux, and determining the difference between the overall average and the
c average value for each thermocouple.
c ..........................................................................
c
c Read in first NAVG values from binary data file (BINARYFIL)
c
OPEN(UNIT=20,NAME=BINARYFIL,STATUS='OLD')
DO J ' l,NSTEPS
K - (J-l)*NCHAN
READ(20,*)(AVGZERO(J,I), I = l,NCHANA)
IF(NCHANB.GT.0.0R.POWER.EO.1)THEN
READ(20,*)(AVGZERO(J,NCHANA+I), I = 1,NCHANB+POWER)
ENDIF
ENDDO
c
c Close binary data file
c
CLOSE (UNITaZO)
c
c ..........................................................................
c
c
c Check weights and recalculate zeros if you find bad thermocouples.
c

WRITE(*,10)’Number','Amplifier','Weighting Factor’
DO I I l,NCHAN
WRITE(*,20)I,AMP(I),WEIGH(I)
ENDDO
TYPE *
WRITE(*,30)'Are corrected weighting factors OK? (Y/N)’

 

 

225

READ(',1)YN
c
c Change weights if necessary
c
IF(YN.EQ.'N'.OR.YN.EQ.’n')THEN
100 WRITE(*,30)’Enter number of weights you wish to change:’
READ ',NWEIGH
DO I = l,NWEIGH
WRITE(*,30)'Enter amplifier number and corrected weight:'
READ *,J,WEIGH(J)
ENDDO
WRITE(*,10)'Number’,’Amplifier','Corrected Weights'
DO J = l,NCHAN
WRITE(*,20)J,AMP(J),WEIGH(J)
ENDDO
WRITE(*,30)'Are corrected weights OK? (Y/N)’
READ(*,1)YN
IF(YN.EQ.'N’.OR.YN.EQ.'n')GO TO 100
ENDIF
SUMAVG - 0.0D0
SUMWGH - 0.0D0
DO I - l,NCHAN
SUMI(I) - 0.0DO
SUMWGH = SUMWGH + WEIGH(I)
ENDDO

Average data over all channels for NSTEPS time steps

DO J a l,NSTEPS
DO I = l,NCHAN
SUMI(I) 3 SUMI(I) + AVGZERO(J,I)*WEIGH(I)
ENDDO
ENDDO

Adjust for gain

0

DO I - l,NCHAN
SUMAVG = SUMAVG + SUMI(I)*GAIN(I)
ENDDO
SUMAVG = SUMAVG/SUMWGH

Calculate zeros

0

DO I = l,NCHAN
CONV(I) = (SUMAVG-SUMI(I)*GAIN(I))/NSTEPS
ENDDO
CALL TRANS_ADC(0,NCHAN,CONV,ZERO)
1 FORMAT(A)
10 FORMATU ’,/,' ',3X,A,3X,A,7X,A)
20 FORMAT(' ',3X,I6,3X,A,3X,F6.3)
30 FORMAT(' ’,/,’ ’,A)

c
c End AUTO_ZERO subroutine.
c
RETURN
END
c

cttttiiiittitittititttt*ttittttttttttttttitittttittttttttttttt*ittittirt
c**t*****wwtttrttttttt*wt*tttttattttttttwtttt*tt*ttttttrtttrrttttwwrwttt
C

SUBROUTINE PROCESS DATA(POWER,HEAT_AREA,RESIST)

c
c This subroutine averages A/D data over each sampling interval, and

 

226

then converts the binary data to millivolts and then to degrees

0

c centigrade.

c
IMPLICIT NONE

c

c Declare integers variables

c
INTEGER.I,J,K,L !Dummy variables
INTEGER NAVG !Total no. of data averaged
INTEGER NCHAN,NCHANA,NCHANB !No. channels (total, board)
INTEGER NCHANPl !Includes voltage channel
INTEGERNDATAP,NDATA !Total no. data points
INTEGERNSAMPL !No. data avg./(samp intrvl)
INTEGER NSTEPS !No. time steps for zeroing
INTEGERPOWER !=l for output to D/A
INTEGERSTATUS !Status returned by LIOS

c
c Declare real variables
c

REAL AVG_DATA(1000) !Data to be avgeraged
REAL'4 GAIN(16) !Gains for amplifiers

REAL HEAT_AREA !Area of heated surf. (m*m)
REAL HEAT_FLUX !Heat flux from heater
REAL RATE,SRATE !Clock rate (Hertz)

REAL RESIST !Resistance of heater (ohms)
REAL SUM(16) !Sums for averaging data
REAL TEMP(16) !Converted temps. from A/D
REAL TIMSTP !Time step for printout
REAL VA(16) !Millivolts from A/D
REAL*4 WEIGH(16) !Weights for amplifiers
REAL*4 ZERO(16) !Zeros for amplifiers

Declare character variable

0

CHARACTER*15 AMP(16) !Ectron amplifier number
CHARACTER BINARYFIL*20 !Binary data file
CHARACTER TEMPFIL*20 !Temperature data file
c
c Common statement
c
COMMON /AMPLIFIER/ GAIN,WEIGH,ZERO,NAVG,NSTEPS,AMP
COMMON /FILES/BINARYFIL,TEMPFIL
COMMON /SETUP/NCHAN,NCHANA,NCHANB,NDATAP,NSAMPL,RATE,SRATE
c

ctiiti‘kti’ttiti’ttti‘kt****it*‘kttfitit*****t‘k‘kt'kti'ktfiitfitffi’titti*i‘tttttttit

c
c Open binary data file (BINARYFIL) and temperature data file (TEMPFIL)
c
OPEN(UNIT=20,NAME=BINARYFIL,STATUSs'OLD’)
OPEN(UNIT=30,NAME=TEMPEIL,TYPE='NEW')

c
c Initialize arrays, etc.
c

NDATA - NDATAP/NSAMPL

NCHANPl = NCHAN + POWER

DO J - l,NCHANPl

SUM(J) - O

ENDDO
C
c ..........................................................................
c

c Read in data values from file

227

D0 L = l,NDATA
DO J a l,NSAMPL
K a (J-l)*NCHANPl
READ(20,*,ERR=99)(AVG_DATA(K+I), I = l,NCHANA),
1 (AVG_DATA(K+NCHANA+I), I = l,NCHANB+POWER)

Average data over sampling interval

DO I = l,NCHANPl
SUM(I) - SUM(I) + AVG_DATA(K+I)/NSAMPL
ENDDO
ENDDO

0

0

Begin processing data

Convert binary data into volts. Binary data is converted to
voltages in subroutine TRANS_ADC assuming a linear relationship
between the binary output and volts based on a calibrated curve.
Voltage values are returned in VA.

 

00000000

CALL TRANS_ADC(POWER,NCHANP1,SUM,VA)

0

DO I a l,NCHANPl

Next, convert thermocouple values to degrees centigrade if channel is
not reading from power supply.

0000

IF(POWER.EQ.1.AND.I.EQ.NCHANP1)GOTO 100

First, adjust thermocouple data for gain and zeros (C), and then con-
vert data to millivolts.

0000

VA(I) = (VA(I)*GAIN(I)+ZERO(I))/1000.0

Voltages are then converted to temperatures using the thermocouple
conversion routine in VAXlab/Lab-Star Programmers Guide. See
Sec. 6.9, pages 6-17 to 6-19; pages 6-62 to 6-63; and Example 6-20
on page 6-119 for details.

Note, this subroutine is for Type E, Chromel-Constantan thermocouples.
Input values are the voltages in microvolts, the number of data
values to be converted, and output values are temperature in
degrees centigrade and the operation status.

