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DESIGN AND VALIDATION OF OPTIMAL EXPERIMENTS FOR ESTIMATING
THERMAL PROPERTIES OF COMPOSITE MATERIALS

By

Ramsis Taktak

A THESIS

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

DOCTOR OF PHILOSOPHY

Department of Mechanical Engineering

1992

/’ "“.~
6/65." (95":

ABSTRACT

DESIGN AND VALIDATION OF OPTIMAL EXPERIMENTS FOR ESTIMATING
THERMAL PROPERTIES OF COMPOSITE MATERIALS

By

Ramsis Taktak

Composite materials have gained an unprecedented interest in the last twenty
years. Their superior strength-to-weight ratios have made them very popular with
aerospace, automotive, boat, biomedical, and even the sporting goods industry. The radar-
absorbing quality of composite materials, together with their strength-to-weight ratios,
make these materials attractive for military applications.

When used in air and space vehicles, composite materials are exposed to harsh
thermal loads. An understanding of the thermal behavior of these materials became
necessary. The thrust of this work is the estimation of two thermal parameters: thermal
conductivity and volumetric heat capacity. There are three main parts to the current
research; each part describes a different aspect of estimating these thermal properties.

The first part compares two experimental research paradigms pertaining to the
effect of the temperature rise on the estimated parameters during transient parameter
estimation experiments.

The second part of this research relates to optimal experiments. Two one-
dimensional experiments for finite and semi-infinite geometries are shown to be superior
to some previously published ones. The two experimental parameters of heating time and

cooling time are especially of interest for these experiments. An optimal value is

obtained for each parameter using a selected optimality criterion. A two-dimensional case
is studied following the same methodology. Design curves relating the optimality
criterion to the geometry and directional properties of the material of interest are obtained.
The analytical results of the optimal experiments pertaining to the finite one-dimensional
geometry are verified using an experimental technique developed by Garnier and Beck.

In the third part of this research, the thermocouple-induced errors, which are
usually small quantities, are quantified using an analytical approach. Finite difference and
finite element methods are supplemented because of their limitations in calculating small
differences. Some experiments are then designed and implemented in order to verify
these analytical results.

The results of the three different parts of this research have great potential for

improving the estimation of composite material thermal properties.

to my parents, Lamjed and Saida Taktak

iv

ACKNOWLEDGEMENTS

I owe a debt of thanks to his major professor, J. V. Beck, for his guidance,
support, and encouragement, and especially for the opportunity to learn from his example
as a researcher, an educator, and an engineer.

Grateful acknowledgement is extended to the other members of the guidance
committee, A. Atreya, C. R. MacCluer, and M. C. Potter.

I would like to thank C. R. MacCluer for his interest and help in understanding
some concepts relating to the current research, A. M. Osman for the fruitful, challenging
discussions, and input throughout this research, E. P. Scott for her meticulous
explanations of different lab procedures, M. Rich for his patience and guidance in the
preparation of different samples, J. Wille for the numerous computer lessons and tips, D.
Delaunay for his support during the stay in France, and S. Herr, M. Loh, and B. Garnier
for the times shared together in the laboratory.

Special thanks are extended to the Laboratoire de Thermocinetique-ISITEM
N antes-France for the opportunity to perform some experiments and to be exposed to a
different approach of conducting experiments.

I am also grateful for the support of the Research Excellence Fund of the State of

Michigan, University of Illinois/Office of Naval Research through Grant No. 86-161 and

the National Science Foundation through Grant No. CBT-88B263.
Last (but by no way least), I am thankful for my family who has given me love,

support, and encouragement throughout my life.

vi

TABLE OF CONTENTS

LIST OF TABLES ............................................ x
LIST OF FIGURES ............................................ xi
LIST OF SYMBOLS .......................................... xiv
CHAPTER 1. INTRODUCTION ............................... 1
CHAPTER 2. BACKGROUND .................................. 4
2.1 Introduction .......................................... 4

2.1.1 Motivation .................................... 4

2.1.2 Meaning of Effective Thermal Properties ............... 5

2.1.3 Carbon Fiber/Epoxy Material ....................... 6

2.1.4 Boundary Conditions ............................. 7

2.1.5 Outline of the Remainder of the Chapter ............... 8

2.2 Description of the Experiment ............................. 8

2.2.1 Experimental Setup .............................. 8

2.2.1a Fabrication of the Specimens ................. 9

2.2.1b Instrumentation of the Specimens .............. 12

2.2.2 Some Experimental Results ............... i: . .' ...... 13

2.3 Description of PROPlD ................................. 23

2.4 PROPlD Experimental Results .................... , ........ 28

vii

2.4.1 Large Pulse Experiments .......................... 28

2.4.2 Small Pulse Experiments .......................... 30
CHAPTER 3 OPTIMAL EXPERIMENTS .......................... 37
3.1 Introduction and Literature Review ......................... 37

3.2 Theoretical Procedure ................................... __ 38
3.2.1 Finite One-Dimensional Composite (X21B50TO) (Case 1) . . . 40

3.2.2 Semi-infinite One-Dimensional Composite (X2OB5TO) (Case

2) ......................................... 47

3.2.3 Finite Two-Dimensional Composite (X22BOOY12BOx5T0)
(Case 3) ..................................... 51
3.4 Results and Discussion .................................. 56
3.4.1 Finite One-Dimensional Composite (X21B50TO) (Case 1) . . . 62

3.4.2 Semi-Infinite One-Dimensional Composite (X20B5TO) (Case
2) ......................................... 65
3.4.3 Comparison of One-Dimensional Results ............... 65
3.4.4 Finite Two-Dimensional Composite ................... 67
CHAPTER 4 THERMOCOUPLE ERROR ANALYSIS ................. 69
4.1 Introduction and Literature Review ......................... 69
4.2 Theoretical Procedure ................................... 73
4.2.1 General Formulation ............................. 73
4.2.2 Green’s Function Approach ........................ 76

4.2.3 The Unsteady Surface Element Method Versus Inverse
Convolution .................................. 83
4.4 Results and Discussion .................................. 85

viii

Cl

CHAPTER 5 EXPERIMENTAL INVESTIGATION OF THE

THERMOCOUPLE ERROR ANALYSIS ...................... 88
5.1 Introduction .......................................... 88
5.2 Some Design Criteria of the Experiment ...................... 89
5.3 Experimental Procedure ................................. 91
5.4 Experimental Setup .................................... 94
5.5 Experimental Results ................................... 98

5.5.1 Steady State Experiment .......................... 98

5.5.2 Transient Experiments ............................ 99

5.5.3 Calculation of the Surface Heat Flux and the Thermocouple

Disturbance ................................. 104
5.5.4 Comparison of the Experimental and Analytical Temperature
Disturbances ................................. 107
5.5.5 Estimation of Thermocouple Locations ............... 108
5.5.6 Sensitivity Analysis ............................. 110
5.6 Discussion of the Results ............................... 113
CHAPTER 6 EXPERIMENTAL VERIFICATION OF OPTIMAL
EXPERIMENTS ....................................... 115
6.1 Introduction ......................................... 115
6.2 Experimental Setup ................................... 116
6.3 Experimental Results .................................. 118
CHAPTER 7 SUMMARY AND CONCLUSIONS .................... 130
BIBLIOGRAPHY ........................................... 135

ix

LIST OF TABLES

Table 2.1 Some High Heat Pulse Experimental Temperature Data for the Surface
Thermocouple

Table 2.2 Some Corrected High Heat Pulse Experimental Temperature Data and
the Corresponding Standard Deviation for the Surface Thermocouple

Table 2.3 Some High Heat Pulse Experimental Temperature Data for the Internal
Thermocouple

Table 2.4 Some Corrected High Heat Pulse Experimental Temperature Data and
Corresponding Standard Deviation for the Internal Thermocouple

Table 2.5 Calculated Confidence Intervals for the Large Pulse Experiments
Table 2.6 Calculated Confidence Intervals for the Small Pulse Experiments
Table 3.1 Comparison of the Maximum Determinant, D, Values for 7 Cases.
Table 5.1 Nominal Versus Estimated Thermocouple Locations

Table 5.2 Sensitivity of Temperature Disturbance to a 10% Error in Thermocouple
Location

Table 5.3 Sensitivity of Temperature Disturbance to a 2.4% Error in Thermal
Conductivity

Table 5.4 Sensitivity of Temperature Disturbance to a 2% Error in Volumetric
Heat Capacity

Table 6.1 Experimental Parameters of the Four Experiments Performed for the
Investigation of Optimal One-Dimensional Designs

Table 6.2 Comparison of the Results of the Four Optimal One-Dimensional
Experiments

19

19

20

20
32

32

66

110

112

112

113

119

128

LIST OF FIGURES

Figure 2.1 Placement of the Specimens with the Heater and Aluminum Blocks

Figure 2.2 Experimental Transient Heat Flux and Temperatures (at two
thermocouples) on DER332/DDS Epoxy for Large Heat Pulse

Figure 2.3 Experimental Transient Heat Flux and Temperatures (at two
thermocouples) on DER332/DDS Epoxy for Small Heat Pulse

Figure 2.4 Transient Residuals Calculated by PROPID for DER332/DDS Epoxy
for Large Heat Pulse (at surface thermocouple)

Figure 2.5 Estimated Thermal Conductivity of DER332/DDS Epoxy versus
Temperature for High and Low Heat Pulse Heating Regimes

Figure 2.6 Transient Residuals Calculated by PROPlD for DER332/DDS Epoxy
for Small Heat Pulse (at surface thermocouple)

Figure 3.1 Finite One-Dimensional Geometry of the Finite X12B05T0 Case
Figure 3.2 Dimensionless Temperature Solution for the Finite X21B50T0 Case

Figure 3.3 Dimensionless Thermal Conductivity Sensitivity Coefficients for the
Finite X21B50TO Case

Figure 3.4 Dimensionless Volumetric Heat Capacity Sensitivity Coefficients for
the Finite X21B50TO Case

Figure 3.5 Determinant, D, for Different Heating Times for the Finite X21B50T0
Case

Figure 3.6 Determinant, D, at Different Sensor Locations for the Finite X21B50T0
Case

Figure 3.7 Determinant, D, for Different Heating Times for the Semi-Infinite

xi

15

22

29

31

34

41

44

46

46

48

48

Fi;

X20B5'I‘0 Case
Figure 3.8 Two-Dimensional Finite Geometry for the X22BO0Y12B0x5T0 Case

Figure 3.9 Dimensionless Temperature Solution for the Two-Dimensional
X22BO0Y12B0x5T0 Case

Figure 3.10 X-axis Directional Thermal Conductivity Sensitivity Coefficient for
the X22BO0Y12B0x5T0 Case

Figure 3.11 Y-axis Directional Thermal Conductivity Sensitivity Coefficient for
the X22BO0Y12B0x5T0 Case

Figure 3.12 Determinant, D, for Different Heating Times for the Finite
X22BO0Y12B0x5T0 Case

Figure 3.13 Determinant, D, for Different Heated Surface Thermocouple
Combinations for the X22800Y12B0x5T0 Case

Figure 3.14 Global Design Curves for the X22BO0Y12BOx5T0 Case
Figure 4.1 Thermocouple Placement Relative to the Heated Surface
Figure 4.2 Theoretical, FE, and FD Thermocouple Disturbances
Figure 4.3 Normalized Theoretical Thermocouple Disturbance

Figure 4.4 Kernel K(t-t) for Different Thermocouple-to-Depth Ratios

Figure 5.1 Thermocouple Numbering for the Estimation of Thermocouple
Disturbance

Figure 5.2 Mold and Different Methods Used in Stretching the Thermocouples
Figure 5.3 Thermocouple Layout in the Mold
Figure 5.4 Experimental Setup for the Estimation of Thermocouple Disturbance

Figure 5.5 Placement of the Epoxy Specimen for the Estimation of Thermocouple
Disturbance

Figure 5 .6 Measured Temperatures of Experiment 2

Figure 5.7 Measured Temperatures of Experiment 3

xii

50

52

54

57

58

59

6O

61

71

82

84

86

90

92

93

95

96

101

102

 

 

Fit
\

a,

F

Figure 5.8 Measured Temperatures of Experiment 4
Figure 5.9 Estimated Surface Heat Flux Using II-ICPlD

Figure 6.1 Experimental Setup for the Investigation of Optimal One-Dimensional
Designs

Figure 6.2 Sensitivity Coefficients for the First Set of Optimal One-Dimensional
Experiments

Figure 6.3 Sensitivity Coefficients for the Second Set of Optimal One-
Dimensional Experiments

Figure 6.4 Residuals of the Four Experiments Performed to Investigate Optimal
One-Dimensional Designs

Figure 6.5 Transient Thermal Conductivity Corresponding to the First Set of
Experiments Performed to Investigate Optimal One-Dimensional
Designs

Figure 6.6 Transient Volumetric Heat Capacity Corresponding to the Second Set
of Experiments Performed to Investigate Optimal One-Dimensional
Designs

Figure 6.7 Confidence Regions Corresponding to the Experimental Investigation
of Optimal One-Dimensional Experiments

xiii

103

106

117

120

121

124

125

126

129

k1, k2

my

LIST OF SYMBOLS

: cartesian coordinates (m)

: cartesian coordinate used with the semi-infinite geometry (m)

: cylindrical coordinate (m)

: radius of the thermocouple wire, and width of the two-dimensional geometry (m)
: thickness of the finite geometry, and thermocouple in-depth location (m)
: non-dimensional spatial position

: thermocouple radius-to-depth ratio

: area (m2)

: volume (m3)

: time (s)

: non-dimensional time

: non-dimensional heating time

2 non-dimensional duration of the experiment

: non-dimensional time used with Green’s functions

: thermal conductivity (W/m°C)

: thermal conductivities at temperatures T1 and T2 (W/m°C)

: directional thermal conductivities (W/m°C)

xiv

pc : volumetric heat capacity (W/m3°C)
pcl, pol: volumetric heat capacities at temperatures T1 and T2 (W/m3°C)

COL, : alternative intercept and slope of the thermal conductivity as a function of
temperature (W/m°C) and (W/m°C2)

C2,C3 : alternative intercept and slope of the volumetric heat capacity as a function of
temperature (W/m3°C) and (W/m3°C2)

pcw : volumetric heat capacity of the thermocouple wire (W/m3°C)
Apc : difference between pc and pcw
0t : thermal diffusivity (m2/s)

q0 : heat flux (W/mz)

q : average heat flux (W/mz)

T : temperature (°C)

T0 : initial temperature (°C)

T : average temperature (°C)

Tw : temperature of the wire (°C)

TL : temperature at x=L (°C)

T. : temperature at the wire location had the wire not been there (°C)
'1" : non-dimensional temperature

TM; : maximum non-dimensional temperature reached between the start and the end
of the experiment

X” : i-th dimensionless sensitivity coefficient

kn, km : eigenvalues

p : number of parameters

In : number of sensors used

XV

"U

.-

t<r<7°C

m

U)

urn-age:

Vb
X(B)

X110)

: number of measurements by each thermocouple

: power (W)

: voltage (V)

: electrical resistance of the heater (Ohm)

: average temperature at time ti from m sensors (°C)

: measured temperature at time ti corresponding to the j-th measurement
: standard deviation

: weighted sum of squares

: (mnxl) column vector of transient measured temperatures
: (mnxmn) weighting matrix

: (mnxl) column vector of transient calculated temperatures
: (pxl) parameter vector

: (pxl) vector estimator of B

: derivative with respect to vector [3

: (mnXp) sensitivity coefficient matrix

: partial derivative of the temperature T at time ti by the j-th sensor with respect

to the k-th parameter

D : determinant of the product of the sensitivity coefficient matrix and its transpose
Ci; : i-th row, j-th column entry of the product of the sensitivity coefficient matrix and
its transpose

ox, oq, CL: variances on k, q, and L

¢(qo,t) : exact temperature at time t for the constant heat flux qo=1

A4)

: temperature disturbance (°C)

xvi

K(t) : thermocouple function kernel

xvii

CHAPTER 1

INTRODUCTION

Special interest in composite materials has been growing in the last twenty years.
This interest has been fueled by the superior dynamic performance of these materials, by
their light weight, and by their radar absorbing quality. The high strength and low weight
properties make composite materials ideal for use in space vehicles (National Aerospace
Plane), airplanes (commercial and military), ships, and in mass-produced goods such as
cars, boats, motorcycles, tennis rackets, and baseball bats. The use of composite materials
in most of these applications means reduced fuel consumption, which in turn leads to
reduced emissions, and eventually to cleaner air, and a better environment.

Composite materials are much more complex than ordinary metallic materials
because they are non-homogeneous. They are made of two ingredients: fibers and matrix.
The fibers can be carbon fibers, aluminum fibers, glass fibers, kevlar fibers, titanium
fibers, etc. The matrix is usually an epoxy (and there are different kinds) made of a resin
and a curing agent. When one or more types of fibers are combined with the epoxy to
obtain the composite material, the strength-to-weight ratio of the composite is higher than

that of the bulk fibers.

2

The use of composite materials in aerOSpace applications and airplanes subjects
them to harsh thermal conditions, which can lead to disasters if not looked at carefully.
To account for these thermal conditions, the study of the thermal behavior of composite
materials became a significant parameter of the evaluation process of these materials. An
important part of the thermal studies of composite materials pertains to the estimation of
their thermal properties: conductivity and volumetric heat capacity.

The process of estimating the thermal properties of materials (or parameter
estimation) combines the two classical approaches usually performed separately:
experiments and theory. This parameter estimation process was the driving force behind
the current research. This research has three main objectives: the investigation of two
experimental paradigms, the analytical investigation and experimental verification of some
Optimal thermal parameter estimation experiments, and the analytical investigation and
experimental verification of thermocouple errors.

The first objective pertains to the investigation of two competing experimental
paradigms which relate to the temperature rise achieved in the parameter estimation
experimental procedure. The effects of low and large temperature rises on the confidence
regions of the estimated temperature dependent thermal properties are investigated using
an experimental approach.