00000000000

CALL LSPSTHERMOCOUPLE_E (VA(I),TEMP(I),1,STATUS)
IF(.NOT.(status)) CALL lib$signal(%val(status))
100 ENDDO
c
c Print out processed temperature data (C) to file (TEMPFIL)
c
TIMSTP=SRATE*L
IF(POWER.NE.1)THEN
IF(NCHAN.LE.8)THEN
WRITE(30,10) TIMSTP,(TEMP(I),I=1,NCHAN)
10 FORMAT(F10.3,1X,8(F6.2,1X))
ELSE
WRITE(30,20) TIMSTP,(TEMP(I).I=1,NCHAN)
20 FORMAT(F10.3,lX,l6(F6.2,1X))
ENDIF
ELSE

228

c Calculate heat flux using measured voltage and resistance of heater

 

c (power - (volts)*(volts)/resistance; heat flux = power/area)
c
HEAT_FLUX - VA(NCHANPI)“VA(NCHANPI)/(RESIST*HEAT_AREA)
c
c Write processed heat flux and temperature data to a file
c
IF(NCHAN.LE.8)THEN
WRITE(30,30) TIMSTP,HEAT_FLUX,(TEMP(I),I=1,NCHAN)
3O FORMAT(F10.3,lX,F8.l,lX,8(E6.2,1X))
ELSE
WRITE(30,40) TIMSTP,HEAT_FLUX,(TEMP(I),I=1,NCHAN)
4O FORMAT(F10.3,1X,F8.1,1X,l3(F6.2,1X))
ENDIF
ENDIF
c
c End data processing
c
c .......................................................................
c
c Initialize summations
c
DO I - l,NCHANPl
SUM(I) = 0.0
VA(I) = 0.0
ENDDO
c
c End loop
c
ENDDO
c .

ctittiii*tttii***t******fittiiititi*ttiii*****ii*********************fi*‘k

c
c Close binary data file and temperature file
c
CLOSE (UNIT=20)
CLOSE (UNIT=30)
c
c End data processing subroutine
c
RETURN
99 STOP 'Error while reading from data file'
END
c

c'k‘tt********************‘ki’i’tiiit‘ktt*‘ktt‘ki‘tt*ttfitittt'ktititttti'i‘kttti'itt

Ctit*‘kfi'i’it'k'kti'fli't***************fi*****t**ttitt*****fl***********itttt'tti

c
SUBROUTINE SETUP_AMP(NCHAN,CAL_ZERO)
c
c This subroutine sets up the Ectron amplifiers; user may change gains,
c weights and/or zeros if necessary.
c
IMPLICIT NONE
c
c
c Define integer variables
c
INTEGER CAL_ZERO !=1 for calculating zeros
INTEGER I,J !Dummy
INTEGER NAVG !Total no. of data averaged
INTEGER NCHAN !Total number of channels
INTEGER NGAIN !No. of gains changed

INTEGER NSTEPS !No. of time steps for zeroing

 

2E29

 

INTEGER NWEIGH !No. of weights changed
INTEGER NZERO !No. of zeros changed

c

c Define real variables

c
REAL'4 GAIN(16) !Gains for amplifiers
REAL'4 WEIGH(16) !Weights for amplifiers
REAL*4 ZERO(16) !Zeros for amplifiers

c

c Define character variables

c
CHARACTER*15 AMP(16) !Ectron amplifier number
CHARACTER‘l YN IYes/no (Y/N) '

c

c Common statement

c
COMMON /AMPLIFIER/ GAIN,WEIGH,ZERO,NAVG,NSTEPS,AMP

c

c Data statements for amplifiers

c Ectron amplifier numbers:

c
DATA AMP(1).AMP(2),AMP(3).AMP(4).AMP(5),AMP(6),AMP(7).AMP(8).
1 AMP(9),AMP(10),AMP(11),AMP(12)/'ECTRON_53661',
1 ’ECTRON_53662',’ECTRON_53663','ECTRON_53664',
1 'ECTRON_53665','ECTRON_53674','ECTRON_5367S',
1 'ECTRON_53676','ECTRON_53677',’ECTRON_53678’,
1 ’ECTRON_53973',’ECTRON_53974'/

c

c Gain of Ectron amplifiers last checked on 3/16/89

c
DATA GAIN(1),GAIN(2),GAIN(3),GAIN(4),GAIN(5),GAIN(6),GAIN(7),
1 GAIN(8),GAIN(9),GAIN(10),GAIN(11),GAIN(12)/1.003,l.003,
1 0.996,0.999,0.998,0.99S,1.002,l.000,0.999,0.999,1.028,
1 1.008/

c

c Weights of Ectron amplifiers

c
DATA WEIGH(1),WEIGH(2),WEIGH(3),WEIGH(4),WEIGH(5),WEIGH(6),
l WEIGH(7),WEIGH(8),WEIGH(9),WEIGH(10),WEIGH(11),WEIGH(12)
l /1.00,1.00,1.00,1.00,1.00,1.00,1.00,1.00,1.00,1.00,l.00,
1 1.00/

c

c Zeros are millivolt readings at 0C; last checked on 3/30/89.

c
DATA ZERO(1),ZERO(2),ZERO(3),ZERO(4),ZERO(5),ZERO(6),ZERO(7),
1 ZERO(8),ZERO(9),ZERO(10),ZERO(11),ZERO(12)/0.002,0.000,
l 0.000,-0.003,0.0005,0.001,-0.0025,-0.0005,0.0005,0.00S,
1 0.00SS,-0.004/

c

cit*****i*********************‘k*****tiii”.*****************ti***iiiiii'kt'k

c
c Set up amplifiers: change gains, weights and/or zeros if necessary:
c

WRITE(*,10)’Begin set up for amplifiers:'

c

c .......................................................................
c

c Change gain of amplifiers if necessary. Gains represent fractional
c output at 10V input (amps should read O-IOV; for example, if the

c amplifier instead read 0-9.99V, the gain would be 0.999)

c

WRITE(*,40)’Check gain of Ectron amplifiers:'
TYPE *

0

0

0

0000

230

WRITE(*,20)’Number’,'Amplifier','Gain'

DO J - l,NCHAN
WRITE(*,30)J,AMP(J),GAIN(J)

ENDDO

You may change any number of gains that you want

WRITE(*,40)'Do you want to change any of the gains? (Y/N)’
READ(',1)YN
IF(YN.EQ.’Y’.OR.YN.EO.'y')THEN
WRITE(*,40)'Enter number of gains you wish to change:’
READ *,NGAIN

Amplifier numbers start from 1 and go to 12 (Ectron Amplifiers)

DO I = l,NGAIN
WRITE(*,40)’Enter amplifier number and corrected gain:'
READ*,J,GAIN(J)

ENDDO

TYPE *

WRITE(*,20)'Number’,’Amplifier','Corrected Gain'

DO J = l,NCHAN
WRITE(*,30)J,AMP(J),GAIN(J)

ENDDO

Re-correct values if necessary

WRITE(*,40)'Are corrected gains OK? (Y/N)’
READ(*,1)YN
IF(YN.EQ.'N'.OR.YN.EQ.'n')GO TO 300

ENDIP

Change weights for amplifiers if necessary. Use a weight of 1.0 for
good thermocouples, and a weight of 0.0 for bad thermocouples.

WRITE(*,40)’Check weights for Ectron amplifiers:'
TYPE *
WRITE(*,50)’Number',’Amplifier’,'Weight’
DO J = l,NCHAN
WRITE(*,60)J,AMP(J),WEIGH(J)
ENDDO
WRITE(*,40)'Do you want to change any of the weights? (Y/N)’
READ(*,1)YN
IF(YN.EQ.'Y'.OR.YN.EQ.'y')THEN

Change as many of the weights as you want.

WRITE(*,40)'Enter number of weights you wish to change:'
READ*,NWEIGH

Amplifier numbers range from 1 to 12 (Ectron Amplifiers)

DO I - l,NWEIGH
WRITE(*,40)’Enter amplifier number and corrected weight:’
READ*,J,WEIGH(J)

ENDDO

TYPE *

WRITE(*,20)’Number',’Amplifier’,'Corrected Weight'

DO J - l,NCHAN
WRITE(*,30)J,AMP(J),WEIGH(J)

ENDDO

 

231

Re-correct values if necessary

c
c
WRITE(*,40)'Are corrected weights OK? (Y/N)’
READ(*,1)YN
IF(YN.EQ.’N'.OR.YN.EO.'n')GO TO 400
ENDIF
c
c _______________________________________________________________________
c
c Change zeros of amplifiers if necessary. Zeros represent the ampli-
c fier readings at 0C. There are three options for the user here:
c 1. use the zeros given; 2. let the program calculate the zeros;
c 3. enter in zeros.
c
WRITE(*,40)’Check zeros (at DC) of Ectron amplifiers:’
TYPE *
WRITE(*,20)’Number','Amplifier',’mV at OC'
DO J =- l,NCHAN
WRITE(*,70)J,AMP(J),ZERO(J)
ENDDO
TYPE *
WRITE(*,80)'Do you want the zeros to be recalculated for you?',
1 ' (Y/N)’
READ(*,1)YN
CAL_ZERO = 0
NSTEPS - O
NAVG - 0
c
c If user wants program to calculate zeros, the voltage values are
c averaged over a number of time steps specified by the user.
c The time step means the actual time step in which data is col-
c lected, not the averaged sampling interval. The temperatures
c should be isothermal over this interval, with no externally
c applied heat flux.
c
IE(YN.EQ.'Y'.OR.YN.EQ.'y')THEN
WRITE(*,90)'Enter number of time steps to average zeros over:'
1 ,'(No heat flux should be applied over this interval.)'
READ *,NSTEPS
CAL_ZERO - 1
NAVG = NSTEPS*NCHAN
ELSE
c
c User may change zeros by himself or herself
c
WRITE(*,40)’Do you want to change any of the zeros? (Y/N)’
READ(*,1)YN
IF(YN.EQ.’Y'.OR.YN.EQ.'y')THEN
500 WRITE(*,40)'Enter number of zeros you wish to change:’
READ*,NZERO
DO I - l,NZERO
WRITE(*,40)'Enter amplifier number and corrected zeros:'
READ *,J,ZERO(J)
ENDDO
TYPE *
WRITE(*,20)'Number',’Amplifier','Corrected mV at OC'
DO J = l,NCHAN -
WRITE(*,30)J,AMP(J),ZERO(J)
ENDDO
c

Re-correct values if necessary

232

c
WRITE(*,40)'Are corrected zeros OK? (Y/N)’
READ(*,1)YN
IF(YN.EQ.'N' .OR.YN.EO.'n')GO TO 500
ENDIF
ENDIP
l FORMA'IHA)
10 FCRMAT(' './/,’ ’,72('*’),//,10X,A,//)
2O FORMATU ',3X,A,3X,A,7X,A)
3O FORMAT(' ',3X,I6,3X,A,3X,F6.3)
4O FORMAT(' ',/,' ',5X,A)
SO PORMAT(' ’,3X,A,3X,A,5X,A)
60 PORMAT(' ',3X,16,3X,A,3X,P6.4)
7O FORMATU ',3X,I6,3X,A,5X,F7.4)
80 FORMATU ',/,' ',5X,A,A)
90 FORMAT(' ',/,' ',5X,A,/,' ',5X,A)
c
c End set up for amplifiers
c
RETURN
END
c

ctttt*ititiitf‘kifiifiitltifiit*ti'k‘iittiittitiit*‘k‘ltitii’ttitit‘kiititiit‘ktt‘kfi

cttt*ttttii*i**i***t***ttti*tititit‘kiIfiitttitt‘kttiitititttitiii**i******

c
SUBROUTINE SETUP_DATA
c
c This subroutine provides the initial set up for data aquisition:
c
IMPLICIT NONE
c
c Define integer variables
c
INTEGER NCHAN,NCHANA,NCHANB !No. channels (total, board)
INTEGER NDATA !No. of data pts per channel
INTEGER NDATAP !Total no. data points
INTEGER NSAMPL !No. data avg/(samp intrvl)
c

c Define real variables

REAL RATE !Clock rate (Hertz)
REAL SRATE !Interval rate

c: Define character variable

c
CHARACTER YN*2 !YeS/no (Y/N)
c
c Common statement
c
COMMON /SETUP/NCHAN, NCHANA, NCHANB, NDATAP , NSAMPL, RATE, SRATE
c
ctiit*iiitiitiiti‘ttitiit*ti‘tittifi'fii*ttii*********i****t*i’iiiiitittttit
c
c IFirst set up sampling interval, and number of samples to be averaged
c over each sampling interval.
c

WRITE(*,10)'Begin set up for data aquisition'
WRITE(*,20)'Data is collected over a sampling interval and',
1 ' averaged for that interval.’
5 WRITE(*,30)'Enter sampling interval in seconds: '
READ *, SRATE
WRITE(*,40)’Enter no. of samples to be averaged per sampling ',

l 'interval:’

233

READ *,NSAMPL
RATE - 1.0'NSAMPL/SRATE
WRITE(*,50)'rate -’,RATE,' Hertz'

Next enter total number of data points for each channel and
the total number of channels.

0000

WRITE(*,40)'Enter total number of sampling intervals',
1 ' for each channel:'
READ*,NDATA
NDATAP = NDATA*NSAMPL
WRITE(*,30)'Enter total number of channels (max=12):'
READ*,NCHAN
IP(NCHAN.LT.8)THEN
NCHANA - NCHAN
NCHANB I 0
ELSE
NCHANA = 8
NCHANB = (NCHAN-8)
ENDIF

Allow user to change input values.

0

TYPE *
WRITE(*,30)'Do you want to change the above input values? (Y/N)’

READ(*,1)YN
IE(YN.EQ.’Y’.OR.YN.EO.’y')GOTO 5

c
c Initial set up complete
c

1 PORMAT(A)

10 FORMAT(' ',////,' ',72('*’).//,' '.20X.A,//.' '.72('*').//)
20 FORMAT(’ ’,10X,A,/,’ ',5X,A,/)

30 FORMAT(' ',5X,A)

4O EORMATU ’,5X,A,A)

50 FORMAT(' ',/,' ',lOX,A,F8.3,A,/)
RETURN
END

c

C**i'k*****‘k'k******i****i******************‘k‘ki’kitt‘ti’kttfi‘kfiiinitit*i******

Ctt*t*****************************i*it*ttttt‘ki’ii’*‘kitii‘k‘kt‘ktitti’i’********

c
SUBROUTINE SETUP_POWER(YN)

c

c This subroutine provides the initial set up for input to D/A for

c power supply.