The second objective of the current research is to design optimal alternative
thermal parameter estimation experiments and implement them in the laboratory. To meet

this objective, three experiments are investigated: two one-dimensional experiments (finite

and semi-infinite), and one two-dimensional experiment. The durations of heating and

3

cooling are the main parameters of interest for the one-dimensional experiments. A non-
dimensional group containing geometry and directional thermal properties of the two-
dimensional case is the main parameter of interest. The experimental implementation is
only applied to the finite one-dimensional experiment. The confidence regions are
compared for the experimental data and are shown to confirm the theoretical results.
The third and last objective of the current research is to quantify the errors caused
by the thermocouples embedded in the specimens to measure internal temperatures. This
was done by Pfahl and Dropkin in 1966 using finite differences. In the current research
however, finite differences and finite elements are avoided because of their limitations in
calculating small differences (such as thermocouple errors); instead, a pure analytical
method is followed. The method employs the Unsteady Surface Element (USE) method,
and is shown to be more flexible than methods using finite elements or differences. The
results of this USE method are also compared with those of a previous investigation by

Beck (1968) based on the evaluation of some convolution kernels.

CHAPTER 2

BACKGROUND

2.1 Introduction

In order to provide a framework for the several analyses in this dissertation, it is
necessary to provide a background outlining the present practice. This chapter provides
this background These analyses involve the investigation of an experimental paradigm,
the investigation of optimal experiments (Chapters 3 and 6), and the investigation of
measurement errors (Chapters 4 and 5). In each case, the analyses have been
supplemented by experiments.
2.1.1 Motivation

Composite materials are important for many advanced applications. They are
much more complex than ordinary metallic materials because they contain fibers and a
matrix supporting the fibers. Fine fibers (10 to 15 microns in diameter) typically have
much improved strength-to-weight ratios compared to the bulk properties of the fiber
material, which is commonly carbon or glass. The combination of the fibers and matrix
makes the composite material.

The thermal properties (thermal conductivity and volumetric heat capacity) of

5

composite materials are needed for a number of reasons. One reason is that such
materials can be subjected to hostile thermal environments during service. Examples are
the proposed National Aerospace Plane (NASP), military airplanes such as the Stealth
bomber, advanced heat exchangers, and jet engines. These properties are also needed to
model the curing and molding of composite materials.

The thermal conductivity of composite materials can be directionally dependent,
as a result of the orientation of the fibers. It can also be affected by the thermal contact
between the fibers and matrix (Cha and Beck, 1989, Beck and Osman, 1987). These
factors make the determination of the thermal pr0perties more varied and difficult to
obtain.

2.1.2 Meaning of Effective Thermal Properties

The meaning of the "effective" thermal properties of composite materials must be
defined. Effective thermal properties are usually needed and also usually measured in the
research to be described By effective thermal conductivity is meant:

k = -_§_
‘3 ( 3'11) (2.1)
6x
where file}: is the average temperature gradient and a is the average heat flux. Both

averages are over the area normal to the direction of flow of heat. The effective

volumetric heat capacity is defined by
1 v

= _ (2.2)
pcaff fpcdv

where V is volume. Notice that the definitions of these properties in Equations (2.1) and

(2.2) are quite different.
The basic equation in which these effective properties are used is (for temperature-

dependent properties)

2 fl: ' 11‘: (2.3)
mimax) ”fiat

where the average temperature is defined by

A
" - 1 2.4
T(X.t) --XfT(X.ae.t)dae ( )

where as and A are the areas normal to x. In the remainder of this dissertation, the
subscript "eff" is dropped.
2.1.3 Carbon Fiber/Epoxy Material

The composite material of interest in this research is a carbon fiber/epoxy material.
The effective thermal properties of a Hercules carbon fibers/Epon 828-DDS epoxy
material were estimated in our laboratory at an early stage of the current research. The
results of the corresponding experiments are not reported in this dissertation however.
The effective thermal properties of such a material are functions of the thermal properties
of the carbon fibers and of the epoxy, as well as the volume fraction of the fibers in the
composite (Ziebland, 1974, Han and Cosner, 1981). One approach for estimating the
effective thermal properties is based on Equations (2.1) and (2.2), and considers the
material to be homogeneous (no inclusions, just one component). Another way of
estimating these effective thermal properties is to obtain the thermal conductivities and
volumetric heat capacities of the different constituents (carbon fibers and epoxy in this

case), and to calculate numerically the effective thermal properties.

 

of

Si:

Si

(A

III

7

Initial average values of the specific heats of the two constituents of the composite
of interest can be obtained by Differential Scanning Calorimetry (DSC) (Barton, 1985).
The densities, on the other hand, can be obtained by following the guidelines of the
standard test method for density of glass by buoyancy. This method was described by
the American Society for Testing and Materials (ASTM, 1990).

Estimation of the thermal conductivity of carbon fibers is very difficult. The
fibers are 10 to 15 microns in diameter, and come in tows of 3,000 to 12,000. The tows
are supplied on spools several hundred yards to a couple of miles long. However, the
samples of use in our laboratory are rigid solids of surface area much larger than a few
square microns. A finite heat. flux and/or temperature are usually imposed at both
surfaces of the solid sample. Thermocouples areusually instrumented at the surface and
inside the samples. The cured epoxy samples used, which are rigid solids of macroscopic
surface area, were analyzed in our laboratory as described below.

2.1.4 Boundfiarv Conditions
The current parameter estimation problem is based on the heat diffusion

differential Equation (2.3) with the following boundary conditions:

at
'5; l*'° =q°m (2.5)

T '1']. = TL(t)

where T is temperature, x is position, and t is time. The heat flux q0(t) is a measured
heat flux at one surface, and TL(t) is the known temperature at the other surface. The
choice of a heat flux boundary condition was partly made because of the practicality and

ease of imposing such a boundary condition. Even more important was the optimal

8

experiment. Such analyses are an important part of this dissertation (see Chapter 3), and
are the second analysis indicated in the first paragraph of this chapter. A heat flux was
generated by applying a voltage across a finite-resistance electrical heater. The power
generated by the heater is obtained using Ohm’s law as

P=U2R (2.6)
where P is power, R is the electrical resistance of the heater, and U is the voltage
imposed across the heater.

The other boundary condition of the sample was a temperature one. This
boundary condition was selected because composite materials tend to have low thermal
conductivities. By attaching these low conductivity materials to high conductivity
materials (such as metals), the temperature boundary condition is easily achieved.

2.1.5 Outline of the Remainder of the Chapter

A brief outline of the remainder of this chapter is given. The next section of this
chapter describes how the problem was implemented experimentally and gives some
experimental data. The third part of this chapter briefly explains the theory and
implementation of program PROPID used to estimate the thermal properties. Finally, the

results of PROPID are shown and discussed.

2.2 Description of the Experiment
2.2.1 Expefimentfiemp
The experimental setup described here was constructed in two major stages. The

first stage consisted of making two epoxy specimens from a commercial epoxy and a

9

commercial curing agent. In the second stage, the specimens were instrumented.
Thermocouples were installed in the specimens, one heater was sandwiched between the
specimens, and a metallic block on the opposite surfaces of the specimens (see Figure
2.1). Finally, the instrumented specimens were placed in an oven, and connected to the
data acquisition system.

2.2.1a Fabrication of the Specimens

The first step of the experimental investigation was to build two epoxy specimens.
For this purpose, the curing agent Diarnino Diphenyl Sulfone (DDS) was added to
DER332 resin in a beaker at a weight ratio of 35.8%. The beaker was then placed in an
oven at 140°C, and the resin was mixed every 5 to 10 minutes until all of the DDS was
diluted. A clean rubber mold was then placed in a vacuum oven along with the beaker
containing the epoxy. The vacuum pump was turned on until all the bubbles in the epoxy
were gone, and new bubbles started to form very slowly. The beaker and the mold were
then taken out of the vacuum oven, and the epoxy was poured in the mold. The mold
was then placed in a programmable oven allowing the control of the curing cycle.

The curing cycle was originally similar to the one usually used to cure dog-bone
shaped DER332/DDS samples that are no more than 1 cm thick. Such samples are
usually used in the Composite Materials and Structures Center at Michigan State
University for structural analysis or optical tests. The curing cycle consists of imposing
a constant temperature of 180°C for 3 hours. This same curing cycle caused our
relatively thick (3 cm) circular samples to burn inside because of the excessive

exothermic reactions taking place during the curing process.

Kapton Heater

Silicone Grease

 

 

 

0.25 mm

 

38.10 mm 31.75 mm

Figure 2.1 Placement of the Specimens with the Heater and Aluminum Blocks

11

The samples were chosen that thick for the following reasons. The first reason
was to allow the installation of the thermocouples at various distances from the heated
surface. The second reason for making the samples relatively thick was to allow them
to behave as semi-infinite regions for only one part of the experiment, then as finite
bodies.

A new curing cycle was then suggested (by M. Rich of the Composite Materials
and Structures Center at Michigan State University) to get around the damaging effect of
the exothermic cure reactions. The new curing cycle was as follows: constant
temperature of 140°C for 2 hours, ramp for 20 minutes to 160°C, constant temperature
of 160°C for 2 hours, ramp for 20 minutes to 180°C, constant temperature of 180°C for
2 hours, and a ramp down to 40°C for one hour. The specimen was then taken out of the
oven and allowed to cool in ambient temperature. Finally, a second identical specimen
was made, following the same methodology used to make the first one. The thickness
. of this second sample was also made equal to that of the first sample.

The choice to make the two specimens identical has an experimental justification.
By making the samples identical in composition and dimensions, then placing a flat heater
between them, a symmetry condition is achieved. The power generated by the heater is
equally shared by the two half-domains made of the identical specimens. For this
particular reason, the measured power of Equation (2.7) is divided by twice the area of

the heater to give the heat flux at each specimen’s surface as

LIKE (2.7)
2A

This of course assumes no side heat losses from the heater to the surroundings, which is

12

a reasonable assumption for our setup.

To reduce the contact resistances between the heater and the specimens, as well
as that between the metallic blocks and the samples, the specimens were polished.
Different grades of sanding paper were used until a nice gloss was achieved. Silicone
grease was then carefully spread between the heater and the samples, and between the
metallic blocks and the samples. By minimizing the contact resistances between the
specimens and the heater, the temperature drop across the surface of contact is minimized,
and most of the heat flux leaving the heater reaches the surface of the specimens.
Minimizing the contact resistance between the specimens and the metallic blocks also
makes the temperatures at the surfaces of contact closer to the temperatures of the
metallic blocks.
2.2.1b Instrumentation of the Specimens

The next step of the experimental investigation was to instrument the cured
samples. The first part of the instrumentation of the samples was to install the
thermocouples needed for temperature measurement. The temperature range to which
these wires are sensitive coincides with the range of temperatures of our experiments.
The plastic insulation was stripped off the Chromel-Constantan wires , which were then
electrically insulated using a ceramic (alumina) sheath; this sheath made the installation
of the thermocouples in the holes drilled in the samples possible and practical.

The next part of the instrumentation of the specimens was to place a heater
between them. A circular Kapton heater (made by Minco of Minneapolis) was

sandwiched between the two identical cured epoxy specimens. The heater was 7.5 cm

13
in diameter, and about 0.025 cm thick. A high conductivity silicone grease (DOW

Corning 340) was used between the heater and each side of the samples to maximize
thermal contact. The non-heated side of each sample was attached to an aluminum block
having a relatively high thermal conductivity compared to that of the samples. As
mentioned above, the temperature of the aluminum block did not change significantly
under the influence of the heat flux imposed by the heater, and thus simulated a constant
temperature boundary condition. This was possible because the low thermal conductivity
of epoxy permits only a small heat flow to the aluminum blocks. During the experiment,
the heat flow to the aluminum blocks was so small that their temperature rise was small.
Any temperature rise is treated in program PROPlD which is described in Section 2.3.

The themocouples were connected to the data acquisition system made of a VAX
station II GPX computer. The heater terminals were connected to a Hewlett Packard
6024A DC power supply which was controlled by the same computer. This VAX.
computer is equipped with a data acquisition board that allows simultaneous temperature
measurements. Indeed, it takes only 10*5 second to scan two consecutive channels of the
board, making the simultaneous measurements assumption a good one. At this stage, the
experimental setup was ready to operate as explained in the next section.
2._2.2 Some Experimental Results

Two series of experiments were initially considered to investigate the effect of the
heating regime on the estimated thermal properties. This relates to the experimental
paradigm mentioned in the first paragraph of this chapter. Each series consisted of

running four independent experiments. In the first series, each experiment consisted of

14

applying a large heat flux (4750 W/m2 on the average) to the samples at uniform room

temperature (25°C). In the second series, a much smaller heat flux (425 W/mz) was
applied to the samples starting at four different uniform temperatures (23°C, 52°C, 78°C,
and 101°C). In the first series of experiments, the heat flux was made large to cover the
total temperature range of interest (20 to 100°C) in an experiment. In the second series,
the same temperature range was covered but each experiment covered a small portion of
it.

Some of the transient temperatures and heat flux data collected for the two series
of experiments are shown in Figures 2.2 and 2.3. Figure 2.2 corresponds to large heat
pulse experiments, and is discussed first. Figure 2.2a shows the transient measured
experimental heat flux. The non-zero values of the heat flux oscillate between 4634.0
W/m2 and 4853.1 W/m2 making the fluctuation in heat flux between -2.44% and 2.17%.

Figure 2.2b shows eight curves corresponding to the transient experimental
temperatures measured by two thermocouples (one at x=0, and the other at 3.9 mm), and
corresponding to four experiments. The upper curves in Figure 2.2b are in fact four
curves corresponding to the surface thermocouple. The two curves indicated by the
continuous line are almost identical. The two other curves indicated by the dashed line
are also almost identical. These upper four curves in Figure 2.2b would have merged into
a single curve if the experiments were all started at exactly 25°C; however, as indicated
by the temperatures measured between time zero and time 10 seconds, the initial uniform
temperature was not exactly the same for all four cases.

Notice that the surface temperatures start rising almost instantaneously

15

 

 

 

 

 

 

 

 

 

 

5000 I. Y I I I 1.1' I ‘I V I I I I V I I I
40004 ..
€‘
fi
3 30004 a
V
X
2
L'- 20004 _
4.1
o
a:
I
10004 -.. exp4q
--- exp}
---- exp2‘
— expi
o...,...,...,.....
O 40 80 120 160 200
Time (sec)
(a)HeatFlux
14o,,,,...,...,...,.r
i
1204
A
O
L 100—
cu .
S
*5 80—
L
o
o.
E 60—
u
i— J
40—
ex 1
20 ...j...,...,...,.,"
0 4O 80 120 160 200

 

Time (sec)

(b) Temperatures

Figure 2.2 Experimental Transient Heat Flux and Temperatures (at two
thermocouples) on DER332/DDS Epoxy for Large Heat Pulse

16

with heat flux, then behave approximately as the square root of time. This behavior is
characteristic of semi-infinite domains subjected to a heat pulse. Based on this
observation, it can be concluded that the heated surface of the specimen does not "see"
the constant temperature boundary during most of the experiment. After the power is
turned off, at about 160 seconds, the heated surface temperature dr0ps abruptly.

The lower curves of Figure 2.2b show a smoother drop than the upper curves after
time 160 seconds. There are again four curves, and they would all have been nearly
identical if the experiments were started at exactly 25°C. These curves correspond to the
internal thermocouple. The delay in the response of this thermocouple is a result of the
distance between the thermocouple and the heated surface. Note also that this set of
curves and the curves described above are nearly parallel to each other for times between
60 and 160 seconds.

There are several purposes in running experiments such as shown in Figures 2.2a
and 2.2b. One relates to replication. Another relates to the investigation of measurement
errors. Yet another relates to the experimental paradigm. These are now briefly
discussed.

Replication must be assured in these experiments (the epoxy material does not
change in composition over the experimental temperature range shown in Figure 2.2b).
The replication is remarkably good since it is within 0.1284°C over the entire range, if '
the initial temperatures are adjusted to be the same. The value 0.1284°C is the largest
standard deviation of the experimental measurements calculated as shown later in this

section (see Equation (2.9)). In calculating this standard deviation, the temperature data

17

obtained from the series of the four large heat flux experiments is considered as data
obtained in one experiment from four identical thermocouples at exactly the same
location. Notice that this type of replication is that termed of the first order by Moffat
(Moffat, 1982, 1985). The same thermocouples and the same specimen are used. A more
rigorous examination would be to run another experiment with different types of
temperature sensors, and a new set of specimens. This was not practical; however, there
are two specimens, each with its own thermocouples.

Running one experiment with those two specimens is similar to a replication of
the second order. In this second order replication, the specimen is changed, and the same
experiment is repeated using the same instruments. Unfortunately though, the
thermocouples of the two specimens were not at exactly the same locations. This
"pseudo" replication of the second order could not beachieved then.

An inspection of the results of Figure 2.2b and also the individual thermocouples
leads to a partial understanding of the measurement errors (all thermocouple readings at
the same time by the same thermocouple could have similar biased errors for the different
experiments). The average temperature at a particular time, t,, for in temperature values

can be calculated using

Yi1 (2.8)

MB

- 1
Y=——
‘ m

be.
g-e

where i is the average at time ti, and Yij is the measured temperature at ti corresponding

to the j-th measurement. The j-th measurement could come from the same experiment

(different specimens) or different experiments. The estimated standard deviation of the

18
error in each Yij at a given ti is

I m _ 1,2
Sr, = [—2 (Yr) ‘ Yr)2

III-11.1

(2.9)

 

The values of sY obtained for the large heat flux experiments range between 0.0427°C and
0.1284°C as shown in Tables 2.2 and 2.4.

The data of Tables 2.2 and 2.4 was obtained as follows. First the temperatures
of Tables 2.1 and 2.3 corresponding to the different thermocouples were averaged at time
zero. Next, the initial temperature offset from this average was calculated for each
thermocouple. Finally, the temperatures at other times were corrected with the obtained
offset, and Tables 2.2 and 2.4 were obtained. The averaging of the temperature at time
zero explains the zero value for the standard deviation at that time. On the average, the
standard deviations of Tables 2.2 are larger than those of Table 2.4. This was expected
based on the knowledge that surface thermocouples cause more disturbance than internal
thermocouples.

The final aspect is related to the experimental paradigm. This is also related to
one of the significant analyses of this dissertation. The underlying question is: is it better
to cover "small" temperature ranges in an experiment or "large" ones? Small and large
are defined with respect to the temperature dependence of the thermal properties. If the
properties change less than some small value (such as 2%) in the temperature range of
the individual experiment, then the experiment has a small temperature range. On the
other hand, if the thermal properties change more than 10% in a given experiment, the

temperature range is said to be large. Figure 2.2 represents a large temperature range, and

Table 2.1 Some High Heat Pulse Experimental Temperature Data for the Surface

19

 

 

 

 

 

 

 

 

 

 

Thermocouple

t(sec) Yi1 Yi2 Yi3 Yi4 Avg Y

0 23.54 22.58 22.33 23.88 23.08

40 71.33 70.40 70.08 71.54 70.84

80 96.08 95.18 94.90 96.28 95.61

120 113.71 112.79 112.56 113.84 113.23

160 122.92 122.05 121.78 123.08 122.46

200 91.26 90.43 90.12 91.50 90.83

 

 

 

 

Table 2.2 Some Corrected High Heat Pulse Experimental Temperature Data and the

 

Corresponding Standard Deviation for the Surface Thermocouple

 

 

 

 

 

 

 

 

 

 

 

 

 

t(sec) Yum, Ymm YB”, Yfim Avg You, sY

0 23.08 23.08 23.08 23.08 23.08 0
40 70.87 70.90 70.83 70.74 70.84 0.0695
80 95.62 95.68 95.65 95.48 95.61 0.0884
120 113.25 113.29 113.31 113.04 113.22 0.1242
160 122.46 122.55 122.53 122.28 122.46 0.1229
200 90.80 90.93 90.87 90.70 90.83 0.0988

 

 

 

Table 2.3 Some High Heat Pulse Experimental Temperature Data for the Internal

20

 

 

 

 

 

 

 

 

Thermocouple

t(sec) Yi1 Yi2 Yi3 Yi4 Avg Y

0 23.54 ' 22.58 22.33 23.88 23.08

40 28.67 27.78 27.56 29.05 28.27

80 42.84 42.01 41.75 43.13 42.43

120 56.20 55.34 55.16 56.44 55.79

160 68.22 67.34 67.13 68.39 67.77

200 72.34 71.50 71.29 72.58 71.93

 

 

 

 

 

 

 

Table 2.4 Some Corrected High Heat Pulse Experimental Temperature Data and

Corresponding Standard Deviation for the Internal Thermocouple

 

 

 

 

 

 

 

fl t(sec) Yu M Yi2.oorr Ymm Yum Avg Yam sY

H 0 23.08 23.08 23.08 23.08 23.08 0

ll 40 28.21 28.28 28.31 28.25 28.26 0.0427

ll 80 42.38 42.51 42.50 42.33 42.43 0.0891
120 55.74 55.84 55.91 55.64 55.78 0.1179
160 67.76 67.84 67.88 67.59 67.77 0.1284
200 71.88 72.00 72.04 71.78 71.93 0.1182

 

 

 

 

 

 

 

 

21

Figure 2.3 represents a small temperature range. The estimation techniques are
sufficiently powerful so the temperature dependence can be found from the analysis of
a single experiment. Even though the properties can be found as functions of temperature
for a given experiment, it does not follow that this is the best procedure.

The competing experimental research paradigms of small and large temperature
variations in experiments have been investigated.’ Such an investigation was one of the
objectives of this research. Unfortunately, the best way to proceed depends greatly upon
the characteristics of the measurement errors, which have not been completely understood.
The investigation of the measurement errors is the third major analysis area in this
dissertation.

Figure 2.3 is now analyzed. This figure is similar to Figure 2.2, and most of the
ideas discussed above apply here. Figure 2.3 corresponds to a much smaller heat flux
(425 W/m2 on the average) imposed at the four different initial temperatures 23°C, 52°C,
78°C, and 101°C. The top part of Figure 2.3 shows that the non-zero values of the heat
flux fluctuate between 396.5 W/m2 and 453 W/m2 making the fluctuations in heat flux
between -6.7% and 6.6%. These fluctuations are larger than the ones corresponding to
Figure 2.2. The possible causes of this behavior are noise in the power supply used to
impose the voltage of Equation (2.7), and/or the behavior of the Minco heater which
might be more accurate for higher voltages.

Figure 2.3b shows the temperatures measured by the same thermocouples used to
obtain Figure 2.2b. The only difference here is that the specimens were initially at

different uniform temperatures, and were then subjected to a smaller heat pulse.

22

 

 

 

 

 

 

 

 

 

 

 

 

 

 

500 1 ‘ j I Y 1 t I T Y I I 1 m I 1 r 1
A
N
E 1
\
g 300— _
V
X
2
“- 200— _
+1
0
Q)
I
100~ -- at 100°C
at 75°C
---- at 50°C
— at 25°C
0 Y 1 7 I 1 r t I 1 ' Tfififi ' Y T .
O 40 80 120 160 200
Time (sec)
(a) Heat Flux
1201111,,2111‘171‘1'
100_--:::Z .......................... _
A
o
o .................
v _____--__-_______,,--_- ---—::::::::::‘
o 80~--.=-.::::: ................................... _
L.
3
+4
0
L I.
E --..°.'.’.'.'.'.'... ...............................................
11’ ‘-- at 100°C
--- at 75°C
40‘.-.. at 50°C -
. f
20 """'I"vtv#r,..,
O 40 80 120 160 200

Time (sec)

(b) Temperatures

Figure 2.3 Experimental Transient Heat Flux and Temperatures (at two
thermocouples) on DER332/DDS Epoxy for Small Heat Pulse

23

A check was performed for the replication of each one of the experiments of
Figure 2.3. This was done by running each experiment twice and checking if the
temperature curves were identical or not. The results of this check are not shown in
Figure 2.3b, but were as good as those of Figure 2.2.

The experimental data of Figures 2.2 and 2.3 was next used to estimate the
thermal properties of the epoxy samples. The computer program PROPlD (Beck, 1989)
was used for this purpose. The logic and theory implemented in PROPlD are briefly

described in the next section.

2.3 Description of PROPlD

The parameter estimation program PROPlD was used to estimate simultaneously
the thermal conductivity and volumetric heat capacity. PROPlD permits the estimation
of any combination of a total of four parameters as functions of temperatures. The
program uses the Gauss minimization method, and the parameters are found by
minimizing the scalar weighted sum of squares S defined by

8 =0! -T(B)]TW[Y-T(i3)l (2°10)

where Y is the column vector of size (mnxl), and whose entries correspond to transient
measured temperatures for each thermocouple used. The dimension m stands for the
number of thermocouples, while the dimension it stands for the number of measurements
obtained by each thermocouple. W is a symmetric (mnxmn) weighting matrix; however,
PROPlD has the capability of treating a diagonal W with the different weights only for

different temperature sensors. The entries of the (mnxl) column vector T are the

24

calculated temperatures obtained using the Crank-Nicolson finite difference numerical
method; p is the (pxl) parameter vector (p being the number of estimated parameters).

The set of estimated thermal properties is obtained by minimizing S with respect

to the parameters of interest. ‘ This is done by setting the matrix derivative of S with

respect to 5 equal to zero:

V55 =2[-X(B)]W [Y -T(i3)l =0 (2°11)
where X(B) is the (mnXp) sensitivity coefficient matrix. Equation (2.11) can be
rearranged to solve a set of p equations for the vector estimator b of 3 using an iterative
procedure needed because of the non-linearity of the problem (indicated by the

temperature (and thus p) dependence of the sensitivity coefficient matrix X):

 

 

bad) =b(i) + (X TQWX (1))-1[X me(Y _ T69] (2.12)
The sensitivity matrix X is computed by PROPlD, and is defined by:
'X(1)'
X(2)
X(p) =[vp'r T(p)]T= ° (2.13)
.Xénh
where
'xua) x126) ..... x,p(i) '
X216) X226) ..... X2p(i)
X(i) g ......... (2.14)

_xm,(1) Xmfi) ..... mefi),

 

 

The entries Xjk(i) of this matrix are the partial derivatives of the dependent variable T

25

measured at time ti by the j-th sensor with respect to the k-th parameter (here there are
two parameters: thermal conductivity, and volumetric heat capacity). These partial
derivatives are referred to as sensitivity coefficients. These sensitivity coefficients are
important when analyzing experiments; they provide much insight into the data as shown
later in Chapter 3.

The entries of the diagonal weighting matrix W are either specified by the user
in the input file of PROPlD, or automatically calculated by the program itself; however,
the weighting is varied only for a given temperature sensor. These weighting coefficients
are larger for thermocouples with smaller residuals than thermocouples with larger
residuals. Smaller residuals imply smaller differences between the measured and the
calculated temperatures; therefore, thermocouples with such residuals measure
temperatures Closer to the exact model, and consequently have smaller errors than other
thermocouples with larger residuals.

Sequential computation of the entries of vector b obtained by PROPlD is now
described. The data over the whole time interval of interest is used with initial guesses
(of the parameters) included in the input file to estimate the thermal properties for the
first iteration. These estimates are used as initial guesses in the next iteration which again
uses the data of the whole time interval. Once the convergence criterion (difference
between consecutive estimated parameters smaller than a specified constant) is met, the
last estimates are used as initial guesses in the last iteration; unlike previous iterations,

this last one uses the data one time step at a time to estimate and print the parameters as

functions of time. The maximum number of iterations is specified in the input file used

26

with the ‘program. Limiting the number of iterations avoids unnecessary computations
that might be due to errors in the input file; however, the parameter estimates usually
converge within five to six iterations. It should be noted that this type of sequential
analysis is not the same as sequential over experiments, which is described in the last
paragraph of Section 2.3 and in Section 2.4.

The equations used for each parameter as a function of temperature are

 

kz'kr
k: + —T (2.153)
k. T2410 0
902-901

c= c + -T (2.15b)

p p 1 —T,-T, (1‘ 1)
instead of

k= C0+C1T (2.163)
pc = C2+C3T (2.16b)

Such a choice (Equations (2.15a) and (2.15b)) gives the parameters the same units as the
property of interest, for which the user has a better feel; furthermore, such a choice allows
the natural extension of the model to more complex ones with still some feel for the
values of the parameters.

Another feature of PROPlD is that the program gives confidence intervals in
addition to parameter values. PROPlD computes two types of confidence intervals of the
estimated parameters: a 95% ellipsoid confidence region, and the Bonferroni square
approximate confidence region (Seber and Wild, 1989). The 95% confidence region is

an ellipsoid; the end points of the major and minor axes of the ellipse obtained by

27

projection on the plane are also computed.

The 95% confidence intervals computed by PROPlD are based on the assumption
that temperature is the only quantity having significant errors. This in itself assumes that
the errors in all other measured quantities are negligible compared to the error in
temperature. Consider for example the steady state case:

q=kfl no k=gli (2.17)
L 6T

where q is the constant heat flux, L is the location of the sensor of interest, and T is the
temperature measured by that same sensor. Assume that q, L, and ST all have random

errors in them, the error in k is approximately
Ak=—°— S—L Aq+— 6 LL AL+ —°— 41‘ ABT
6151 GT 6L 6T BOT 6T

=—Aq+—— ‘1 AL- ‘1’“ ——A6T
6T 8T (5 n2

(2.18)

Assume independent errors so the covariances of the different quantities are zero.

Then we have

L22 (122 quz 2.19
0:: —=V(Ak) -[6-—T) oq+ (6T) OL+[(8T)2] 0T ( )

 

where of is the variance of k (also denoted V(Ak)). <qu is the variance of q, GL2 is the
variance of the thermocouple location L, and GT2 is the variance of the measured
temperature T. PROPlD handles only errors in the measured temperatures. The other

I

sources of errors such as the measured heat flux, and the thermocouples locations are not

accounted for in PROPlD; therefore, PROPlD assumes 0:30.21 << 0:.

Finally, another one of the many features of PROPlD is that the program permits

28

the use of sequential-over-experiments analysis. This type of analysis allows the user to
combine information gained from one experiment (as prior information) to estimate the
thermal properties using the data of the next experiment (Beck, Hollister, and Osman,
1990). The added information coming from more experiments improves the quality of
the estimated parameters by reducing their confidence regions. This is exactly what is

needed to analyze the series of experiments of Section 2.2.

2.4 PROPlD Experimental Results

Some of the results of program PROPlD are shown in Figures 2.4, 2.5, and 2.6,
and are discussed in this section.
2.4.1 Large Pulse Experiments

When the data of Figure 2.2 (for large heat pulse) was analyzed with PROPlD,
the u'ansient thermal properties were calculated as functions of temperature. They were
determined at two temperatures. At 20°C, the thermal conductivity k was found to be
0.1657 i 0.0139 W/m°C, and pc to be (0.1179 1: 0.0082)x107 J/m3°C. At 120°C, k was
found to be 0.2395 :1: 0.0107 W/rn°C, and pc to be (0.2240 :1: 0.0205)x107 J/m3°C.

The transient residuals were also computed by PROPlD, and are shown in Figure
2.4 for two thermocouples. The surface thermocouple (sensor 1) shows residuals as large
as 3°C for a maximum temperature rise of 104°C. The corresponding error is then 2.9%,
which is considered large for our purposes. Note that 3°C is much larger than the
0.1284°C found in connection with Table 2.4. The curves of the same Figure show that

the errors are not zero on the average (they do not oscillate around zero), indicating a

29

 

4 r i 77 I T I l i r
, o—o SENSOR 1

[Si—{l SENSOR 2

 

 

 

Resdiuols (°C)

 

 

 

‘4 ' ’ . T ° 7 .
O 40 80 120 160 200

 

Time (sec)

Figure 2.4 Transient Residuals Calculated by PROPlD for DER332/DDS Epoxy
for Large Heat Pulse (at surface thermocouple)

 

 

bi.

30

bias. Finally, the residuals of Figure 2.4 seem to be highly correlated. This correlation
is observed by looking at consecutive portions of each curve, and noting that the data
follows an increasing (or decreasing) trend. If the errors were uncorrelated, the behavior
of a portion of the data could not be predicted from an adjacent portion.

The estimated confidence region of the thermal conductivity corresponding to the
four large pulse experiments is shown by the dashed lines of Figure 2.5. Each one of the
four experiments has information about the whole temperature range. In the process of
using the sequential-over-experiments analysis, the data sets of the four experiments were
combined together. This combination of the data sets of the four experiments was needed
to compare the parameter estimates of the two sets of large and small pulse experiments;
it was necessary to have the same amount of information (same number of measurements)
for the large and small pulse experiments. The combination process minimally improved
the confidence regions at the different temperatures (see Table 2.5).

2.4.2 Small Pulse Experiments

The experimental data of Figure 2.3 (for small heat flux) was next analyzed using
PROPlD. The transient thermal properties were calculated as functions of temperature.
They were estimated at two temperatures. At 20°C, k was found to be 0.1977 i 0.0095
W/m°C, and pc to be (0.1327 i 0.0019)x107 J/m3°C. At 120°C, k was found to be
0.2316 :t 0.0113 W/m°C, and pc to be (0.1861 :1: 0.0027)x107 J/m3°C.

The transient residuals were also computed by PROPlD and are shown in Figure
2.6. These residuals were as large as 03°C for a maximum temperature rise of 126°C.

The corresponding error is then 2.4%, which is of the same order as that of Figure 2.4

31

 

 

 

 

 

0.26 .
8 q _
0
E 0.24— —
\ -i
E
43‘ 0.22— —
IE °
1‘3 _ _
3 0.20
D d .1
C
O
O 0.18e _
B _
E 016
a) Q -‘ .—
[E --- high pulse ~
-— low ulse
0.14 I I I r I r r I I I p I
20 40 60 80 100 120 140

Temperature (°C)

Figure 2.5 Estimated Thermal Conductivity of DER332/DDS Epoxy versus

Temperature for High and Low Heat Pulse Heating Regimes

32

Table 2.5 Calculated Confidence Intervals for the Large Pulse Experiments

 

 

 

 

 

 

 

data of k(20°C) k(120°C) pC(20°C) pC(120°C)
exp.
1 0.16511: 0.238921: (0.1 173:1:0.0166)x (0.2240i0.0406)x
0.0278 0.0207 107 107
2 0.16611: 0.23923: (0.1 179:0.01 15)x (0.2239i0.0288)x
0.0196 0.0149 107 107
3 0.1659: 0.23961: (0.1 180:1:0.0094)x (0.2241i0.0236)x
0.0159 0.0123 107 107
4 0. 1 657:1: 0.239521: (0.1 179:1:0.0082)x (0.2240:t0.0205)x
0.0139 0.0107 107 107

 

 

 

 

 

Table 2.6 Calculated Confidence Intervals for the Small Pulse Experiments

 

 

 

 

 

 

data of k(20°C) k(120°C) pC(20°C) pC(120°C)
exp. at
20°C 0.1575i 0.65991: (0.1 129i0.0338)x (0.5210:|:0.5859)x
0.0444 0.4632 107 107
20, 50°C 0.19703: 0.23181: (O.1315:i:0.0023)x (0.1925i0.0096)x
0.01 17 0.04348 107 107
20, 50, 0.1965i 0.23501: (0.1321:t0.0020)x (0.1886i0.0047)x
75°C 0.0101 0.0188 107 107
20, 50, 0.19771: 0.2316i (0. l 327:1:0.0019)x (0. l 861:1:0.0027)x
75, 100°C 0.0095 0.0113 107 107

 

 

 

 

 

 

33

(for large heat pulse). Note that 03°C is of the same order of magnitude as the 0.1284°C
found in connection with Table 2.4. Finally, the same correlated and biased behaviors
of the errors observed for Figure 2.4 apply to Figure 2.6.

When a small heat pulse was used, the temperatures inside the specimens
increased by no more than 10°C from the initial steady state temperature. PROPlD,
however, estimated the thermal properties at temperatures between 20°C and 120°C. The
confidence intervals of the thermal properties at the temperature closest to the initial one
were the smallest (see Table 2.6). This is true because most of the experimental
information was available at and around that particular initial temperature. The thermal
properties estimated at the initial temperatures were then expected to be more accurate
than the other properties at higher temperatures. The knowledge gained from one
experiment was used as prior information (Beck, Hollister, and Osman, 1990) in analyzing
the next experiment. In doing so for the first experiment of Figure 2.3, the confidence
region obtained at 25°C was much smaller than that at 125°C. The knowledge gained at
25°C was then used (as prior information) in the analysis of the next experiment. Using
this prior information with data of the experiment at 50°C helped estimate the parameters
at both 25°C and 50°C. The same idea was used for 75°C, 100°C, and 125°C. The
confidence region for the thermal conductivity was then obtained as shown by the
continuous lines of Figure 2.5.