0
IMPLICIT NONE

c

<: Define integer variables

c
INTEGER*2 BUFF_DA_AXB(SOO) !500 word buffer for D/A
INTEGER.) !Dummy variable
INTEGER*4 NVOLT !No. voltages converted
INTEGERPOWER !Indicates use of D/A

c

c Define real variables

REAL HEAT_AREA !Area of heated surf. (m*m)
REAL RESIST !Resistance of heater (ohms)
REAL TIMES(100) !times(s) for power supply
REAL VOLTS(100) !volts(v) for power supply

 

 

234

Define character variables

c

c
CHARACTER POWERFIL'ZO !Data file for D/A input
CHARACTER YN'Z !Yes/no (Y/N)

c

c Common statement

COMMON /POWER/ BUFF_DA_AXB,NVOLT,POWER,HEAT_AREA,TIMES,VOLTS,
l RESIST

C
cit*ttitttiiiifitii‘kiiiti"kit!*iittitiititifl'fi**************t***t**fi*******

c
c Set up program for heat flux:
c
WRITE(*,10)'Begin set up for D/A to power supply:'
POWER = 1
c
c Description of input file.
c
WRITE(*,20)'A data file is created which contains the ',
l 'desired input ','voltages for the power supply. ’,
1 'Set up the file as follows:',
1 'line 1: Surface area of the heating unit (m*m)',
l ' (Divided by two for symmetrical case.)',
1 ’line 2: Total number of voltage data values (NVOLT)',
1 'line 3: Time (sec), volts',’.','.',
1 'line NVOLT+2= Time (sec), volts ',
1 'For, example, if you want to read 10 volts to the power',
1 'supply at 20 sec and then read 0 volts at 40 sec from',
1 ' the ',‘start of the experiment, enter in from line ',
1 ’2 to line NVOLT+2:',
1 'line 2: 2',
1 'line 3: 20.,10.',
1 'line 4: 40.,O.'
c
c Enter file name for input voltage data for power supply
c
IF(YN.EQ.'F'.OR.YN.EQ.'f')THEN
WRITE(*,30)'Enter input file name for power supply input:'
READ(*,1)POWERFIL
OPEN(UNIT=10,NAME=POWERFIL,STATUS='OLD’)
c
c The heat flux area is area of heated surface (divided by two for
c the symmetrical case) (m*m).
c
READ(10,*)HEAT_AREA
c
c Read the resistance of the heater in ohms.
c
READ(10,*)RESIST
c
c Read the total number of voltage D/A input values.
c

READ(10,*)NVOLT

Start DO loop for voltages. The times (sec) and voltages (volts)
are read in for the total number of voltage values.

The actual desired output voltages from the power supply are to
be read in. These values are converted to the output range of
the computer (0 to 10 volts), and then a 2:1 divider is used to
scale the voltage to the input required voltage for the

HP 6024A DC power supply (0 to 5 volts). The power supply then

0 0 0 0 0 0 0 0 O

00000000

000000

0

0

0000

SK35

scales this input to O to 60 volts output to the heater.

The voltage will commence at the designated time and the signal
will continue until changed. The dummy variable starts from '2'
because an initial zero voltage is added later. A final zero
voltage is also added to turn off the power supply at the end
of the run.

DO J = 2,NVOLT+1
READ(10,*)TIMES(J),VOLTS(J)
ENDDO
ELSE

Or, enter voltage data by keyboard.
The variables are entered in the same order as described above.

First, enter heat flux area (m*m).

WRITE(*,30)'Enter area of heated surface (divide by 2) (m*m):'
READ(*,*)HEAT_AREA

Next, enter the resistance of the heater (ohms).

WRITE(*,30)'Enter resistance of heater (ohms):'
READ(*,*)RESIST

Enter the total number of voltage D/A input values.

WRITE(*,30)'Enter total number of voltage data values:'
READ(*,*)NVOLT

Now start the DO loop for D/A voltage data. See notes above.

WRITE(*,30)'Enter time (sec) and voltage (volts):'
DO J = 2,NVOLT+1
WRITE(*,40)J-l
READ(*,*)TIMES(J),VOLTS(J)
ENDDO
ENDIF

Initialize and shutdown power supply by imposing 0.0 volts at
time equal to 0 sec and at end of run.

NVOLT - NVOLT+2
VOLTS(l) = 0.0
VOLTS(NVOLT) = 0.0

Convert voltage input to integer values for D/A;
CALL TRANS_DAC subroutine to convert volts in to binary form.

CALL TRANS_DAC(NVOLT,VOLTS,BUFE_DA_AXB)
End set up for power supply

FORMAT(A)

FORMAT(' ’,/,' ',72('*’),/,10X,A,/)
FORMAT(11X,A,A,/,6X,A,A,/,7(/,16X,A),//,llX,A,/,6X,A,A,
1 /6X,A,A,/,3(/,16X,A))

FORMAT(' ’,/,' ',5X,A)

FORMAT(' ',I3,’:')

FORMAT(' ’,F10.2,5X,F10.2,5X,I10)

RETURN

 

 

C

SK36

END

ct‘t'tt'iif'it'itittit‘ltilit'tttit!tittiitiiii'ii’ttittttit*tiiiiiiitt'kti

citIifliiiitfitiliiiitiiii*tt‘l'ii*tit'itti't't'ttfliit*itiitiiittiititiiiit

C

0000000

00

0

C

SUBROUTINE TRANS_DAC(NVOLT,VOLTS,BUFP_DA_AXB)

This subroutine first scales the input voltage to the output voltage
of the VaxII/GPX (0 to 10V), and then converts the voltage
values to the binary form required for the D/A converter, using
a calibration chart by M. Loh, 7/30/89. The chart may be
found in the data aquisition notebook in RM A-32, RCE, MSU.

IMPLICIT NONE

Define integer variables

INTEGER'Z BUPF_DA_AXB(SOO) !buffer for D/A
INTEGER.J !Dummy variable
INTEGER NVOLT !Number of D/A voltage data

Define real variables

REAL VOLTS(100) !volts(v) for D/A
REAL A !Y—intrcpt for volt/bin conv
REAL B !Slope for volt/binary convr

ct!iiiflttttt‘lfitttitiiittttttiitiiti*itttIt*tii*tiitittiitiiitittiitiittt

0000 0000000000000

0

0

Based on calibration by M. Loh, scaling factor between the
output voltage signal from the VaxII/GPX and the output signal
from the HP 6024A DC power supply was found to be 6.012559. The
voltage was found to vary linearly with binary output, over the
voltage range from 0 to 10 volts. The conversion equation is:
binary - A + B*volts, where A - 2048.2235, and B - -34.02974.

Note: 10V is the maximum output for the VaxII/GPX computer. If more
than 60 V are read in as output to the power supply, it will be
scaled to 10 V for the D/A, and a warning statement will be
shown on the screen.

A = 2048.2235
B = -34.02974
DO J = l,NVOLT

Check to see if input voltage is greater than 60.0V and set to 60.0V
if it is. (60V is maximum voltage for HP 6024A power supply.)

IF(VOLTS(J).GT.60.0)THEN

VOLTS(J)=60.0
WRITE(*,*)’WARNING: INPUT VOLTAGE GREATER THAN 60V: SET TO',

1 ' 60V!’
ENDIF

Convert to binary format
BUFF_DA_AXB(J) = A + B*VOLTS(J)
ENDDO
RETURN
END

End subroutine TRANS_DAC

 

 

237

cttiiititit‘l'kii’iititittitiiiiiitiftittitii‘kiiiti‘iii’i*ii’tittittt‘ktitiiit‘k

c!t*tittii’tii‘itifti‘i‘kitt’ttitt*t‘lftt******'**iti*t****t*t*tit*‘Iitt‘ltittt

c
SUBROUTINE TRANS_ADC(POWER,NCHANP1,SUM,VA)
c
c This subroutine converts binary output data to volts using a cali-
c bration chart by M. Loh, 7/30/89. The chart may be found in
c the data aquisition notebook in RM A-32, RCE, MSU. If the HP
c power supply is used, the voltage is scaled from approximately
c 10 to 60 volts for heat flux calculations. (0 to 10 volts is
c the range for the VaxII/GPX, while the actual voltage output
c from the power supply is 0 to 60 volts.)
c
IMPLICIT NONE
c
c Define integer variables
c
INTEGER.) !Dummy variable
INTEGER NCHANPI !Total number of channels
INTEGERPOWER !Indicates use of D/A
c
c Define real variables
c
REAL B !Slope for volt/binary convr
REAL SCALE !Scaling factor
REAL SUM(16) !Sums for averaging data
REAL VA(16) !Millivolts from A/D
c

ctttittitttiitiii*titfiitttttiii’i*iiti‘ki'lti'kiiii’i'*itiiiiiiitiii’i’iiifiii’ii'k

c
c Based on calibration by M. Loh, the voltage was found to vary linear-
C ly with binary output from 0 to 4095. The conversion equation
c is: volts - B*binary, where B = 0.00243503.
c
SCALE 8 6.52543
B = 0.00243503
DO J = l,NCHANPl
VA(J) = B’SUM(J)
IE(J.EQ.NCHANP1.AND.POWER.EO.1)VA(J) = VA(J)*SCALE
ENDDO
RETURN
END
c
c End subroutine TRANS_ADC
c

ci’*‘kt*tit‘ki’i’ti‘k*i*****i********i****iittitttti*ii‘*tiiittii’i'fii‘ktii‘kiitiit

c
c \ ___

c@@@@@@@@@@@@@@@@@@@@@@@@@@@ \ /___\ @@@@@@@@@@@@@@@@@@@@@@@@@@@@@
c ........ ... ............... / \|\___/ .............................
c!I!!!!!!!!!!!!!!!!!!!!!!! / / \ !l!!!!!!!!!!!!E!!!!!!!!!!!!!
c//////////////////////// I l @ -< I \\\\\\\\\\\\\\\\\\\\\\\\\\\
c\\\\\\\\\\\\\\\\\\\\\\\\ l | \_/ | ///////////////////////////
c!! !I!!!!!!!!!!!!!!!!!!!!! \ \ / !I!!!!!!!!!!!!!!!!!!!!!!!!!!
c.. ........................ \ \ / ........... ... .......... ..EPS

 

c@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@
c
c

ct************************t******i*****t******************************‘k'k

APPENDIX C

APPENDIX C

PARAMETER ESTIMATION RESULTS FROM EXPERIMENTAL

REPETITION S USING CURED COMPOSITE SAMPLES

This appendix contains the results from the estimation of the thermal properties.
namely the estimated effective thermal conductivity perpendicular to the fiber axis and
the estimated efl‘ective density-specific heat product. of the cured [0°]2 4 AS4/EPON 828
composite samples for each repetition with approximate initial temperatures of 25°C.
50°C. 75°C. 100°C. and 125°C in Tables C.1-C.5 respectively. The results for the es-
timated thermal properties for each repetition of the experiments using cured [0° .:|:30° .
i60°. 90°]2(sym) samples with initial temperatures of approximately 25°C. 50°C. and

75°C are given in Tables C.6-08. respectively.

238

239

Table C. 1. Estimated Effective Thermal Conductivity. k. Perpendicular to
the Fiber Axis and Density-Specific Heat. pc . of Cured [0°] AS4/EPON
828-mPDA Composites from Experiments with an Initial Temperature of
Approximately 25 ’ C.

 

 

 

 

 

 

 

 

 

 

 

 

Exper . Repet . Temperature k pc RMSb
No.a No.a Range (°C) (W/m°C) (kJ/m?°C) (°C)
1 26 - 58 0 803 1,590 0.440

2 26 - 57 0 805 1,580 0 480

3 26 - 57 0.803 1,570 0.480

4 32 - 53 0.780 1,580 0.173

1.1 5 35 - 54 0.776 1,590 0.210
6 26 - 45 0.769 1,570 0.299

7 30 — 49 0 817 1,590 0 259

8 30 - 48 0.818 1,550 0.250

9 30 - 49 0.805 1,600 0.232

1 27 - 45 0.794 1,590 0.249

2 29 - 47 0.792 1,550 0 255

3 29 - 48 0.808 1,540 0.240

1.6 4 25 - 44 0.798 1,590 0.284
5 25 - 45 0.799 1,590 0.234

6 25 - 45 0.790 1,580 0.340

1 26 - 46 0.767 1,560 0.190

2 26 - 46 0.748 1,560 0.146

3 27 - 47 0.742 1,580 0.138

1.11

4 26 - 46 0.730 1,560 0.127

5 26 - 46 0.737 1,550 0.130

6 26 - 46 0.744 1,540 0.147

a. Experiment and repetition numbers refer to those in Table 4.2.

b. Root mean squared error of calculated versus experimental temperatures.

240

Table C.2. Estimated Effective Thermal Conductivity. k. Perpendicular to
the Fiber AXIS and Density-Specific Heat. pc . of Cured [0°] AS4/EPON
828-mPDA Composites from Experiments with an Initial Temperature of
Approximately 50 ° C.

 

 

 

 

Exper . Repet . Temperature k pcp RMSb
No.a No.a Range (°C) (W/m°C) (kJ/m?°C) (°C)
1 52 - 76 0.797 1,650 0.334

2 53 - 76 0.789 1,640 0.363

1.2 3 52 - 75 0.783 1,670 0.431
4 54 - 74 0.794 1,620 0.202

5 54 - 74 0.795 1,650 0.240

6 54 - 74 0.797 1,660 0.252

1 50 - 67 0.855 1,700 0.242

2 50 - 67 0.868 1,660 0 223

3 50 - 67 0.866 1,660 0.209

4 50 - 67 0.877 1,740 0.575

5 50 - 67 o 870 1,740 0.336

1.7 6 50 - 67 0.868 1,680 0.222
7 52 - 70 0.850 1,620 0.145

8 52 - 69 0.897 1,670 0.198

9 49 - 66 0.852 1,680 0 202

10 50 — 68 0.875 1,640 0.203

11 51 - 68 0.847 1,640 0.159

12 51 - 68 0.895 1,710 0.229

1 52 - 71 0.775 1,620 0.117

2 52 - 71 0.773 1,620 0.108

1.12 3 53 - 73 0.797 1,610 0.168
‘ 4 53 - 72 0.805 1,620 0.205

5 52 - 71 0 818 1,550 0.208

6 52 - 71 0.771 1,610 0.141

 

 

 

 

 

 

 

 

a. Experiment and repetition numbers refer to those in Table 4.2.
b. Root mean squared error of calculated versus experimental temperatures.

241

Table 03. Estimated Effective Thermal Conductivity. k. Perpendicular to
the Fiber Axis and Density-Specific Heat. pc. of Cured [0°] AS4/EPON
828-mPDA Composites from Experiments with an Initial Temperature of
Approximately 75 ° C.

 

 

___

a!

 

 

Exper. Repet. Temperature k pcp RMSb
No.a No.a Range (°C) (W/m°C) (kJ/m?°ci (°C)
1 78 - 99 0.796 ' 1,670 0.408
2 78 - 99 0.801 1,680 0.420
3 76 - 98 0.809 1,680 0.414
1.3 4 78 - 98 0.815 1,720 0.323
5 78 — 97 0.827 1,710 0.323
6 78 - 97 0.811 1,700 0.356
1 75 - 92 0.924 1,830 0.220
2 75 - 91 0.900 1,820 0.203
3 75 - 91 0.926 1,760 0.209
1.8 4 75 - 91 0 919 1,820 0 211
5 75 - 91 0.915 1,830 0.208
6 75 - 91 0.922 1,820 0.209
1 77 - 94 0.825 1,780 0.126
2 75 - 93 0.811 1,740 0.146
3 77 - 95 0.802 1,760 0.206
1.13
4 77 - 95 0.782 1,770 0.190
5 77 - 95 0.779 1,760 0.139
6 77 - 95 0 783 1,760 0.129

 

 

 

 

 

 

 

 

a. Experiment and repetition numbers refer to those in Table 4.2.
b. Root mean squared error of calculated versus experimental temperatures.