The possible sources of the residuals of Figures 2.4 and 2.5 are errors in
temperature measurements, errors in the times at which the heat flux starts and ends,

errors in the measurement of the positions of the thermocouples, non-uniform initial

 

0.4. . , . I , r , I ,
G—O SENSOR 1

8—8 SENSOR 2

,.

"l s I“.

a .3!
o

 

l
.-
C -
)
,'_.
~, g '
C‘:
E ‘ I ~ '
‘ i
‘ l..’ I...
t . ‘ D
, \
c :1 ~ ‘ ,
A. J . )
. , e _
ev- .
a - ’
: ( .
. )
(‘ ’
o)
I
7 r‘
(Ii-<1, ‘ ‘ 3
_, ,
'3 . ‘ 3
c, ,.'
.
‘ J 3 ‘ _)
'. r"
. t
- a
. I
I_ ) r ’
.; ‘2’
'7’.
3.;
”f '
‘
_._r

 

 

 

 

 

 

 

 

 

—.l~

_.2_

“-3 r l ' r T l ' 1 ' mi
0 40 80 120 160 200

Time (sec)

Figure 2.6 Transient Residuals Calculated by PROPlD for DER332/DDS Epoxy
for Small Heat Pulse (at surface thermocouple)

35

temperature, side heat losses, and the existence of contact resistance between the
thermocouple and the surrounding material, and between the heater and the samples (even
with the use of silicone grease). Most of the possible sources of errors were kept in mind
when designing the experiment, and setting it up. Some of these sources could not be
tightly controlled for a number of reasons. One of the reasons was that temperature
measurement, for example, involves the thermocouple wires, the alumina sheath, the
drilled holes, and the contact resistance between the junction and the surrounding
material. The possibility of conu'olling all these experimental conditions, and others not
identified, was not practically possible.

The above focused on the thermal conductivity as indicated by Figure 2.5, and
ignored volumetric heat capacity. The reason for doing so is that volumetric heat capacity
is not as sensitive to temperature as thermal conductivity. It should be noted though that
program PROPlD estimates both thermal properties, and provides the same information
for both of them.

Each one of the two confidence regions of Figure 2.5 is, in reality, made up of
five confidence intervals at 25°C, 50°C, 75°C, 100°C, and 125°C computed by PROPlD.
The two confidence regions of Figure 2.5 are similar in size. These confidence regions
were considered large, i.e., the parameter estimates were not accurate enough. Using
different heat pulse magnitudes then proved to be insufficient in improving the transient
estimation of thermal properties, and another strategy was needed. This other strategy
consists of designing optimal experiments, and is applied in Chapter 3. By the same

token, since the residuals of Figures 2.4 and 2.5 seemed large, it was necessary to analyze

36

the errors due to thermocouples; this is done in Chapters 4 and 5. Chapter 4 represents
an analytical study of the transient behavior of thermocouple errors, while Chapter 5 is
the experimental investigation corresponding to this analytical study. If such analyses did
not show the possibility of reducing these residuals, then another temperature
measurement technique was needed. Chapter 6 investigates an alternative method of

measuring temperatures developed by Garnier, Delaunay, and Beck (1991).

CHAPTER 3

OPTINIAL EXPERIMENTS

3.1 Introduction and Literature Review

In performing experiments, the researcher would like to gain as much insight and
information from the results as possible. To reach this goal, experiments have to be
designed properly. The design of experiments has been the topic of a number of papers
in the fields of statistics (Biggers, 1961, Box and Lucas, 1961, Draper and Hunter, 1966,
Draper and Hunter, 1967, Kenward and Stone, 1969, Hill and Hunter, 1974, Atkinson,
1981, Atkinson, 1982, Seber and Wild, 1989), chemical engineering (Hunter et al., 1969),
and mechanical engineering (Beck, 1966, Beck, 1969, Beck and Arnold, 1977,
Balakovskii et al., 1988, Artyukhin et al., 1988, Artyukhin, 1989, Vigak et al., 1989).

This chapter focuses on the analytical design of optimal transient heat conduction
experiments performed in our laboratory on orthotropic materials (composite materials can
be modeled as such). These experiments have been specifically designed to estimate the
thermal properties of cured carbon-fiber/epoxy-matrix composite materials. The thermal
properties of interest are conductivity and volumetric heat capacity for the one-

dimensional case, and directional thermal conductivities for the two-dimensional case.

37

38

Three different cases were considered: one-dimensional heat conduction
experiments in cured composite materials of finite thickness, one-dimensional heat
conduction experiments in thick cured composites which can be approximated by a semi-
infinite geometry, and two-dimensional heat conduction in finite cured composite
materials. The Optimal heating durations were considered for all three cases. The results
of this analysis were applied in our laboratory on composite materials; however, this
analysis can also be applied to metals.

A brief outline of this chapter is given. In the first section of the chapter, the
theory pertaining to the design of optimal experiments for the estimation of thermal
properties is reviewed. The temperature solutions and the sensitivity coefficients are
derived and plotted for the three cases mentioned above. The optimality criterion is
calculated and plotted for different experimental parameters. The results are discussed

and compared with those of previous studies. Finally, some conclusions are stated.

3.2 Theoretical Procedure

The Choice of an optimal design must be based on some criteria. Three optimality
criteria are given by Beck and Arnold (1977); all of these criteria are related to the
sensitivity coefficient matrix, X, which is described below. The entries of this matrix are
the dimensionless sensitivity coefficients. There are two parameters under consideration:
thermal conductivity, k, and volumetric heat capacity, pc. The two dimensionless

sensitivity coefficients associated with these parameters, X,+ and Xz“, are defined as:

39

k 6T,

 

Xi. = m55 i=1.2.....n (3.1)
. _ c 31‘: ._ (3.2)
X1L2 -— pqL/k ape 1—1,2,...,n

where q0 is a constant heat flux, T is temperature, L is the sample thickness, and n is the
number of measurements.

The optimality criterion chosen in this study is discussed by Beck and Arnold
(1977); it is based on the maximization of the determinant, D, of the sensitivity
coefficient matrix and its transpose. It is subject to a maximum temperature rise, a fixed
number of measurements, and the eight standard statistical assumptions (Beck and Arnold,
1977). These assumptions are summarized as additive, uncorrelated normal errors with
zero mean and constant variance, with errorless independent variables, and no prior
information. This criterion of maximizing D was selected because it minimizes the
hypervolume of the confidence region of the parameter estimates.

In equation form, the determinant, D, for the case of two parameters is:

D = (191(er = C1: C1: = CriC22‘(Cr2)2 (3'3)
C12 022
where C,,", Cn”, and C1; are defined for uniformly spaced measurements in time between

0 and tn”, and for large number of time steps as (Beck and Arnold, 1977)

O

‘-
4, = l 1 111 6T: 6T, + (3.4)
l 010;: {[‘tiap—iiia—ozidt

where B,=k, and Bzzc.

 

40

Numerical values for the temperature distribution in the geometry of interest are
needed; for the investigated cases, these solutions are derived by the method of separation
of variables or obtained from (Carslaw and Jaeger, 1959); finite element (FE) or finite
difference (FD) methods could also be used. The sensitivity coefficients are then
computed by differentiating the temperature solutions with respect to thermal conductivity
or volumetric heat capacity. Finally, the determinant, D, is calculated from Equation
(3.3). The Optimal experimental conditions are then established through the comparison
of the values of D obtained for different experimental parameters, such as the duration
Of the heating time, the duration of the experiment (or equivalently, the duration after
heating), and the sensor placement within the composite.

Three, cases were considered in this study. First, one-dimensional heat transfer
was considered in a finite cured carbon/epoxy composite, with the heat transfer in the
direction perpendicular to the fiber axis. In the second case, one—dimensional heat
transfer was studied in a thick cured carbon/epoxy composite thermally behaving as a
semi-infinite body. Finally, the finite composite was considered with the boundary heat
flux imposed on half of the surface causing a heat flow both parallel and normal to the
fiber axis.
£1 Finite One-Dimensitml Composite QQIBSOTO) (Case 1)

Carbon-fiber/epoxy-matrix composite materials tend to have low thermal
conductivities for which an isothermal condition can be readily approximated at the
unheated surface. The experiment shown in Figure 3.1 and analyzed here allows for the

measurement of this relatively low thermal conductivity. The specimen is a slab of finite

41

 

 

 

Figure 3.1 Finite One-Dimensional Geometry of the Finite X2lBSOT0 Case

[bid
for

term
adv
the
59.0
by

101

(A

42

thickness, L, with one boundary subjected to a heat flux produced by a heater (constant
for a prescribed time and then zero), and the second boundary having a constant
temperature (the number for this case is XZIBSOTO; see Beck et al., 1992). The
advantages of this experiment include the ability to obtain the heat flux, experimentally,
the simplicity of the experimental procedures, and the relative ease of the composite
sample preparation due to its simple geometry; for example, the heat flux is controlled
by simply turning it on and off. The heat conduction equation, boundary conditions, and

initial conditions in non-dimensional form are:

 

 

 

2 + +

a T = 6T , 0<x’<1,t‘>0 (3.53)
Bx+2 6V

6T* = —1 0*<t+Sth ’ at x*=0 (3.5b)
6x1 0 th<tn
T‘ = 0 at x*=1, t*>0 (35°)
T+ = 0 , for Osx’sl , t*=0 (35d)

The dimensionless time th+ is the dimensionless heating duration. The dimensionless

variables are defined as:

 

, t4. = (II/L2, t1: = Gib/L2, X+ = X/L (3.6a,b,c,d)

Equation (3.5b) gives the heat flux condition, and Equation (3.50) gives the isothermal
condition.
One method for the solution to this problem involves the use of the method of

superposition. Up to time tf, the temperature solution of Equation (3.5) is obtained for

43

a heat flux condition starting at time zero using the method of separation of variables

(Carslaw and Jaeger, 1959, Ozisik, 1980):

 

T*(x*,t*) = (l-x“) —2 Z: (-l)nsin(ln(1—x r)) 64:", 0<t*st; (3.7a)

n-O A:

where 3.“ is equal to (2n+1)rt/2. For t > th“, superposition is employed, and the solution
for a heat flux condition starting at dimensionless time t,“ is subtracted from the solution
for the heat flux condition starting at time zero, shown above in Equation (3.7a). The

resulting solution is

r:(x*,t*) = 42 5512—) sin (An(l-x‘)) [ e“i"— e"i“°"§’], tgqtst; (3.71:)
r

n-O n

where t: is the final time for taking measurements. This solution (Equations (3.7a,b))
is shown in Figure 3.2 as a function of time for four different x" values.

The next step is to compute the dimensionless sensitivity coefficients defined by
Equation (3.1) and Equation (3.2). The differentials in these equations are found by
differentiating the temperature solutions shown in Equations (3.7a,b) with respect to the
parameters, k and pc. The i subscript in Equations (3.1) and (3.2) is dropped in the
remainder of the chapter for simplicity of notation. The resulting sensitivity coefficients

for thermal conductivity are

 

X1. . Lilhuqqd 2 (-12).. sin(1,(1-x*>)e":" (1+ 13*) (3.83)

for 0 < t+ S th+, and

 

 

 

 

 

 

10 I I ’vI'_--T"—"I I I I I I I I I
<1) ‘ ‘
L— ,
B 0.84 —
8 ’I .......................................................................
0) . ,' -
C). I
I
E .
(1) 0.6— I, .—
l— ; 1
U)
U)
.0.)
C
.9
(n -4
C
(l)
E --- x*=0 fl
5 x*=0.25
-------- x =0.5 _
-— x*=O.75
I I I I T I I I I
3 4 5 6 7

 

Dimensionless Time

Figure 3.2 Dimensionless Temperature Solution for the Finite XZIBSOTO Case

45

x; =2; figuring -x ’))[e "3T1 +131 .) -e "‘i“"‘i’(1 +131: 4.3)] (3.8b)

for th+ < t+ S tn“. Likewise, the sensitivity coefficients for the volumetric heat capacity
are:

 

. aT " . -121: .
x = 9" —=-2 -1°sm(1. 1- . t: - , o<t: t (3-93)
2 qOIJkapc 112-o( ) n( x )) e S h

X; = 22(-1)“sin(A.(1-x*>)i-t+e"i" + (t+-t;)e"i“°"". t;<t*st..‘ (3'9”)

n-O

The sensitivity coefficients for thermal conductivity and volumetric heat capacity are
shown as functions of time for t; = t; in Figures 3.3 and 3.4. The magnitude of the
thermal conductivity sensitivity coefficient is about equal to that of T", while the
sensitivity coefficients for the volumetric heat capacity are smaller, and they approach
zero for t+ greater than 2. Also note that the shapes (except at very early times) of the
thermal conductivity and volumetric heat capacity sensitivity coefficient curves are quite
different. Finally, note that the sum of the dimensionless sensitivity coefficients X,+ and
X2+ is equal to the negative of T“. These observations verify that the two sensitivity
coefficients, Xf, and XZ“, are "large" (i.e. on the order of T"), and uncorrelated (different
shapes), which are desirable conditions for parameter estimation.

The final step of the analysis requires the determination of the determinant, D,
shown in Equation (3.3). The maximum temperature rise, Tmu‘”, is first determined from
Equations (3.7a,b) with x*=0, and the sensitivity coefficients, X1+ and X2", are found from

Equations (3.8a,b) and Equations (3.9a,b). The Cij matrix coefficients are then found from

46

 

 

 

 

 

 

 

 

I I I I I ' 1 I
. .. — x'=0-75
x' 0.0m '“ ”=05
. ------ x°=0.25
rt: """ ,.___
a) .1
O -0.2-‘
0
g t“ -.
cu - 0. 4 -* i‘. _
m t‘ r“ ...................
m ‘.
U) ‘1
2 —O.6—‘ ‘\ fl
C \
.9 4 “
m ‘\ ---------------------------------------------------
C ‘. _
0.) —O-8—‘ ‘\
E “
a 1 ‘ ~.
-10 I INHN"? ' I ‘ l I r
o 1 2 3 4 5 6 7

Dimensionless Time

Figure 3.3 Dimensionless Thermal Conductivity Sensitivity Coefficients for the
Finite X21B50T0 Case

 

0.1 . I v I v I r I I l I i

 

 

Dimensionless Sens. Coeff. X;

 

 

 

----- x'=0.75
------ x'=0.5
x’=0.25
—— x°=
_O.5 I I I I I I ’—f I I I r I r
O l 2 .3 4 5 6 7

Dimensionless Time

Figure 3.4 Dimensionless Volumetric Heat Capacity Sensitivity Coefficients for
the Finite XZIBSOTO Case

47

Equation (3.4), using the calculated values of ij, X“, and x; and integration. Finally,
the determinant, D, is calculated from Equation (3.3). The determinant was found and
compared using different heating times (th‘) and different experiment durations (tj)
(Figure 3.5), and different sensor locations (Figure 3.6).
3_.2_;2 Semi-infinite One-Dimensional Composite (XQOBSTO) (Case Q

The second case considered is similar to the first case, with the exception that the
sample is thick; thus, it behaves as a semi-infinite body with a constant heat flux at its
surface (the number in this case is XZOBSTO; see Beck et al., 1992). In this case, the

heat conduction equation, boundary conditions, and initial conditions are

2
kill = peg , o<x<m , t>0 (3.103)

6x2

_ 6T _ qo 0<tsth _ (3.1%)
kg {0 t>th, at X-o
T = T0, at x20, t=0 (3-10C)

For convenience, the following dimensionless groups are defined for this case:

T-T
= 0 t’ = .91, x :

, (3.lla,b,c)
W x:

T+

 

x
x0
where x0 can be any given location inside the body (not on the surface). If the
temperature at the surface is of interest, then T is non-dimensionalized with respect to the
position of the internal thermocouple.

The temperature solution for this problem was obtained from Carslaw and Jaeger

(1959). The dimensionless temperature is given by

48

 

0.020 v i v r Y

Determinant D

 

 

 

 

Dimensionless Time

Figure 3.5 Determinant, D, for Different Heating Times for the Finite X21B50T0
Case

0.012

 

0.010~

0.008-

0.006—

Determinant 0

0.004—

0002 n

 

 

0000 _ .~"" "'.:..T--=%‘==::r:::;;;:_-:l,::;;:;:;1:;:: ?—‘—== i=—-w—-' ' ' ' ' ==fu=? :4 "i
0 l 2 3 4 5 6 7

 

Dimensionless Time

Figure 3.6 Determinant, D, at Different Sensor Locations for the Finite XZIBSOTO
Case

49

T*(t*)=2/F ierfc " ,0<t*st; (3.1221)
2.]?

 

4.

T*(t*)=2i/t_*ierf={ x ]-2 v-tgierf x , t;<t*_<.t; (3.1213)

74/; 2‘/t +-t,:

 

 

Upon differentiation of T (not T‘) with respect to the parameter k, the thermal

conductivity sensitivity coefficient X,* can be written as

+ +2 +
Xl+=__E___a_I=-— Lexp[—x ]+erfc[x

4t ’ 2F

, O<t * 5th+ (3.133)

 

 

 

 

 

 

 

 

 

 

 

 

 

(3.13b)
t’—t+ / +2 + + +
hexp -——x , “If x » th<t‘<t..
7‘ ( 4(t._th) 2‘/t”-t;
In a similar way, the volumetric heat capacity sensitivity coefficient x; is
+ +2
X2. = pc aT = _ L exp _x , “v“; (3,143)
quo/k 390 1: 4t+
+ t‘ x“ "’9: l x+2 + 3 14b
X2 = "' — exp - + emi- , tT>th (. )
n 4t* n ( 4am

 

The determinant, D, is again calculated from Equation (3.3) as described for the
first case involving a finite geometry. The solutions corresponding to different heating

times t; are found andishown in Figure 3.7.

50

 

0.006 . I 1 I . T . 1 .
— t*.=0-5

‘ i ........ t+h=1 .0 ‘

0.005— T i+,=1.s _
E; \\ T. "' t+h=2'0

 

Determinont D

 

 

 

 

 

 

Dimensionless Time

Figure 3.7 Determinant, D, for Different Heating Times for the Semi-Infinite
XZOBSTO Case

51
_3_‘_._2.3 Finite Two-Dimensional Composite (XZZBOOYIZBOXSTO) (Case 3)

The third case considered is that of two-dimensional heat conduction through a
carbon- fiber/epoxy-matn'x composite material. The experimental set-up and boundary
conditions used for the first case are also considered here, with the exception that the
boundary heat flux is imposed on half of the surface instead of all of it as shown in
Figure 3.8. This type of boundary condition causes the heat to flow both parallel and
normal to the axis of the fibers. The specimen then behaves as an orthotropic material
with directional thermal conductivities kx and k,; the subscripts x and y refer to the
directions parallel and normal to the fibers respectively. The volumetric heat capacity is
assumed known in this study; it could have been obtained from the above-described finite
one-dimensional cured composite study, or from Differential Scanning Calorimetry (DSC)
(to get the specific heat), and the standard test method for density of glass by buoyancy
(the American Society for Testing and Materials (ASTM, 1990)) (to get density). The

energy equation, boundary conditions, and initial conditions for this case are then

az'r + a”r* a'r“

 

 

 

 

 

= (3.153)
Bx“ 6y+2 6t*
T*=0 at y+=o (3151»
HI” =0 at x’=0, x‘=-§- 5— (315°)
Bx’ b k;

+ 3
6T =1 at y*=1, Dan‘s—l 5— (3'1“)

52

 

 

 

 

 

 

V
q 0
\ i i i i i i i i i/////////////////
i 31 /
\ /
\ /
\ /
\ /
\ /
\ /
\ é
\
\ /
\ ?
E /
\ g .. X
a

\\\

Figure 3.8 Two-Dimensional Finite Geometry for the X22BOOY12B0x5TO Case

art ’31 k, . a K, ' (3.15e)
aw b k. b 1:.

 

T*=O a. M (3450

where

T-T

The temperature solution to this problem is made up of two components: a steady

 

state one, and a transient one. The steady state part of the temperature solution is given

for x'=x/a and y'=y/b by

Ts;(x 2y +) = 81.), . +

29. k,"1. . . bk... (3.17)
fiBJE-g E;sm(n1ta1)cos(mtx )mnb[nn;\J;y ]

The transient part of the temperature solution is given by

 

" _ n _ 2 .
T.‘(x*.y*.t*)=2a;£‘ 1,) sina,y'>e ‘n‘ +£-
n=l

 

(3.18)

where An=(2n-1)1c/2, lap—mu. The transient temperature solution is shown in Figure 3.9

for different values of x at the heated surface (y=b). Examination of Figure 3.9 reveals

54

 

 

 

 

 

1.0 I I l I T T I T T
- - - x=0
a) T ---- x=0.250 ‘
L ----- x=0.50
3 0 8— _
*5 ' ------ x=0.750
L- _ _
a) . ____________________ "_‘_° _____ -
Cl :7. ......................................
E I -—1
(D
I...
(n -----------------------------------------------------
U)
i) —l
C
.9 -------------------------------------------------------------------------------
m 1
C
(D
._E_ -
0 Cl
r T r i I T 1
4 6 8 10

 

Dimensionless Time

Figure 3.9 Dimensionless Temperature Solution for the Two-Dimensional

XZZBOOYIZBOXSTO Case

55

that the sum of the non-dimensional temperatures at steady state at x=O, x=a, and y=b is
unity; the same result is obtained when temperatures at locations on opposite sides of the
midpoint (x=a/2) are added. This is a check on the validity of the solution; the problem
can be looked at as the sum of two "half" problems, one insulated and the other heated.

The steady state and transient components of the sensitivity terms in Equation
(3.2) were found by differentiating the dimensional forms of Equations (3.17) and (3.18)
with respect to the parameters thermal conductivity, k,, and thermal conductivity k,, then
premultiplying by kx/(qob/ky) and k,/(qob/ky) respectively. The resulting expressions for

the steady state components are

+ 18 HI‘ -1a
X1=——=—— nrr 1cos(nrtx 1t—
gob/158g“ Knlnmi) a )mhin :fiy i (3.193)

_2_ lsin(n1tal)cos(n1tx ‘)sech nrt-a-J—éy'

1th.] 11

+=-4-k’—-§£=—-—1--%J:Z:nflzacos(nrtx‘)umh[mt—\J—Ey‘]-
X2 (lob/18618 11.1-12.8!!!“ I) (3.1%)

_2_ lsin(n1tal)cos(n1tx ')sech MEJ’ZY -a;’y

1111.1 11
The transient components of the sensitivity coefficients are

Xf=41r22 m(-1)n+12moos(l x )sin()i mal)si.rr(l.ny)e ( _
m-ln-l 2b 5
lg

 

1:- 4+1}!
ls 1!.l)

(3.20a)

56

ad 1!

o t. In . ~1:t’ l 4. 4
X2 =2312(-1) lsrn(lny')e [Tin ]+_*

 

 

.. .. _ N ‘ agar“
232? ( 1,) 005(1mx‘)sinllmaxlsinllny‘le ( ‘ '5 *
m=in=1 m[A2b_ 5+ *2) (3.201»
11182 n
1’ K1
1- m +13‘
12 +1”:—

These sensitivity coefficients are plotted in Figure 3.10 and Figure 3.11. The zero value
of the k, sensitivity coefficient at the midpoint of the flux boundary implies that no
information is gained by placing the sensors there. Placing the sensors at x=a/2 on the
heated surface will then decrease the value of the determinant D.

The determinant, D, is then calculated from Equation (3.3). The matrix
coefficients defined by Equation (3.4) are determined using the maximum temperature
rise, Tm", found from Equations (3.17) and (3.18) with x+ = O, and the sensitivity
coefficients, Xfi and X;, are found from Equations (3.19a,b and 3.20a,b). In this case,
the solutions resulting from different heating times, th", as well as different thermocouple
combinations were found and are shown in Figures 3.12 and 3.13 respectively. Finally,
six global design curves (Figure 3.14) were generated for different values of the ratio of

the heated surface to the whole surface; these curves involve the geometry and the

thermal properties of the sample to be tested.

3.4 Results and Discussion

The determinant, D, was compared for different experimental conditions, such as

57

 

 

 

 

 

 

 

0.15 I I I I I I I T I
+'_ -1
X
H—. 0.10“ __________________________________ {(O_::____:
1+_ I
0) _ ’, ________________________________________________________ a
o 'I
II X/O=O 75
O 005—; 4
US l
g 1 ~
m ....................................................................... 119???? ...... _.
U)
(I) .
3.)
(C) ._
._ =D.25
m x/o
C
OE x/o=0 —
Q ‘ _
”0.15 I I i I i I i I
O 2 4 6 8 TO

Dimensionless Time

Figure 3.10 X-axis Directional Thermal Conductivity Sensitivity Coefficient for
the X22BOOY12BOx5TO Case -

58

 

 

 

 

DimenSIonless Sens. Coeff. X;
I
.0
07
I
I
I
l
l
l
l
I
I
I
l
I
l
I
I
l
I
I
I
I
l
l
|
I
I
I
|
l
I
I
I
l
I
I
u

L
l

 

 

|
.0
\l
-
_.
-
_
.
_
.

 

O
N
is
07
CID-1
S

Dimensionless Time

Figure 3.11 Y-axis Directional Thermal Conductivity Sensitivity Coefficient for
the X22BOOY12BOx5TO Case

59

 

Determinont D

 

 

 

 

Dimensionless Time

Figure 3.12 Determinant, D, for Different Heating Times for the Finite
X22BOOY12BOx5TO Case

60

 

 

 

 

{‘3
(\j 0.015 0.015
ll
g R\ 1943 (IX/‘1‘ ____________
«E- * ‘.‘\\\ ............. ..
m 0.01 — \ / « 0.01
g . “\\ /’
.\ j
Lo“ / ‘\\\ ' ,/ 0.25,x/a
O O.75,X/a ‘R\ / i
3; 0.005 — \t\ / ~ 0.005
E O .5.x/a \\ /'
Q ~ ~ _______ ’17. ______________
E \ \‘ . / I \\\\\\ ,/ I 1
9 x >«\
E 0 m 1 /.'I’ 1 \ w / 1 $\._..1__ A -- 0
£193 0 0.2 0.4 0.6 0.8 1
Q x/a Position

Figure 3.13 Determinant, D, for Different Heated Surface Thermocouple
Combinations for the X22300Y12B0x5T0 Case

0.03 _

9
O
[\J

Determinant D
O
O
a

Figure 3.14 Global Design Curves for the XZZBOOYIZBOXSTO Case

61

 

l

’—

 

[a1/a = 0.66 x x X
0.025 f

\o°

0

OX

81/8 = 0.75

/

X

81/8 = 0.9 :

A

—4

l

1 L 1 l L l l

l

0.03

1 0.025

0.02

0.015

0.01

0.005

 

 

a/b sqrt(ky/kx)

62

different heating times (tf), total measurement time (ff), and sensor locations, to
determine the optimal experimental conditions. In the first case, a finite one-dimensional
composite was considered, and the experimental variables included heating time, total
experimental time, and sensor location. In the case of the thick composite, the experiment
was optimized with respect to heating time and total experimental time. In the third case,
a finite two-dimensional composite was considered; the experimental parameters of
interest for that case were the heating time as well as the sensor locations. Finally, some
optimal solutions were obtained for four simple experiments.

3.4.1 Finite One-Dimensional Composite (X21B50T0) (Case 1)

The Optimal criterion used in this study is based on the determinant, D, which
involves the sensitivity coefficients, X,‘ and Xz“. Therefore, investigation of the
sensitivity coefficients can be useful in providing insight into the optimization procedure.
For this investigation, the heating time, If, is considered to be equal to the total
experimental time. Figure 3.3 corresponds to the transient change of the thermal
conductivity sensitivity coefficient, XE. Each curve starts at zero and goes to a non-zero
negative, steady state value; the magnitude of the sensitivity coefficients is largest at the
heated surface. Figure 3.4 shows that the volumetric heat capacity sensitivity coefficient,
Xz“, becomes essentially zero shortly after the dimensionless time, t“, equals 2. This
4 indicates that little additional information is obtained using values of t+ greater than two
for the estimation of the volumetric heat capacity. The sensitivity coefficients, as shown
in Figures 3.3 and 3.4 are not linearly dependent on each other; consequently, k and pc

can be simultaneously and independently estimated.

63

The first experimental variable investigated was the heating time, th“, for the heat
flux boundary condition at x+ = 0. Five different dimensionless heating times were
considered, and the results for the determinant, D, are shown in Figure 3.5 for a single
sensor at x+ = O. The curve having the highest peak represents the maximum value of
the determinant, D; this value of D equal to 0.0195 corresponds to a dimensionless
heating time of about 2.5, and a "cooling" time of 0.73. A heating time of 2.25 results
in a slightly higher maximum value of D; in this case, the maximum value is
approximately equal to 0.020 which occurs at 3.

An interesting aspect of Figure 3.5 is that the optimal heating time curve obtained
by joining the peaks of the four different adjoining curves has a rather flat peak between
dimensionless heating times of 1.5 and 3.5; this implies that any values used within this
range will be close, in terms of the optimum, to the optimal value. Hence, the choice of
the optimal time does not have to be precise. Notice that the choice of the duration of
the experiment after heating, tn*-th*, is crucial. The four high-peak curves show a sudden
drop in the value of D, which means that taking the data longer than the time at which
D is maximum lowers the value of D and degrades the quality of the sought thermal
properties k and 00 (it is assumed that the same number of measurements is taken
regardless of the experiment duration). The maximum value of D occurs a constant 0.73
dimensionless time interval after the heatin g time; this implies that the total dimensionless
duration of the optimal experiment is about 3. Note also that for a given heating time,
an error of say 10% in the chosen optimal duration of the experiment has less effect on

driving the value of D away from the maximum one than a 10% error in an experiment

64

lasting longer, and for which the value of the determinant is away from the peak. Note
that information regarding the rapid degradation (if the same number of measurements is
spread over a large time) is lost if an optimizing program is used and only the maximum
is found.

Another clarification might help. More measurements invariably contain more
information if the same size time step is taken, such as going to time 125 seconds rather
than 100, both with time steps of one second. In such cases, the confidence interval
should decrease with increasing number of measurements. That is not what is being held
constant in this analysis; the number of equally spaced measurements is held constant.
If the total duration of the experiment is allowed to become large for a fixed number of
measurements, then it is possible that the finite duration of heating experiment gives
poorer results than if heating occurs over the total experiment (see for example Figure 3.5
for th*=2 and t” greater than 5).

The next factor considered was the sensor location. Figure 3.6 shows four curves
corresponding to four different sensor locations. The maximum value of the determinant,
D, corresponds to the sensor at the heated surface. This is because, as shown in Figure
3.3, the sensitivity coefficients at the heated surface have the greatest magnitude. It is
then concluded that, when using a single sensor, it is best to place it as close to the
heated surface as possible. Placing a sensor at the heated surface can have its
disadvantages in practice; it may be difficult to place a sensor at the heated surface, and
doing so can magnify errors in the sensor’s reading caused by contact resistance and

temperature disturbances resulting from the sensor’s size.

65
3.4.2 Semi-Infinite One-Dimensional Composite (X20B5TO) (Case 2)

For the semi-inifinite body with a constant surface heat flux, the experimental
variable investigated was the duration of the heat pulse referred to as the heating time,

4.

th . The determinant, D, with one sensor at the heated surface and a secOnd one at an
internal location, x0, was calculated for six different dimensionless heating times. The
resulting curves are shown in Figure 3.7. This figure shows that the optimal
dimensionless heating time is approximately 1.5 which corresponds to a value of D of
0.0055. The Optimal duration of the experiment is shorter for this geometry than for the
finite body geometry; indeed, the total optimal experiment lasts only a heating time period
plus about 0.22, making the optimal dimensionless total time equal to 1.7. The abrupt
increase after heating stops, and rapid decrease after the maximum illustrate the
importance of plots such as Figure 3.7. If the duration of a fixed number of equally-
spaced measurements is extended, the value of the determinant decreases so much as to
become smaller than that one corresponding to collecting measurements only over the
heated period.
3.4.3 Comparison of One-Dimensional Results

At this point, a comparison of the current study with other published results is
needed. Two geometries are described by Beck and Arnold (1977), a finite and a semi-
infinite geometry. These two geometries are subjected to boundary conditions different
from the ones used in the current study. Table 3.1 lists the boundary conditions, the

locations of the temperature sensors, and the values of the determinant D for the different

cases. Cases I, III, IV, and V come from Beck and Arnold (1977), while cases 11, VI,

Table 3.1 Comparison of the Maximum Determinant, D, Values for 7 Cases.

66

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Geom Boundary Sensor Max. Opt t; Opt tn“
Condition(s) Location(s) D
semi- (X20B1T0) heated 0.00263 tn+=1.5 tn*=l.5
inf. surface and
in-depth
semi- (XZOBSTO) heated 0.0055 1.5 1.72
inf. surface and
in-depth
finite (X22B 10T0) heated 0.00098 tn+=1.2 tn*=1.2
surface
finite (X22B10T0) x = O and 1 0.0058 tn*=0.65 tn+=0.65
finite (X22B50T0) x = 0 and 1 0.0088 0.4 0.6
finite (X12B05T0) heated 0.020 2.25 2.98
surface
finite (X12B01T0) heated 0.012 tn+=7 tn"=7
surface

 

a. Beck and Arnold (1977)

b. This study

 

 

67

and VII are from the present study. For the finite geometry, cases VI and VII show a
larger value of D (0.02 and 0.012 respectively) than cases III, IV, or V. This implies
that, for a finite geometry, to estimate the thermal properties of interest, it is better to
have a constant temperature boundary than to insulate one side with a finite duration heat
pulse on the other side. For the semi-infinite geometry, a finite duration heat flux
boundary condition with measurements lasting 0.22 after the end of the heat flux (with
D=0.0055) makes a better experiment than a continuous heat flux that lasts during the
whole experiment (where D is 0.00263).

3.4.4 Finite Two-Dimensional Composite

The experimental variables considered for the two-dimensional composite were the
duration t; of the heat pulse, as well as the locations of the thermocouples. The
determinant, D, was calculated for five different heating times using five different
thermocouples spread evenly along the surface that is half-heated and half-insulated. This
determinant was then plotted as shown in Figure 3.12. The time dependence of this
determinant is very different from that of the one-dimensional cases. For the two-
dimensional case, the time to achieve the optimal experiment corresponds to steady state,
and the optimal duration of the experiment is the same as the heating time period.

The next factor of interest was the locations of two thermocouples. Figure 3.13
shows that two thermocouples are best placed at the two ends of the flux boundary
surface; i.e. x=0 and x=a. This Figure also shows that the determinant D is larger when
the thermocouples are placed at the insulated half rather than at the heated half. Finally,

some global design curves were developed for two-dimensional experiments with setups

68

similar to the one described above. The curves were obtained for different values of the
ratio of the heated surface to the whole surface. These curves involve the geomeu'y and
the thermal properties of the sample to be tested; the curves are shown in Figure 3.14,
and represent the determinant, D, as a function of the non-dimensional quantity
(a/b)sqrt(ky/k,). The quantity (a/b)sqrt(k,/kx) combines the composite’s geometry
parameters, a and b, as well as the directional properties of interest, k, and k,. The
curves show that for the design of an optimal experiment, (a/b)sqrt(k,lk,) needs to be
around 2.25. In other words, to achieve an optimal two-dimensional experiment on an”
orthotmpic material, the specimen needs to be cut in such a way that (a/b)sqrt(k,/k,) is

around 2.25 and three quarters of its surface need to be heated.

CHAPTER 4
THERMOCOUPLE ERROR ANALYSIS

4.1 Introduction and Literature Review

With the development of new materials for automotive and aerospace industries,
power generation, and medicine, there is a need for the quantification of the thermal
behavior of these materials and various transient thermal phenomena. By studying this
thermal behavior, knowledge about the safety and feasibility of the materials and about
thermal processes can be gained. These thermal studies might involve the estimation of
the thermal properties thermal conductivity and volumetric heat capacity. Another related
problem is the inverse heat conduction problem (IHCP), which involves the estimation
of the surface heat flux history from interior transient temperatures. In problems such as
estimation of thermal properties, investigation of chemical reactions in solids and the
inverse heat conduction problem, the results can be extremely sensitive to measurement
errors. This chapter presents a study of the measurement errors motivated, at this point
by the estimation of thermal properties from transient measurements (Scott and Beck,
1991, Loh and Beck, 1991, and Garnier, Delaunay, and Beck, 1991). Transient

measurements were used in part because of characterizing composite materials during the

69

70

cure cycle.