 

242

Table C.4. Estimated Effective Thermal Conductivity. k. Perpendicular to
the Fiber Axis and Density-Specific Heat. pc . of Cured [0°] AS4/EPON
828-mPDA Composites from Experiments with an Initial Temperature of
Approximately 100°C.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Exper. Repet. Temperature k pc RMSb

No.a No.a Range (°C) (W/m°C) (kJ/m", °C) (°C)
1 102 - 119 0.812 1,780 0.289 ,

2 102 - 122 0.844 1,910 0.303

3 102 - 123 0.844 1,900 0.312

4 102 - 120 0.830 1,980 0.237

5 102 - 120 0.829 2,020 0.274

1.4 6 102 - 120 0.847 2,000 0.241

7 101 - 119 0.784 1,850 0.330

8 103 - 119 0.804 1,830 0.282

9 103 - 120 0.822 1,840 0.258

10 102 - 119 0.841 1,830 0.300

11 102 - 118 0.876 1,910 0.249

12 102 - 118 0.847 1,950 0.555

1 100 - 115 0.969 1,950 0.178

2 100 - 115 0.959 2,000 0.178

1.9 3 100 - 115 0.963 1,930 0.187

4 100 - 115 0.979 1,960 0.195

5 100 - 115 0.961 1,960 0.179

6 100 - 115 0.996 1,970 0.183

1 100 - 115 0.806 1,850 0.112

2 100 - 115 0.808 1,850 0.115

1.14 3 100 - 115 0.815 1,850 0.116

4 100 - 115 0.816 1,810 0.142

5 100 - 115 0.825 1,850 0.146

6 100 - 115 0.824 1,790 0.145

a. Experiment and repetition numbers refer to those in Table 4.2.

b. Root mean squared error of calculated versus experimental temperatures.

243

Table C.5. Estimated Etiective Thermal Conductivity. k. Perpendicular to
the Fiber Axis and Density-Specific Heat. pc . of Cured [0°] AS4/EPON
828-mPDA Composites from Experiments with an Initial Temperature of
Apprwdmately 125°C.

 

 

 

 

 

 

 

 

 

 

 

 

Exper. Repet. Temperature k pc RMSb
No.‘3 No.3 Range (°C) (W/m°C) (kJ/m‘; °C) (°C)
1 126 - 145 0.814 2,150 0.447
2 126 - 145 0.820 2,220 0.481
3 126 - 145 0.834 2,170 0.436
4 127 - 144 0.870 1,980 0.465
1.5 5 127 - 145 0.894 2,000 0.454
6 127 - 145 0.874 1,940 0.398
7 128 — 144 0.965 1,960 0.297
8 128 — 144 0.911 2,000 0.288
9 128 - 144 0.934 1,990 0.239
1 126 - 141 1.006 2,070 0.189
2 126 - 141 0.998 2,130 0.156
3 126 - 141 1.004 2,110 0.163
1..10
4 126 - 141 1.013 2,110 0.157
5 126 - 140 1.014 2,090 0.167
6 126 - 140 1.028 2,050 0.206
1 126 - 140 0.862 1,990 0.093
2 126 - 140 0.880 1,910 0.104
3 126 - 140 0.870 1,930 0.121
1.15
4 126 - 140 0.875 1,920 0.100
5 126 - 140 0.874 1,940 0.099
6 126 - 140 0.881 1,920 0.104
a. Experiment and repetition numbers refer to those in Ta: . 4.2.

b. Root mean squared error of calculated versus experimental temperatures.

244

Table C.6. Estimated Effective Thermal Conductivity. k. Perpendicular to
the Fiber Axis and Density-Specific Heat. pcp. of Cured
[0°.130°.160°.90°] AS4/EPON 828-mPDA Composites from

2(syni)

Experiments with an Initial Temperature of Approximately 25° C.

 

 

 

Exper. Repet. Temperature k pcp RMSb
No.a :40." Range (°C) (W/m°C) (kJ/m’ °C) (°C)
1 25 - 32 0.697 1,780 0.922
2 25 - 32 0.690 1,800 0.991
3 25 - 32 0.667 1,720 0.950
4 25 - 37 0.660 1,680 0.767
5 25 - 45 0.685 1,830 0.897
1.16 6 25 - 42 0.648 1,670 0.637
7 28 - 45 0.716 1,680 0.166
8 28 - 45 0.727 1,680 0.168
9 28 - 49 0.659 1,520 0.176
10 29 - 49 0.696 1,560 0.183
11 29 - 48 0.713 1,590 0.200
1 26 - 49 0.716 1,540 0.246
2 29 - 53 0.754 1,610 0.194
3 25 - 48 0.765 1,610 0.268
1.19
4 . 25 - 49 0.750 1,610 0.250
5 31 - 55 0.813 1,640 0.247
6 30 - 53 0.800 1,630 0.206

 

 

 

 

 

 

 

 

a. Experiment and repetition numbers refer to those in Table 4.2.
b. Root mean squared error of calculated versus experimental temperatures.

245

Table C.7. Estimated Effective Thermal Conductivity. k. Perpendicular to the
Fiber Axis and Density-Specific Heat. pCp. of Cured [O°.:t30°.:t60°.90°]2 H”)
AS4/EPON 828-mPDA Composites from Experiments with an Initial Temperature
of Approximately 50 ° C.

 

 

 

Exper. Repet. Temperature k pcP RMSb
No.a No.a Range (°C) (W/m°C) (kJ/m?°C) (°C)
1 52 - 76 0.723 1,660 0.134
2 53 - 76 0.708 1,640 0.183
3 52 - 75 0.709 1,620 0.176
1.17
4 54 - 74 0.724 1,650 0.302
5 54 — 74 0.726 1,630 0.165
6 54 - 74 0.709 1,690 0.413
1 50 - 67 0.809 1,710 0.326
2 50 - 67 0.802 1,700 0.383
3 50 - 67 0.805 1,700 0.352
1.20
4 50 - 67 0.767 1,680 0.403
5 50 - 67 0.766 1,680 0.353
6 50 - 67 0.766 1,640 0.222

 

 

 

 

 

 

 

 

a. Experiment and repetition numbers refer to those in Table 4.2.
b. Root mean squared error of calculated versus experimental temperatures.

246

Table C.8. Estimated Effective Thermal Conductivity. k. Perpendicular to the
Fiber Axis and Density-Specific Heat. pop. of Cured [O°.i30°.160°.90°]2”ym
AS4/EPON 828-mPDA Composites from Experiments with an Initial Temperature
of Approximately 75 ° C.

 

 

 

 

 

 

 

 

 

 

 

 

Exper. Repet. Temperature k pc RMSb
P
No.a No. Range (°C) (W/m°C) (kJ/m?°C) (°C) '
1 78 — 96 0.735 ' 1,740 0.151
2 78 - 96 0.743 1,730 0.153
3 78 - 97 0.738 1,730 0.177
1.18
4 78 - 96 0.747 1,700 0.155
5 78 - 97 0.759 1,680 0.194
6 78 - 97 0.745 1,720 0.151
1 76 - 98 0.721 1,750 0.415
2 77 - 96 0.884 2,240 0.623
3 76 - 97 0.757 1,720 0.490
1.21
4 76 - 96 0.744 1,740 0.423
5 76 - 97 0.748 1,740 0.432
6 76 - 97 0.749 1,730 0.426
a. Experiment and repetition numbers refer to those in Table 4.2.

b. Root mean squared error of calculated versus experimental temperatures.