Thermocouples can be placed either parallel or normal to the heated surface as
shown in Figure 4.1. In both cases, the temperatures measured by thermocouples are
those of the junction and not of the specimen. In this chapter, the sensor is parallel to
the heated surface (Figure 4.1b). For the case of low conductivity composite materials,
the thermocouple has a relatively high thermal conductivity and volumetric heat capacity
resulting in a temperature measurement error. Certain systematic errors were noted.

A number of papers dealt with thermocouple errors; for example Beck (1962)
used finite differences to analyze the case of a thermocouple normal to the heated surface,
and showed that the disturbances can be excessively large. Pfahl and Dropkin (1966)
used an implicit finite difference approach to study the case of a thermocouple parallel
to the heated surface in low conductivity materials; the authors assumed the thermocouple
to have a square cross-section and considered different parameters affecting the
thermocouple disturbance. This configuration inherently introduces smaller errors than
that normal to the heated surface, provided the location of the sensor is known. Beck
(1968) also computed some correction kernels in a form of Duhamel’s superposition
integral to determine the undisturbed temperatures from the thermocouple measurements;
the case analyzed was that of a thermocouple normal to the heated surface. Larrain and
Bonilla ( 1968) calculated the error due to leakage current flowing through the electrical
insulation between thermocouple wires for metal-sheathed swaged thermocouples used at
high temperatures; the authors considered temperature errors for sheathed and unsheathed

thermocouples with step change and linear temperatures. Dutt and Stickney (1969)

71

 

 

 

 

 

 

 

 

 

 

 

 

(a) Normal Configuration

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(0) Parallel Configuration

Figure 4.1 Thermocouple Placement Relative to the Heated Surface

72

considered the error due to conduction through the wire when measuring fluid or solid
temperatures; their analysis used fin theory and energy balance ideas. Yoshida,
Yamamoto, and Yorizane (1982) performed a finite difference calculation to determine
the error due to the radial insertion of the thermocouple in a thermal conductivity
measuring device; they found that the insertion of the thermocouple led to a higher
temperature at that location as compared to without the thermocouple. Balakovskii (1987)
considered the inverse problem of estimating the surface heat flux using thermocouple
temperature measurements. He studied the sensitivity of the estimated heat flux to errors
in the thermocouple location; he showed that location uncertainties cause an error in the
estimated heat flux mostly at early times. Balakovskii and Baranovskii (1987) performed
a similar study; they considered a thermocouple normal to the heated surface, and
included the thermocouple effect in their model for the estimation of the surface heat flux.

The goal of the current study is to extend the analysis of Pfahl and Dropkin
(1966). Longer dimensionless times are investigated, larger thermocouple depths are
considered, and the circular cross-section is treated The circular cross-section was treated
using the finite element (FE) method; however, the FE or finite difference (FD) method
is not completely satisfactory for this problem. For the problem of small disturbances
(some less than one percent), the FE and FD methods can have difficulty. Even though
the errors may be small, they are not random and have amplified effect on the estimation
of thermal properties, and upon the II-ICP. For these and other reasons, this study utilizes
the Unsteady Surface Element (USE) method with a single element. The method is

closely related to the Boundary Element Method (BEM) (Brebbia, 1978).

73
The USE method was introduced by Keltner and Beck (1981), and used by

Litkouhi and Beck (1986), by Beck and Keltner (1987), and by Sobolik, Keltner, and
Beck (1989). One advantage of the USE method is that only the interface nodes need to
be considered initially and later the temperature at any interior or boundary location can
be obtained using a convolution equation; as a result of the much smaller grid, the USE
method uses less computer time than the FE or FE method. Another advantage is that
the USE method treats semi-infinite bodies more readily than the two other methods. FE
and FD methods are potentially much more accurate for determining the temperature level
than for determining small differences in temperature at two locations (such as at the
thermocouple and far away from it). The USE method is also more powerful than the
two above-mentioned methods when the quantity of interest is a small difference in
temperature (which is what the thermocouple disturbance is). A further very important
characteristic of the USE method is that many times it leads to relatively simple algebraic
solutions which cover a wide range of conditions; this permits greater insight, such as the
behavior for infinite times. Such simple solutions are given in this chapter.

An outline of the chapter follows. First, a general formulation of the problem is
given. Second, a Green’s function solution to the problem is provided, and its results are
compared to those of the FE and FD methods’ solutions. Third, the form of the solution
is compared to that of obtaining the undisturbed temperatures using thermocouple kernels

as shown by Beck (1967). Finally, some conclusions are given.

4.2 Theoretical Procedure

74
4.2.1 General Formulation

Q

The thermocouple wire is assumed to extend infinitely parallel to the heated
surface of a semi-infinite solid representing the surrounding material (Figure 4.1b).
Perfect contact is assumed to exist between the thermocouple and the surrounding
material, but the method can also readily treat the case of imperfect contact. The

describing differential equation and its boundary conditions are:

_a_(k§Iy-_§_(k§r_) = peg —oo<x< 00,0 gy< an, t 20 (4.13)

axaxayay at

31‘
-k— = y=0,t>0 (4.1b)
fly so
Ttx.y.0)=o (4.1c)
T(X,y,t) *0 x—ooe,x-+ —oo,y-ooo (4.1d)

where T is the space-dependent temperature, k and pc are the constant thermal properties
outside the thermocouple and different but constant inside the circular region of radius,
a, centered at y=L and x=0 (Figure 4.1b), and q0 is the imposed constant boundary heat
flux.

For simplicity, the thermal properties are assumed to be independent of
temperature. This is an important case; certainly thermal properties vary with
temperature, but the main error is caused by local conditions. Hence for small sensors,
these results may also have some validity for temperature-variable properties. It is also
true that small errors are being considered. An inaccuracy of say ten percent in an error

which itself is only two percent leads to an error in the correction of only 0.2%, which

75

is much smaller than the uncorrected value of two percent.

The temperature disturbance caused by the presence of the thermocouple is mainly
due to the difference in volumetric heat capacities of the wire and the surrounding
material. The thermal conductivity of the wire is very large compared to that of the
surrounding material, making the lumped capacitance assumption an acceptable one for
the wire. A control volume approach is then used to obtain an expression for the driving
force behind the disturbance. The procedure is based on three observations. First, the
relatively high thermal conductivity, kw, of the wire (or thermocouple assembly) causes
it to be nearly uniform in temperature; it can however vary with time. Second, the heat
capacity of the wire minus that of the material that would be there (if the wire were not)
is the main cause of the temperature disturbance. Third, the effect of lumped wire is the
same as that of a volume heat sink, which is proportional to the rate of change of its
temperature. In other words, a control volume just outside the wire should have the same
temperature and same net integrated heat flux for the actual problem of a relatively high
thermal conductivity wire as for a volume energy sink. This USE analysis permits easy
extension to larger dimensionless time and provides an analytical form that yields insight
more readily than purely numerical solutions.

The excess energy entering a control volume about the wire of radius a is

 

2" ‘ 4.2
[(‘q.fi)ad6 =f(pc-pcw) arw(t)21trdr ( )
o 0 at

where a is the radius of the thermocouple wire, (I is the inward pointing heat flux, and

fl is the outward pointing normal. pc and pcw are respectively the volumetric heat

76

capacities of the region surrounding the thermocouple, and of the thermocouple wire. Tw
is the temperature of the wire. Since the temperature of the wire is independent of
position (lumped capacitance), Equation (4.2) becomes

dTw(t)
dt

1:82 (4.3)

 

21:
f(-q.fi)ade =(pc-pcw)
0

If both sides of Equation (4.3) are divided by the cross-sectional area 1ra2 of the wire, the

average volumetric sink is found to be

dTw(t)
dt

O<r<a (4.4)

 

g(t) = (pc - 96,.)

The derivative of temperature with respect to time is a total one rather than a partial one
because of the lumped capacitance assumption. The describing partial differential

equation then becomes

821‘ + 621‘] +g(x,y,t) = peg, -oo<x< 00,0 gy< on, tZO (405)

k[
6X2 ayz at

 

 

where g(x,y,t) = g(t) as given by Equation (4.4); g(t) is non-zero only at the location of
the wire which is located at x=0, y=L (see Figure 4.1b).
4.2.2 Green’s Function Approach

In this analysis, Green’s functions are a natural tool because the disturbance is
caused by a heat source. The current Green’s function analysis is based on the work of
Beck (1983), and on the book by Beck, et a1. (1992).

The Green’s function corresponding to an instantaneous, cylindrical source
surrounded by an infinite medium is denoted GRoo(r,t|r’,t). The R subscript denotes the

radial coordinate, the first zero indicates a natural boundary condition at r=0, and the

77

second zero is for the condition r —9 00. This notation is described more fully by Beck
and Litkouhi (1988), and Beck, et al. (1992).

The temperature of the wire is assumed uniform with position, i.e., lumped, due
to the relatively large thermal conductivity of the wire. At early dimensionless times, the
temperature of the wire is affected by the temperature, time, and space variations near the
wire and this is better investigated using the FE and FD methods. The dimensionless
time period of interest can be conveniently broken into three parts. The first part is
before the temperature penetrates from the heated surface to the thermocouple and shortly
thereafter; this corresponds to dimensionless times less than about at/L2=O.06. In such
times, the assumption of uniform temperature in the wire may not be accurate, compared
with the temperature rise. However, the temperature rise is extremely small at these
times. It is not difficult to analyze times of at/L2=0.06 using the FE and FD methods.
The second part of the dimensionless time period is the period of Ott/L2 between 0.06 and
0.31; in this time period, the heated surface at y=0 does not cause any thermal
"reflections" at y=L. During this period of time, the disturbance caused by the
thermocouple is similar to that caused by a heat sink in an infinite medium. This can be
modelled by using the GRoo(r,t |r’,‘t) Green’s function. The third part of the dimensionless
time period is for at/L7'>0.31. In this time period, the effect of the-surface at y=0 is
important at the location of the thermocouple. The effect of this boundary is equivalent
to having a second fictitious source a distance twice the depth of the thermocouple from
the wire, 2L. The Green’s function contribution describing this second source is given

by Gxooyoo(0,2L,t|0,0,1:) as

78

~(2L>’

_ 1 a -r (4-6)
G O L,t 0’0, ‘— 4 (t )
WM ’2 I 1’ «ma-1:)”

 

where the notation XOOYOO denotes a line source in Cartesian coordinates.

We first obtain the average wire temperature as (Beck, et al., 1992):

T, (t) = T..(t) +

r=a r’=n

t=t
if ‘1—2f f 8(7)Gm(r.tlr’.1)21tr’dr’21rrdr dr, (0.06<atlL2<0-31)
"0 11:8 r -O r’-0

(4.7)

This equation has two basic parts. One of these is T-(t), which is the undisturbed
temperature at the same y=L location as the wire but a large x distance away. The other
term in Equation (4.7) is caused by the volume energy.

There are two integrations over space in Equation (4.7) because one (on r’) is
needed for integrating over the annular source, and the other (on r) is for finding the
_ average temperature from r=0 to a. It should be noted here that the analysis assumes the
whole wire as a cylindrical source of uniform temperature and not just one line; this leads
to averaging the temperature over the cross sectional area of the wire which, in turn, leads

to the use of the avera e' Green’s function G (t-r) as
8 R00

m r’q
Gm(t-r)=-—13f f Gm(r,t|r’,r)21tr’dr’ 21trdr
753

"° "'° (4.8)

l
=_1_ _ ’35 _1_. . _1_ =1 _
m2{1 e (Inch) 14211)]. 820 r)

where 10 and II are the modified Bessel functions of the first kind of orders 0 and 1.

Equation (4.8) is denoted Gama—T) and is the average of the integrated GRoo over r’

79

(weighted with r’). This relation was derived by D. Amos of Sandia National

Laboratories and is contained in Beck, et al. (1992). The quantity 7:32 Gama—T) starts

at the value of 1 for u=O and decreases gradually to zero with increasing u. Using the

definition given by Equation (4.8), Equation (4.7) can be written as

1:!

tht) -T.(t ”E f GEO-ogtomzdr (49)
1-0

The expression for the wire temperature disturbance for times Ott/L2>0.31 then

 

 

becomes
ar=t az _ (2L)z
T t-T t=— G t- a2+ ————-——e “0") d
.0 -0 do m( on am“) (1) r (410)
T=t L2
_ -— dT r
=Apcf Gm(t-t)1ta2+———a w") -——"( )dr
pc "0 4a(t-r) dt

 

 

Notice that Equation (4.10) can be considered as an integral equation for the unknown
function Tw(t) which is valid for any time greater than say at/LzzOJ.

One major simplification to the solution of this equation is the result of small
temperature disturbances. For such cases, the rate of change of the wire temperature is

nearly equal to the rate of change of the undisturbed temperature, T..(t); in equation form,

 

we have
dTw .. £- (4.11)
dt dt

This undisturbed temperature, T..(t), is the temperature distribution in a semi-infinite solid

subjected to a constant heat flux. At the thermocouple location x=L, it is given'by

80

(4.12)

2J7>=2%r71ifi22fl

where ierfc is the integral of the complementary error function. Then,

“10:32? leg; (4.13)
dt k rt J;

This simplifies the solution of Equation (4.10) in the sense that the equation becomes an

     

T (t)= 2—fifi ierfc(

 

 

explicit expression of Tw(t); it is no longer an integral equation. Equation (4.13) is then

used to obtain the normalized temperature disturbance Ad) such that

 

 

 

 

A rzt 2 -i qo __I:2_
T..(t)—T..(t) =—pc f Gnu (t-t)1ta2+——a e “(W — .35.: “rd: (4°14)
pc "0 4a(t-1:) k 1:1:
01‘
t 2 -1 _-_r_
M=_A_P_c_ fG ma_ T)1L'82 +_I:I___:_e t°-t’]__1_e4t’dt+ (4.153)
pc 1"0 4(t "T ) ”1+
where
k at at a
A =_ t -T. t" , t" =_, 1+=.._’ H=— (4.15b,c d C)
d> «101-W ‘) ( )1 L2 L2 L , ,

Equation (4.15a) reveals that the temperature disturbance is directly proportional to the
ratio Apc/pc, which represents the difference of volumetric heat capacities between the
thermocouple wire and the surrounding material, normalized with respect to that of the
surrounding material. As a result, dividing both sides of Equation (4.15a) by C = Apc/pc
eliminates the explicit dependence of the disturbance on the difference in volumetric heat

capacities. This fact is not apparent from FE and FD analyses. The numerical evaluation

81
of Equation (4.15a) was done using the IMSL library.

The normalized temperature disturbance is computed for non-dimensional times
as large as 1000, and for thermocouple radius to depth ratios, H, of 0.25, 0.125, 0.0625,
and 0.03125; numerical integration using some IMSL subroutines was performed for this
purpose. The temperature disturbance is shown in Figure 4.2 as a function of
dimensionless time. The data generated by Pfahl and Dropkin (1966) (Figure 8) for the
case of H = 0.25, k/kw=0.025, and Apc/pc=-9, as well as the data obtained using the FE
code TOPAZZD are also shown in Figure 4.2 which is discussed below.

In order to decide which method was more accurate, the competing results were
compared to the FE extrapolated results corresponding to a grid size equal to zero. This
was done by setting up three different FE grids, one with 374 nodes, another one with
825 nodes, and a third one with 1452 nodes. A (Ax)2 extrapolation was then used to
obtain the expected FE results corresponding to (Ax)2=0. For reasonably sized grids, the
dimensionless temperatures varied in a linear fashion with (Ax)2. The results of the
extrapolation are shown in Figure 4.2 along with the FD results and the USE ones. That
figure shows that, for dimensionless times greater than 3, the thermocouple disturbances
calculated by the USE method are the same as the FE extrapolated results. For
dimensionless times less than 3, the USE and FE results are different; this difference was
expected since, as mentioned above, the USE method does not apply to early times.

It should be noted here that, in generating the FE results, the calculated
undisturbed temperatures were up to 1.6% different from the exact temperatures obtained

by Equation (4.12). This limitation, added to the fact that the FE-calculated temperatures

A¢/c

82

 

 

LOOE—OS:
i

1

[3—8 H=0.0625

P&D (Fig. 8)
extrop. FE
H=0.25

  

 

 

I] I T frtrrrf 1 T

1. 10. 100.

0—0 H=0.03125
1.00E-O4 . r .

Dimensionless Time

Figure 4.2 Theoretical, FE, and FD Thermocouple Disturbances

 

83

depend on the grid size, made the FE method not reliable for our purposes of estimating
thermocouple disturbances.

The USE curves of Figure 4.2 show linear dependence on HB ; a least squares fit
showed that B is about 1.9. Larger values of H lead to larger values of the temperature
disturbance. This allowed the normalization of the curves with respect to H”, and Figure
4.3 was then obtained.

4.2.3 The Unsteady Surface Element Method Versus Inverse Convolution

The above method of obtaining the thermocouple induced temperature disturbance
is now compared to the method described by Beck (1968). The latter uses thermocouple
function kernels, K(t), which need to be computed by numerical inversion of a
convolution integral. That inversion method employs a least squares procedure similar
to the inverse problem of estimating the surface heat flux from internal temperature

measurements. The expression for the disturbance given by that method is

 

Tw(t)-T.(t)= - f K(t—r)ar‘a't(t)d~c (4.16)
1-0

where Tw(t) is the measured function of time. Notice that this equation does not require
any inverse convolution calculations to find the error in the measurements, with K(t)
known. Note also that K(t-t) is a universal function in the sense that it is independent
of time variation of Tw(t); it does however depend upon the geometry.

Comparing Equation (4.16) to Equation (4.10), the kernel K(t-t) is related to

Green’s functions; more precisely, for the present problem, we have

A¢/(C*H1'9)

84

 

1.00

 
 
 

L A. Anti l I

———_——_—_1

 

H=0.03125
H=0.0625
H=0.125
H=0.25

  

 

 

0'01 I I I I I IIrI I I I r1 rrT] I
l. 10. 100.