 

 

BBLIOGRAPHY

 

BIBLIOGRAPHY

Acitelli. M. A.. R. B. Prime. and E. Sacher. 1971. "Kinetics of Epoxy Cure: (1) The
System Bisphenol-A Diglycidyl Ether/m-phenylene Diamime." Polymer. 12:445-
463. ' .

Alifanov. O. M. and N. V. Kernev. 1981. "Determination of External Thermal and Load
Parameters by Solving the Two-Dimensional Inverse Heat-Conduction Problem."
translated from Inzhenerno-Fizicheskii Zhurnal, 1981. 41(4):581-586. Journal of
Engineering Physics. 41(4): 1049-1053. .

Al'ifanov. O. M. and V. V. Mikhailov. 1979. "Solution of the Nonlinear Inverse Thermal

' Conductivity Problem by the Iteration Method." translated from Inzhenerno-

Fizicheskii Zhurnal. 1978. 35(6):1123-1129. Journal of Engineering Physics.
35(6): 1 50 l - l 506.

Alifanov. O. M. and S. V. Rumyantsev. 1987. "Formulas for the Discrepancy Gradient in
the Iterative Solution of Inverse Heat-Conduction Problems. II. Determining the
Gradient in Terms of a Conjugate Variable." translated from Inzhenerno-Fizicheskii
Zhurnal. 1987 , 52(4):668-675. Journal of Engineering Physics. 52(4):489-495.

Barton. J. M.. 1985. 'The Application of Differential Scanning Calorimetry (DSC) to the
Study of Epoxy Resin Curing Reactions." Advances in Polymer Science 72 Epoxy
Resins and Composites 1. Springer-Verlag. New York. pp. 1 1 1-154.

Beck. J. V.. 1966. "Iransient Determination of Thermal Properties." Nuclear
Engineering and Design. 32373-381.

Beck. J. V.. 1987. "PROPID". Parameter Estimation Computer Program. Michigan State
University. East Lansing. MI 48824

Beck. J. V.. 1989. "PROPIDMA". Parameter Estimation Computer Program (revised
version). Michigan State University. East Lansing. MI 48824

Beck. J. V. and S. Al-Araji. 1974. "Investigation of a New Simple Transient Method of
Thermal Property Measurement." Journal of Heat Transfer. 96(1):59-64.

Beck. J. V. and K. J. Arnold. 1977. Parameter Estimation in Engineering and Science.
John Wiley & Sons. New York.

Borchardt. J. and F. Daniels. 1957. "The Application of Differential Thermal Analysis to
the Study of Reaction Kinetics." Journal of American Chemical Society. 79:41-46.

Boyer. G. T. and W. C. Thomas. 1985. "An Analytical Investigation of Chan-ing
Composite Undergoing Thermochemical Expansion," American Society of
Engineers Paper N o. 85-HT-54.

Brennan. J. J.. L. D. Bentsen, D. P. H. Hasselman. 1982. "Determination of the
Thermal Conductivity and Diffusivity of Thin Fibres by the Composite Method."
Journal of Materials Science. 17:2337-2342.

247

248

Chem. C.-S.. and G. W. Poehlein. 1987. "A Kinetic Model for Curing Reactions of
Epoxides with Amines." Polymer Engineering and Science. 27(1 1):788-795.

Courville and J. V. Beck. 1988. "Measurement of Field Thermal Performance
Parameters of Building Envelope Components." ASHREA Transactions. 94(Part
2):1595-1612.

Digital Equipment Corporation. 1986. VAXlab/LabStar Programmers Guide. AA-HXO4B-
TE. Digital Equipment Corporation. MA.

Dul’nev. G. N.. M. A. Eremeev. Y. P. Zarichnyak. and E. N. Koltunova. 1977. "A
Combined Numerical Method for Determining the Conductance of Composite
Bodies." translated from Inzhenerno-Fizicheskii Zhumal. 1977. 32(2):284—291.
Journal of Engineering Physics. 32(2): 174-180.

DuPont Company Instrument Systems. 1985, 910 Ditferential Scanning Calorimeter
Operator’s Manual. DuPont Company Instrument Systems. Wilmington. DE
19898.

Ectron Corporation. 1982. Insb'uction Manual 687 DC Amplifier and Options A/O.
Ectron Corporation. San Diego. CA.

Farnia. K.. 1976. Computer-Assisted Experimental and Analytical Study of Time /
Temperature-Dependent Thermal Properties of the Aluminum Alloy 2024-7351.
Ph.D. Dissertation. Department of Mechanical Engineering. Michigan State
University. East Lansing. MI 48824.

Fava. R. A.. 1968. "Differential Scanning Calorimetry of Epoxy Resins." Polymer. 9: 137-
151.

Golovchan. V. T.. and A. G. Artemenko. 1987. "Heat Conduction of Orthogonally
Reinforced Composite Materials." translated from Inzhenemo-Fizicheskii Zhumal.
51(2):260-297. 1986. Plenum Publishing Corporation.

Griifis. C. A. R. A Masumura. and C. I. Chang. 1981, "Thermal Response of Graphite
Epoxy Composite Subjected to Rapid Heating." Journal of Composite Materials.
15:427-442.

Hagnauer. G. L.. B. R LaLiberte. and D. A. Dunn. 1983. "Isothermal Cure Kinetics of an
Epoxy Resin Prepreg." Epoxy Resin Chemistry II. R. S. Bauer. ed.. American
Chemical Society Symposium Series 221. pp. 229-244.

Han. L. S. and A. A. Cosner. 1981. "Effective Thermal Conductivities of Fibrous
Composites." Journal of Heat Transfer. lO3(5):387-392.

Harris. J. P.. B. Yates. J. Batchelor. and P. J. Garrington. 1982. "The Thermal
Conductivity of Kevlar Fibre-Reinforced Composites." Journal of Materials Science.
17(9):2925-2931.

Hasselman. D. P. H. and L. F. Johnson. 1987. "Effective Thermal Conductivity of
Composites with Interfacial Thermal Barrier Resistance. Journal of Composite
Materials." 21:508-515.

Hawley. M. C.. L. T. Dizal, A. Asmussen. Jr.. and J. V. Beck. 1988. "Study of
Interface/Interphase in Thick Section Composites Annual Technical Report 1986-
1988." submitted to: University of Illinois / Office of Naval Research.

Henderson. J. B. and J. A. Wiebelt. 1987. "A Mathematical Model to Predict the Thermal
Response of Decomposing. Expanding Polymer Composites." Journal of Composite
Materials. 2 1:373-393.

249

Henderson. J. B.. J. A. Wiebelt. and M. R. Tant. 1985. "A Model for the Thermal
Response of Polymer Composite Materials with Experimental Verification." Journal
of Composite Materials. 19:579-595.

Huag. E. J. and Arora. J. S.. 1979. Applied Optimal Design. John Wiley and Sons. NY.

Ishikawa. T.. 1980. "Analysis and Experiments on Thermal Conductivities of
Unidirectional Fiber-Reinforced Composites." Heat Transfer - Japanese Research.
9(3):41-53.

James. B. W.. G. H. Wostenholm. G. S. Keen and S. D. Mclvor. 1987. "Prediction and
Measurement of the Thermal Conductivity of Composite Materials." Journal of
Physics: D: Applied Physics. 20(3):261-268.

Kama]. M. R. S. Sourour. and M. Ryan. 1973." SPE 3lst Annual Technical Conference
Proceedings." pp. 187. In Barton. J. M.. 1985. 'The Application of Differential
Scanning Calorimetry (DSC) to the Study of Epoxy Resin Curing Reactions."
Advances in Polymer Science 72 Epoxy Resins and Composites 1. Springer-Verlag.
New York. pp. 1 1 1-154.

Lamm. P.. 1989. "Notes on the Adj oint Method for Estimating Functions." Personal com-
munication.

Lee. H. J. and R E. Taylor. 1975. "Thermophysical Properties of Carbon/Graphite
Fibers and mod-3 Fiber-Reinforced Graphite." Carbon. 13:521-527.

Lee. W. I.. A C. Loos. and G. S. Springer. 1982. "Heat of Reaction. Degree of Cure. and
Viscosity of Hercules 3501-6 Resin." Journal of Composite Materials. 16:510-520.

Loh. M.. 1989. "Two-Dimensional Heat Transfer Studies in Carbon Composite
Materials." Diplomarbeit. Michigan State University. East Lansing. MI: Rheinisch-
Westfaelisch Techische Hochschule Aachen. Aachen. FRG.

Loos. A. C.. and G. S. Springer. 1983. "Curing of Epoxy Matrix Composites." Journal of
Composite Materials. 17(3): 135- 169.

Mijovic. J .. 1986. "Cure Kinetics of Neat Versus Reinforced Epoxies." Journal of Applied
Polymer Science. 3 1:1 177- l 187.

Mijovit. J.. J. Kim. and J. Slaby. 1984. "Cure Kinetics of Epoxy Formulations of the
Type used in Advanced Composites." Journal of Applied Polymer Science. 29: 1449-
1462.

Mijovic. J.. and H. C. Mei. 1987. "An Apparatus for Measurement of Thermal
Conductivity of Neat Epoxies and their Composites." Polymer Composites. 8(1):53-
56.

Mijovit. J. and H. T. Wang. 1988. "Modeling of Processing of Composites Part II -
Temperature Distribution during Cure." SAMPLE Journal. March/April. pp. 42-55.
191.

Mijovic. J. and H. T. Wang. 1989. "Cure Kinetics of Neat and Graphite-Fiber-Reinforced
Epoxy Formulations." Journal of Applied Polymer Science. 37 :2661-2673.

Moroni. A.. J. Mijovi'. E. M. Pearce. and C. C. Faun. 1986. "Cure Kinetics of Epoxy
Resins and Aromatic Diamines." Journal of Applied Polymer Science. 32:3761-
3773.

250

Mottram. J. T. and R Taylor. 1987a. "Thermal Conductivity of Fibre-Phenolic Resin
Composites. Part I: Thermal Difl'usivity Measurements." Composites Science and
Technology 29:189-209.

Mottram. J. T. and R Taylor. 1987b. ”Thermal Conductivity of Fibre-Phenolic Resin
Composites. Part 11: Numerical Eyaluation." Composites Science and Technologr
29:21 1-232.

Osman. A. M.. 1987. Estimation of Transient Heat 'D‘ansfer Coefficients in Multi-
Dimensional Problems by Using Inverse Heat TI'ansfer Methods. Ph.D. Dissertation.
Department of Mechanical Engineering. Michigan State University. East Lansing.
MI 48824.

Pappalardo. L. T.. 1977. "DSC Evaluation of Epoxy and Polyimide-Impregnated
Laminates (Prepregs)." Journal of Applied Polymer Science. 21:809-820.

Parker. W. J.. R J. Jenkins. C. P. Butler. and G. L. Abbott. 1961. "Flash Method of
Determining Thermal Difi‘usivity. Heat Capacity. and Thermal Conductivity."
Journal of Applied Physics. 32(9):1679-1684.

Prime. R B.. 1970. Analytical Calorimetry. 2:201. in Barton. J. M.. 1985. "The
Application of Difl'erential Scanning Calorimetry (DSC) to the Study of Epoxy Resin
Curing Reactions." Advances in Polymer Science 72 Epoxy Resins and Composites
1. Springer-Verlag. New York. pp. 1 1 1-154.

Prime. R B.. 1973. "Difl'erential Scanning Calorimetry of the Epoxy Cure Reaction."
Polymer Engineering Science. 13:365-371.
PLOTitR

Rabek. J. F.. 1980. Experimental Methods in Polymer Chemistry: Physical Principals and
Applications. John Wiley and Sons. NY.

. 1989. Scientific Programming Enterprises. Haslett. MI 48840.

Rich. M.. 1987. Personal Communication.

Ryan. M. E. and A. Dutta. 1979. "Kinetics of Epoxy Cure: a Rapid Technique for Kinetic
Parameter Estimation. Polymer. 20(2):203-206.

Scientific Programming Enterprises. 1987. "Plotl'I‘R Reference Manual". Scientific
Programming Enterprises. Haslett. MI 48840.

Schechter. L.. J. Wynstra. and R P. Kurkjy. 1956. "Glycidyl Ether Reactions with
Amines." Industrial and Engineering Chemistry. 48:94-97.

Schimmel. Jr. W. P.. J. V. Beck. and A B. Donaldson. 1977. "Effective Thermal
Diffusivity for a Multimaterial Composite Laminate." Journal of Heat TYansfer.
99:466-470.

Sichina. W. J .. "Autocatalyzed Epoxy Cure Predictions using Isothermal DCS Kinetics."
DuPont Application Brief No. TA-93.

Sichina. W. J.. 1989. "Efl‘ects of Annealing on the Observed Tg of a Glassy Polymer."
The TA Hotline. DuPont Company Instrument Systems. Fall.

Smith. I. T.. 1961. 'The Mechanism of the Crosslinking of Epoxide Resins by Amines."
Polymer. 2:95-108.

Sourour. S. and M. R Kama]. 1976. "Differential Scanning Calorimetry of Epoxy cure:
Isothermal Cure Kinetics." Thermochemica Acta. 14:41-59.

251

Springer. G. S. and S. W. Tsai. 1967. "Thermal Conductivities of Unidirectional
Materials." Journal of Composite Materials. 1: 166-173.

Tant. M. R. J. B. Henderson. and C. T. Boyer. 1985. "Measurement and Modelling of
the Thermochemical Expansion of Polymer Composites." Composites. 16(2):121-
126.

Taylor. R E. and B. H. Kelsic. 1986. "Parameters Governing Thermal Diffusivity
Measurements of Unidirectional Fiber-Reinforced Composites." Journal of Heat
Transfer. 108:161-165.

Tortorelli. D. A.. R B. Haber. and S. C-Y. Lu. 1989. "Design Sensitivity Analysis for
Nonlinear Thermal Systems." accepted for publication in Computer Methods in
Applied Mechanics and Engineering.

Tu. J. S.. 1988. Solution of General Tim-Dimensional Inverse Heat Conduction Problems
and OneDimensional Inverse Melting Problems. Ph.D. Dissertation. Department of
Mechanical Engineering. Michigan State University, East Lansing. MI 48824.

Vinson. J. R and R L. Sieakowski. 1987. The Behavior of Structures Composed of
Composite Materials. Martinus Nijhoff Publishers. Hinghan. MA.

Walpole. R. E. and R H. Myers. 1978. Probability and Statistics for Engineers and
Scientists. 2nd Edition. Macmillan Publishing Co. NY.

Waterbury. M. 1988. "Optimal Numerical Volumetric Analysis". computer software.
Michigan State University. East Lansing. MI 48824.

Ziebland. H.. 1977. 'The Thermal and Electrical Transmission Properties of Polymer
Composites." in Chapter 7. Polymer Engineering Composites. M. O. W. Richardson.
ed.. Applied Science Publishers. LTD.. London.

7111111111