Dimensionless Time

Figure 4.3 Normalized Theoretical Thermocouple Disturbance

IfIIIII

1000.

85

L2

G— t-r rta2+—a—e-“("‘)
3"“ ) 4a(t-r)

A pc (4.17)

K(t-t) =
pc

 

which is shown in Figure 4.4 along with G 3006-1“) for different a/L ratios. The kernel

K(t-t) in Equation (4.16) then represents a Green’s function. This is an important
observation and has significant applications; it means, for example, that the error using
Equation (4.16) can be found provided the Green’s function K(t) can be found. Since
K(t) represents a Green’s function, it is not necessary to solve the complete problem as
has been done herein; instead, one can concentrate on obtaining K(t). No inverse
convolution is now needed to obtain the kernel K(t-t); instead, this kernel can now be
computed from known Green’s functions which can be obtained using exact expressions

available in the literature such as (Beck, et al., 1992).

4.4 Results and Discussion

The restriction to relatively large values of thermal conductivity of the
thermocouple compared to that of the surrounding material is not severe for measurements
in low conductivity materials. For example, a surrounding material might be epoxy or
composite material whose thermal conductivities are about 0.2 W/m°C. The thermocouple
assembly can be an alumina sheath through which two wires pass; the thermal
conductivity of alumina is about 40 W/m°C. The ratio of the two thermal conductivities
is about 200, which justifies the assumption of uniform temperature inside the wire.

The major simplification used with the Green’s function solution (Equation (4.11))

Kernel K(t—T)

86

 

  
 
  
  

 

 

 

 

1.0 I I IIIIIII I I I IIIIII I rI IIIIII _T TI IIIIII
— C(o/L=1/2)
* K(o/L=1/2)‘
6(o/Lz1/4)
08‘ -- K(o/L=1/4)‘
‘ -- G(o/L=1/8)d
—- K(o/L=1/8)
0.6— —
0.4—
0.2—
0.0 I I jfrIIIT I I I I IIIrT I I I I IIIII I I I ITTII
lE—03 lE—02 lE—01 1 l0

Dimensionless Time (at/oz)

Figure 4.4 Kernel K(t-t) for Different Thermocouple-to—Depth Ratios

87

was checked using FE computations (TOPAZZD). The transient temperatures at the wire
location and at a "distant" location were obtained; the rates of change of the temperatures
at the two locations with time were then found to be the same.

Figure 4.3 shows the temperature disturbance normalized with respect to C=Apclpc
and H”. The dimensionless disturbance is of the order of 0.4 for dimensionless times
between 0 and 100. In some experiments performed in our laboratory on a DER332/DDS
epoxy sample, a heat flux of 4700 W/m2 causes a maximum temperature rise of about
100°C. The thermal properties of this epoxy are about 0.2 W/m°C for k and 1.5)(106
J/m3°C for pc. Using this information along with Equation (4.15a), the dimensionless
temperature disturbance of 0.4 corresponds to about 1°C. Since Equation (4.15a) was
derived for an infinitely long thermocouple, it can be induced (by linearity of the
problem) that the disturbance for the finite length thermocouple is only 05°C. The
maximum temperature disturbance is therefore about 0.5 percent of the maximum
temperature rise. This is a small value; however, as shown in Chapter 2, when estimating
the thermal properties from transient measurements of temperature and heat flux, the
residuals (difference between the calculated and measured temperatures) for this case were
about 2%; this means that the disturbance caused by the thermocouple is 25% of the

discrepancy between the model and the experiment.

CHAPTER 5

EXPERIMENTAL INVESTIGATION OF THE THERMOCOUPLE ERROR

ANALYSIS

5.1 Introduction

The process of estimating thermal properties, and finding ways to improve
transient parameter estimation experiments is a combination of two interrelated tasks: an
analytical one and an experimental one. The analytical task was presented earlier in this
dissertation in Chapters 3 and 4. Part of the experimental task is presented in this
chapter. This experimental task consists of "measuring" the temperature disturbance
caused by the measuring device (here a thermocouple). To achieve this goal, a series of
experiments was designed and then performed in the Laboratoire de Thermocinetique-
ISITEM Nantes-France. The material of interest is a cured epoxy, similar to the one of
. Chapter 2. The specimen has an imposed finite-duration constant heat flux on one side
and a 40°C constant temperature on the other side.

This chapter describes the design, implementation, and results of the series of
experiments. The first part of the chapter briefly describes some criteria used in

designing the experiments, and making the specimen. The next part of this chapter

88

89

explains the procedure followed. The third part of the chapter describes the setup used.
The fourth part of the chapter describes the experiments performed on the specimen, and
compares the results to those of Chapter 4. In that same part, the thermocouple locations
are estimated, and the corresponding disturbances are calculated. Unfortunately, the
position errors seemed to mask much of the errors caused by the thermocouples
themselves. A sensitivity analysis on the importance of different parameters was also
performed. The last part of this chapter discusses the results of the experimental

investigation.

5.2 Some Design Criteria of the Experiment

In designing the thermocouple error analysis experiment, four criteria were
satisfied. First, the thermocouple-induced disturbance had to be large. Second, the
different experimental parameters such as heat flux, temperature, and thermocouple
locations had to be known as accurately as possible. Next, the sample had to be thick
enough to behave as a semi-infinite body during the experiments; in other words, none
of the thermocouples "sees" the back boundary of the sample. Finally, the thermocouples
had to be placed far enough from each other to minimize thermal interaction, and to not
disturb each other.

To satisfy these criteria, thermocouples of three diameter sizes were selected:
0.16mm, 0.5mm, and 2mm and were numbered as shown in Figure 5 .1. The 0.16mm and
0.5mm-diameter wires were used because those were the largest diameter wires available

in the laboratory. The 2mm-thermocouple was a little different. It was made from a

90

 

O

 
 
 

0000

00 01 02 03 04 05 06 07 08 09 1o

 

 

* distances between thermocouples not shown to scale

Figure 5.1 Thermocouple Numbering for the Estimation of Thermocouple
Disturbance

91
Chromel-Alumel male plug. The leads of the plug were cut, then welded tip-to-tip at the

junction. A platinum gage was used to measure the reference temperature that was
needed in the measurement of all the experimental temperatures. The 140mmx140mm
DER332 epoxy-based specimen (described with more detail in the next section) was
18mm thick, insuring its semi-infinite thermal behavior. The thermocouples were placed

10mm away from each other to make sure they do not interact during the experiments.

5.3 Experimental Procedure

The first step of this experimental investigation was to build the specimen with
the chosen themocouples. The mold shown in Figure 5.2 was used to make this specimen
with the installed thermocouples. The spring system shown in that same figure kept the
smallest thermocouples in tension to improve— the accurate positioning of the
corresponding junctions. The mold was sprayed with a mold-release compound to
facilitate the separation of the sample from the mold after the epoxy was cured. The
thermocouples were cleaned with an acid-based solution to eliminate any impurities that
might introduce some contact resistance between the thermocouples and the epoxy. These
thermocouples were then stretched across the mold as shown in Figures 5.2 and 5.3. The
DER332-based epoxy was mixed with the curing agent, then poured inside the mold. The
mold was then placed in a vacuum chamber to eliminate all the air bubbles in theepoxy.
The mold was then placed inside an oven to cure the epoxy. Later, the cured sample was
removed from the mold and allowed to cool.

The next step of the experimental investigation was to polish both surfaces of the

92

Thermocouple Wires

 

 

—}/I))i '

 

 

 

 

 

 

 

 

Figure 5.2 Mold and Different Methods Used in Stretching the Thermocouples

93

 

 

 

 

 

 

 

 

 

 

 

 

Figure 5.3 Thermocouple Layout in the Mold

 

94

sample. In doing so, thermal contact between the specimen and the bounding heater on
one side, and between the specimen and the rear plate on the other side was improved.
After polishing the specimen, it was clearly visible that most of the thermocouples had
moved inside the sample. This movement was due to the release of the tension that was
imposed by the spring system. The locations of the thermocouple junctions were thus
altered. The new locations were estimated as described in Section 5.5.6.

The third step of this experimental study was to use the finished sample with a
setup already in place at the Laboratoire de Thermocinetique-ISITEM Nantes-France.

This setup is described in the following section.

5.4 Experimental Setup

The experimental setup was previously built for estimating the thermal
conductivities of low conductivity materials. A schematic of the setup is shown in Figure
5.4. It mainly included a Hewlett Packard (HP)-based data acquisition system, a pair of
Lauda temperature control units, and a hydraulic unit in which the finished sample was
placed.

The HP-based data acquisition system consisted of an HP terminal, an HP scanner
(model 3455A), an HP digital voltmeter (model 3490A), and a computer controlled power
supply unit. The next part of the setup was a pair of Lauda temperature control machines
pumping oil into a closed loop to maintain a constant boundary temperature. The oil
leaves the reservoir of the Lauda machine through a rubber hose to reach the plates

shown in Figure 5.5, then returns back through another rubber hose to the reservoir.

95

 

 

 

 

 

 

  
     
   
 

Thickness Gage Temperature

 

 

 

 

 

 

 

 

 

 

 

HP Terrrinal Control
"232?.I?Z??332773i.:::-. Um I
Vonmeter _. ; m
- raulic Unit
Power Supply -—~ “Yd Temperature
Control
Unit ll
Rubber Hoses

 

 

 

Figure 5.4 Experimental Setup for the Estimation of Thermocouple Disturbance

96

Holding Plate

Cold Source

Epoxy Sample

TC. 1
\’\’\’\’\’\’\’\’\l\ \,‘/‘,\,\/\/\/\/\/\/\ Heat'ng Element
«iii/:IPPI/Z’P . ‘ ”2’: ‘ ‘ :1
Insulation
Hot Source
Holding Plate

 

Figure 5.5 Placement of the Epoxy Specimen for the Estimation of Thermocouple
Disturbance

97

Finally, the hydraulic unit of Figure 5.5 consisted of two plates (labeled hot source and
cold source in the figure) through which the constant temperature oil passes, an electric
heater embedded in an aluminum block, an electronic caliper, a compressed air system,
and finally some side insulation. The electronic caliper allowed the measurement of the
average thickness of the sample. The compressed air system was needed to raise and
lower the upper constant temperature plate. This air system also held the specimen, the
heating plate, and the constant temperature plate together. Good contact was thus
maintained by the constant pressure applied throughout the experiment.

The scanner had thirteen channels and each temperature was read at a different
time. One channel recorded voltage for the power input. Another channel recorded the
current (which is really voltage converted to current by means of a 10-ohm resistor). A
third channel allowed the recording of the reference temperature measured by a platinum
gage (which is an accurate temperature measurement device). A fourth channel was
connected to the electronic caliper, allowing the measurement of the average thickness of
the sample. Two other channels recorded the temperatures behind the heating plate (T C3
and TC8 of Figure 5 .5). Two more channels recorded the temperatures of the specimen’s
boundaries (ICC and TCl of Figure 5.5). Finally, six more channels acquired the
temperatures inside the specimen. The time was recorded every time a channel was
scanned. In doing so, it was possible to circumvent the limitation of sequential reading
of the channels and to minimize any possible timing errors.

A thin layer of silicone grease was applied to the aluminum plate covering the

heating element, and to the back surface of the specimen. The specimen was then

98

carefully placed on top of the heating element. Insulating material of the same thickness
as the sample was placed all around to eliminate side losses. The upper part of the unit
was lowered hydraulically. The thermocouples were then connected to the data

acquisition system, and four successful experiments were run.

5.5 Experimental Results

The thermal properties were needed in the estimation of the thermocouple
disturbance, and had to be obtained first. The thermal conductivity of the specimen was
estimated by running a steady state experiment. The volumetric heat capacity was
assumed known as indicated in Chapter 2. In that chapter, the volumetric heat capacity
of DER332/DDS epoxy was found to be about 1.5)(106 J/m3°C.

The first one of the four successful experiments was steady state and allowed the
estimation of the thermal conductivity of the specimen. The three other experiments were
transient ones performed in order to investigate thermocouple disturbance. More than one
experiment was needed (three here) because the specimen had eleven thermocouples and
the data acquisition system had only six channels available for thermocouple hookup.
5.5.1 Steady State Experiment

The steady state experiment allowed the system to reach a linear temperature
distribution. One boundary was at 30°C and the other at 40°C. The 30°C boundary was
possible by the cold plate of Figure 5.5. The 40°C boundary was achieved by the heating
element and the hot plate of the same figure. The heater was turned on and off by the

data acquisition system every time the temperature difference between TC3 and TC8 of

99
Figure 5.5 was larger than 0.025°C (which is really lpV). The thermal conductivity, k,

of the sample was calculated by

x=a./Li (5.1)

AT

where L is the thickness of the specimen, AT is the difference in average temperature
between the two boundaries of the specimen, and q is the average heat flux imposed on
the specimen. The software . for this experiment was previously developed in the
laboratory; a brief description is given here. The data acquisition system kept sampling
continuously in time; when the difference between the readings of thermocouples TC3 and
TC8 of Figure 5.5 was less than 0.025°C (which is really lpV), the current, voltage, and
temperatures were stored. The process of sampling and storing the different quantities
was repeated sixty times. An average of the stored data was taken and the thermal
conductivity was calculated using Equation (5.1). It was found to be about 0.237W/m°C
at 40°C; this value is 12% larger than the upper limit of the confidence region of Figure
2.5 obtained for a DER332-based epoxy with a different curing agent.
5.1; Transient Experiments

At this point, the thermal properties of the epoxy were known, and the
temperatures recorded by the different size thermocouples were needed to estimate the
disturbance. These temperatures were obtained in a series of three experiments. First,
the system was allowed to reach the constant uniform initial temperature of 40°C. A
lower temperature was not chosen because of the difficulty of controlling temperatures
near the ambient temperature. Once an initial steady state was reached, data collection

started. The heater was turned on around time 33 seconds to impose a constant heat flux.

100

At time 153 seconds, the heater was turned off, and data collection continued until the
end of the experiment (at time 219 seconds).

The temperatures recorded during the three experiments are shown in Figures 5 .6,
5.7, and 5.8. The two temperature curves of Figure 5.6 labeled T3 and T3 correspond to
thermocouples TC3 and TC8 at the back of the heater (recall Figure 5.5). These
temperatures T3 and T8 increased with time from 395°C to about 47°C and 43°C
respectively. The increase in T3 and T8 is an indication that it was incorrect to assume
that 100% of the heat supplied by the heating plate goes into the specimen. The heat flux
at the heated surface of the specimen was then unknown. Knowledge of this boundary
heat flux was crucial for the estimation of the thermocouple temperature disturbance. It
was then necessary to estimate the heat flux. This is done in the following section.

The temperatures recorded by thermocouples 05 (of Figure 5.7) and 06 (of Figures
5 .6 and 5 .8) are inaccurate. The curves T05 and T06 corresponding to these thermocouples
are not as smooth as the other curves; curves T05 and T06 show some unusual oscillations
throughout the whole experiment, even at early times before the heating starts. These
early time oscillations between time 0 and time 33 seconds indicate that thermocouples
05 and 06 are not reliable ones. As a result of the non-reliability of those two
thermocouples, the data they collected had to be ignored in the remaining part of the
dissertation.

Figure 5.6 shows that the experimental heated surface temperature curve, T1, and
the curve Tam for the calculated surface temperature (calculated as shown later in this

chapter) are similar. The maximum difference between the two curves is less than 1°C,

101

 

o
‘
F
.P
r

58— 9.3,
0-0
. ra-a
54— 33-53
. I I
(Am-A
so ""

. e-e
~ 0 o

8838
>\
h
b
b
1’
b
I
b
I
b
l
i
D
I

O

—l—l—l—l—l-—l—i—l—i—-la

  

Temperature (°C)

 

 

 

r T
0 50 100 150 200

Time (sec)

‘

Figure 5.6 Measured Temperatures of Experiment 2

102

 

Temperature (°C)

 

l l l J

1 l l l l l J l l l l

l

l

 

 

 

A I
0 50 1 00

Time (sec)

Figure 5.7 Measured Temperatures of Experiment 3

r
200

103

 

 

 

 

 

 

f T I l' I l I I
de-orm 2
53:34:11,, —:
,A-AT“ q
.o-or, .
8 54— v-vr“ T
\o/ :Ble-JIET,o l
e . .
3 50- ..
+1 ...) 4
r « .
i 'i
g 464 a
m . .
l— 4 «
423 5 a

381 ' r T I I I v r

0 50 100 150 200

Time (sec)

Figure 5.8 Measured Temperatures of Experiment 4

104

which is of the same size as the temperature error at the surface caused by a 10% error
in heat flux.

Notice that thermocouples 04, 09, and 10 were used in more than one experiment.
Thermocouple 04 was used in experiments 2, 3, and 4. Thermocouple 09 was used in
experiments 2 and 4, and thermocouple 10 was used in experiments 2 and 4. This
repeated use of the same thermocouples was needed to check for replication. The
replication sought here was of the first order (recall Chapter 2); it was remarkably good
for all three thermocouples 04, 09, and 10.

5.5.3 Calculation of the Surface Heat Flux and the Thermocouple Disturbance

The calculation of the surface heat flux was motivated by the existence of a heat
flow at the insulated back of the heater. This back heat flow implied that the heat flux
at the surface of the specimen was unknown, not as thought initially to be known from
voltage, current, and surface area (recall Chapter 2). This surface heat flux was expected
to be a constant finite duration pulse. The experiments of interest here were the ones that
led to Figures 5 .6, 5.7, and 5 .8. All three experiments had the same boundary conditions;
the only difference was that different thermocouples were used. One boundary of the
specimen (finite body) was heated by the unknown heat flux, and the other boundary was
held at a constant 40°C temperature. Ideas of the inverse heat conduction problem
(IHCP) (Beck, Blackwell, and St Clair, 1985) were implemented in order to obtain this
heat flux. The software IHCPID (Beck, 1990) was used for that purpose, and the
transient heat flux was estimated from the data of Figure 5.6, using the surface

thermocouples TCO and TCl of Figure 5.5. The calculated heat flux is shown in Figure

Heot Flux q" (W/m’)

105

 

1 200 . . . ,

1000—

800 ~

600 -

400-

200~

 

 

0 50 100

Time (sec)

Figure 5.9 Estimated Surface Heat Flux Using IHCPlD

 

200

106

5.9. This figure shows that the heat flux going into the sample varied with time, and was
not constant as thought initially.

To account for the time dependence of the calculated surface heat flux, a
convolution was needed in the calculation of the exact surface temperature. This

convolution is written as

_ N
“‘25:” ’d directives-Master) (52)

1-1

T(t) =fq()————-—

where the time interval t was divided into N equal time steps, and each time step was
labeled t,; q,- is the constant heat flux at time ti, N is the index corresponding to the time
of interest t, and ¢(q0,t,) is the exact temperature calculated at time t, for the constant heat
flux q0 equal to unity. This temperature ¢(qo,ti) was calculated for a semi-infinite
geometry for the following reason. For dimensionless times less than 0.3 (based on the
thickness of the specimen), the heated surface of the finite thickness specimen does not
"see" the constant temperature boundary, and behaves as the heated surface of a semi-
infinite region. The dimensionless time for the specimen based on the thickness of the
specimen (0.018m) and the assumed known thermal properties (k=0.2W/m°C,
pc=1.5x10°J/m3°C) for the maximum time of the experiment (233 seconds) was only
0.095 (which is smaller than 0.3).

The temperature for a semi-infinite geomeUy with constant heat flux is

T _(qo,t)= -TO +—(/_t ierfc( ) (5.3)

 

2m

where T0 is the initial temperature, q0 is the constant surface heat flux, x is the distance

from the heated surface. The corresponding temperature T(q(t)) for a time dependent heat

107

flux was computed at the heated surface with the convolution of Equation (5.2), then
plotted in Figure 5.6 (as Tam) with the experimental temperatures.

As mentioned above, the temperatures used in the computation of the discretized
convolution integral are evenly spaced in time. The data acquisition system did not allow
this even time spacing, and a linear interpolation was used to compute experimental

temperatures, T

”mm, for each thermocouple at every two seconds. These interpolated

temperatures, T were then used to compute the finite difference convolution of

exp,interp’
Equation (5.2). The calculated exact temperature, Tma(q(t)), for the experimental time-

dependent heat flux was obtained at the specified locations of the thermocouples. Finally,

the temperature disturbance was computed as

i N
Artt>=fqm g(Mtqu-thdt ~2: q,(A¢>(q,. .,.,> - Ad>(qo.t,._,)) <52)
0 {II

where ¢(qo,ti) is the exact temperature calculated at time tI for the constant heat flux q0
equal to unity. This equation gives the experimental temperature disturbance for a time
dependent heat flux. This disturbance was next compared with the analytical temperature
disturbance of Chapter 4. This comparison is done in the next section.
5.5.4 Comparison of the Experimentwd Anglvticgl Tempergture Disturbances
Unfortunately, the values and sign of .the experimental temperature disturbances
were not in agreement with those obtained using the USE method (recall Chapter 4). The
possible sources of this disagreement had to be determined. The first one of these sources
was the error in thermocouple locations. The cause of this error was the movement of

the thermocouples noted after releasing the tension on the wires. The error in

108

thermocouple locations was found to affect the temperature disturbance a great deal.
Knowledge of the exact location of the thermocouple junctions then became crucial. X-
ray could have been used to measure the thermocouple locations. This was not done
though because the importance of the thermocouple location was quantified only later
when the specimen was not available. A numerical estimation of the thermocouples
locations was then the only method to use. This estimation was carried out as described
in the next section.

5.5.5 Estimation of Thermocouple Locations

The calculation of the thermocouple locations was made possible, in part, by the
parameter estimation program PROPlD (see Chapter 2). The first step in the estimation
of the junctions’ locations was to calculate the analytical disturbance using the USE
method as shown above for a constant surface heat flux, then to apply the convolution of
Equation (5.5) to account for the time dependence of the heat flux. At each time step,
the value qj of the heat flux was assumed constant, and the difference of disturbances
A¢(qo,t,(.,m) and A¢(qo,tx,i) was multiplied by that constant heat flux. The analytical
disturbance for the time dependent heat flux was thus obtained.

Ideally, the experimental temperatures used in the least squares minimization of
PROPlD should not have thermocouple-induced errors (recall Equation (2.19) of Chapter
2); however, based on the current study, experimental temperatures are known to have
such errors (which are in fact calculated by the USE method and the convolution of
Equation (5.4)). To accommodate for these thermocouple-induced errors in PROPlD

(when estimating thermocouple locations), the experimental temperatures were corrected

109

before being used with PROPlD. This was accomplished by subtracting the errors
calculated by the USE from the experimental temperatures.

Recall from Chapter 2 that PROPlD allows the estimation of the thermal
properties k and pc. To estimate the thermocouple depth d, the heat equation was

transformed as:

 

kg = (ac-gt! - kaazfz 4:12”)? (5.53)
(a)

T(x=0,t) =T1(t) (5.5b)

T(x=d,t) =T2(t) (5.5c)

T(x,t=0) = To (5.5a)

where k is the known thermal conductivity, and pcd2 is the unknown "new" volumetric
heat capacity. The new space variable is not just x, but x/d; the thickness of any region
was thus divided by an initial guess of the thermocouple depth, d. The weighted
volumetric heat capacity estimated by PROPlD was divided by the original one to get d2.
This new thermocouple location was used to compute the undisturbed temperature and
then the disturbance (by the USE method). The whole process was repeated until the
calculated thermocouple location stopped changing. This took no more than two or three
iterations.

The new estimated thermocouple locations are listed for the thermocouples of
interest in Table 5.1 along with the locations specified before making the sample. The

absolute error in the location of the thermocouples was as high as 26% (for thermocouple

110

Table 5.1 Nominal Versus Estimated Thermocouple Locations

 

 

Thermocouple Nominal Estimated Approx .
Location Location Error

(turn) (Inna) (mm
00 1 1.021 0.02
02 1 0.739 -0.26
04 2 2.251 0.25
06 2 1.823 -0. 18
07 2 1.851 -0. 15
08 2 2.120 0.12
09 4 3.523 -0.43
10 4 4.336 0.33

 

 

 

 

 

 

 

02). One should not be misled by this large value; it should be kept in mind that the
nominal thermocouple depths were only 1, 2, and 4mm. Taking into account all the
errors involved with preparing the mold, the degree to which the thermocouples were
stretched across the mold, and the effect of the release of the tension on the
thermocouples, an error of 0.26mm out of lm (for thermocouple 02) was not too
surprising. The same conclusion can be drawn for the other thermocouples.
5.5.6 Sensitivity Analysis

The goal of the final part of this experimental investigation was to quantify
analytically the importance of the effect of the possible sources of disagreement between
the experimental and analytical studies. Some of the possible sources of discrepancy are
the thermocouple location, the thermal conductivity of the epoxy, and its volumetric heat
capacity. To reach the goal of quantifying the importance of these sources, a sensitivity

analysis was performed. The first step of this analysis consisted of differentiating the

111

exact undisturbed temperature T. given by Equation (5.3) with respect to the parameters
of interest (here the sources of discrepancy) one at a time; next, this derivative was
multiplied by the expected error of each parameter as well as the parameter. Finally, the
resulting quantity was compared to the disturbance at the same depth. For the

thermocouple location, the sensitivity gives

l’i’irlzial

The sensitivity to thermal conductivity results in

-,|—t-ierf x +3—erfc 1:
kpc 2M 21‘ Nat—t

and the sensitivity to volumetric heat capacity provides

_il_§_ierf x - x eff x
pc kpc zJa—t mm 276:?

Three values of radius-to-depth ratio, H, were considered (H=0.0625, H=0.125, and

a (5.6)
x

 

31‘
AT.=X—.£ :-
6:: x

 

 

Ak (5.7)

k

 

 

 

 

3r
A13. pc_- A_p°. =qo
6pc pc

 

 

 

 

Apc (5.8)
pc

H=O.25) at two dimensionless times 5 and 10. The results are shown in Tables 5.2, 5.3,
and 5.4. Table 5.2 was generated for a 10% error in thermocouple location. Table 5.3
was generated for a 2.4% (not 10%) error in thermal conductivity. This was done
because, as reported by Garnier (1991), the error in the estimation of the thermal
conductivity using the same setup was less than 2.4%. Finally, Table 5.4 was generated
for a 2% error in volumetric heat capacity; this was based on the results of Chapter 2.
The confidence intervals of the estimated volumetric heat capacities in that chapter reflect

an error less than 2%.

Table 5.2 Sensitivity of Temperature Disturbance to a 10% Error in Thermocouple

112

 

 

 

 

 

 

 

Location

9* H Aux) (°C) Arum (°C) ATJArmu)
0.0625 -0075 -0.003 25

5 0.125 -0.075 -0.009 8.3
0.25 0075 -0.030 2.5
0.0625 -0.083 00023 36

10 0.125 -0.083 -0.0075 11.1
0.25 -0.083 -0.028 3

 

 

 

 

 

 

Table 5.3 Sensitivity of Temperature Disturbance to a 2.4% Error in Thermal

 

 

 

 

 

 

 

Conductivity
t+ H AT__(x) (°C) ATm(x) (°C) ATJATm(x)
0.0625 -0.01 1 -0.003 3.67
5 0.125 -0.01 1 -0.009 1.22
0.25 -0.01 1 -0.030 0.367
0.0625 -0.022 -0.0023 7.334
10 0.125 -0.022 -0.0075 2.44
0.25 -0.022 -0.028 0.733

 

 

 

 

 

 

 

 

113

Table 5.4 Sensitivity of Temperature Disturbance to a 2% Error in Volumetric
Heat Capacity

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

II H AT..(X) (°C) AT....(X) (°C) ATJAT....(X)
0.0625 -0.0239 -0.003 7.987
5 0.125 -0.0239 -0.009 2.662
__ 0.25 -0.0239 -0.030 0.799
—— 0.0625 -0.0351 -0.0023 15.261
10 0.125 -0£0351 -0.0075 4.68
0.25 -0.0351 -0.028 1.254

 

 

5.6 Discussion of the Results

The estimated heat flux obtained by IHCPlD was based on measured
temperatures. These measured temperatures have thermocouple-induced errors as shown
in Chapter 4, which means that the estimation of the heat flux should have been an
iterative procedure. The iterative procedure would first involve estimating the heat flux,
the thermocouple locations, the temperature disturbance, then correction of the measured
temperatures using the obtained locations and disturbance values, and finally estimation
of the heat flux. The same steps are repeated until the heat flux stops changing from one
iteration to the next one. This whole iterative process on the heat flux was avoided
because, even with the assumption of an error-free heat flux, the experimental
thermocouple disturbance was found to be quite different from the analytical one.

Tables 5.2, 5.3, and 5.4 show that thermocouple location affects the temperature

disturbance calculations more than the thermal properties. The third column of Table 5.2

114

indicates that a 10% error in thermocouple location causes errors in the exact undisturbed
temperature from 2.5 to 36 times in the value of the calculated temperature disturbance.
The thermocouple-induced temperature disturbance (as shown in Chapter 4) is less than
5% for dimensionless times larger than 3. A 10% error in the thermocouple location then
causes a net error between 12.5% (2.5x5%) for H=0.25, and 180% (36x5%) for H=0.0625
on the measured temperature. Such an error is too large because it is much greater than
the residuals of Chapter 2. The stretching of the thermocouple before curing the epoxy
and the release of the tension after the epoxy was cured proved to be an inefficient
technique when estimating thermocouple temperature disturbance. Larger thermocouples
proved to be more consistent with analytical results than smaller thermocouples. To
insure the future success of this experimental technique, the specimen should be x-rayed
to determine the exact locations of the thermocouples. The specific heat of the specimen
should be obtained by DSC (Chapter 2), and its density should be obtained by the
standard method for density of glass by buoyancy (Chapter 2). The setup could be
modified to have the heater sandwiched between two identical specimens as in Chapter

2, so that the imposed heat flux is equally shared by the two identical specimens.

CHAPTER 6

EXPERIMENTAL VERIFICATION OF OPTIMAL EXPERIMENTS

6.1 Introduction

A unique component of this dissertation is the development of optimal
experiments, as discussed in Chapter 3. The statistical assumptions in that chapter are
quite idealized, including no bias and uncorrelated errors. It is known that these
conditions are not satisfied in our experiments. Nevertheless, the general guidelines
provided in that chapter should lead to better experiments than if they were not used. It
is extremely important that the optimal experiments concepts have practical application
and thus can be experimentally verified.

The objective of this chapter is to demonstrate that the optimal experiment designs
lead to better parameter estimates. One indication that one experiment is better than
another experiment is that the better experiment has a smaller confidence interval.
Fortunately, program PROPlD can provide such intervals. However, a number of the
basic assumptions in Chapter 3 are not experimentally satisfied. One of these is that the
errors are uncorrelated; PROPlD models these errors as a first order autoregressive

process. There are also some other conditions that are not satisfied. Nevertheless, it is

115

116

believed that the recommended optimal experiments in Chapter 3 are superior to other
related experiments. This can only be demonstrated through some experiments.

The experimental verification of the analytical results of Chapter 3 was originally
made possible by a setup different from the one described in Chapter 2. This setup was
develOped by Garnier, Delaunay, and Beck (1991). The setup uses resistance
thermometer devices (RTD) and thermocouples to measure temperatures. Using this
setup, the X21B50T0 case of Chapter 3 was implemented in our laboratory. The material
of interest was DER332/DDS epoxy. The thermal properties of the sample were
estimated for four different experimental conditions, then confidence intervals were

obtained and compared.

6.2 Experimental Setup

The first step of this experimental investigation was to cut two specimens out of
the circular DER332/DDS epoxy specimens of Chapter 2; each of the specimens was
5.62mm thick. Both specimens were polished to a high gloss, and a RTD/heater
combination was sandwiched between them as shown in Figure 6.1. Silicone grease was
used to achieve good contact between the RTD/heater combination and the samples. The
back surface of each specimen was in contact with an aluminum block (simulating a
constant temperature boundary condition because of its higher conductivity relative to
epoxy) to which a thermocouple was attached. The whole assembly of epoxy specimens,
RTD/heater, and aluminum blocks was placed in an oven, and the thermocouples and

RTD/heater were connected to the data acquisition system described in Chapter 2.

117

Heating Element

Kapton Kapton

[—— Smcone Grease ' r——— Sflicone Grease fl

....................................................

 

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_ ‘ ,~ . . . . . . .. _ -.- . . . ._ . .__ .. -. -._i. -. -. o..-._'-. --------------------
......................................................
........................................
.............
..............................................................
..........................
v. .. .' ._ .‘ . .,1 . ., -. -. -. -. ., -. -. -. -. -. ., -._.. -------------------------
...............................................................

................................

EPOXY Epoxy TT'C'

_ . . _.. .' .' .‘ .‘ .' .‘ .' ... - '- ‘ . . -.--. -.". -. ._._-.'-. -..-..-._-. -. .................
..........................................................................
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.....................................................
..........................................
, a L x A x L a . A

 

 

Resistance Thermometers

0.08 mm W.

0.08mm
0.25 mm —d ~—

 

 

 

 

 

 

5.62 mm

 

 

Figure 6.1 Experimental Setup for the Investigation of Optimal One-Dimensional
Designs

118

Two sets of experiments were run to show the effect of optimal heating, and the
effect of the duration of the experiment on the optimality of the experiments. Each set
consisted of two experiments, and all four experiments were based on the ideas of
Chapter 3. The first experiment was an Optimal one ending when the heating was turned
off (at th*=2.5). The second experiment was a non-optimal one with a short heating time
(th+=0.5). The third experiment was chosen Optimal with a duration longer than the end
of heating (at th*=2.5). Finally, the fourth experiment was non-optimal; it was similar to
the third one except for its shorter heating time (th+=0.5). The same number of
measurements (250) was taken for all four experiments, and for each optimal/non-optimal
combination, the same maximum temperature rise was achieved as required by the

constraints of the analytical study.

6.3 Experimental Results

The different experimental parameters corresponding to the four experiments
mentioned above are summarized in Table 6.1. The calculated voltage was deduced from
the boundary heat flux which was, in turn, calculated from a preselected maximum
temperature rise. Table 6.1 shows also the measured heat flux which is calculated from
the measured voltage and current, and from the heated surface area. Notice that the
column of non-dimensional heating times shows numbers slightly different from the ones
indicated above. This difference is due to the round off in the calculated experimental
times needed to satisfy the same number of measurements condition.

The parameter estimation program PROPID (Beck, 1989) described in Chapter 2

119

Investigation of Optimal One-Dimensional Designs

Table 6.1 Experimental Parameters of the Four Experiments Performed for the

 

 

 

 

 

exp. time time non— time non- meas. meas. meas.
step heat dim. exp. dim volt. curr. heat
off off cooling flux
(sec) (sec) t; (sec) time (V) (A) (W/mz)
1 2.2 550 2.8 550 0 17.5 0.176 337.7
2 0.44 1 10 0.7 1 10 0 20.3 0.204 452.9
3 2.7 558.9 2.7 675 0.59 22.4 0.223 547.7
4 0.53 109.7 0.7 132.5 0.14 26.4 0.262 758.4

 

 

 

 

 

 

 

 

 

 

 

 

was used to estimate the thermal properties, the corresponding 95 % confidence intervals,
the sensitivity coefficients, and the residuals. The sensitivity coefficients and residuals
are examined now at the heated surface. The magnitudes and relative shapes of the
curves of the sensitivity coefficients of the different parameters of interest are indicators
of how good a particular experiment is to estimate the parameters (recall Chapter 3). The
two sets of sensitivity coefficients for the two sets of experiments described above are
shown here in Figures 6.2 and 6.3. The "k" in the labels of the curves stands for thermal
conductivity, and the "c" stands for volumetric heat capacity. The curves of Figure 6.2
labeled "kl" and "cl" correspond to the first experiment, while the curves labeled "k2"
and "c2" correspond to the second experiment. The curves of Figure 6.3 labeled "k3" and
"c3" correspond to the third experiment, while the curves labeled "k4" and "c4"
correspond to the fourth experiment. The shapes of the optimal thermal conductivity

curves ("kl" and "k3") are very similar to those of the curves of Figure 3.3 of Chapter

120

 

 

 

 

 

' 1 ' 1 ' T r l ' l ' r '
- -— (kl) opt (tufth) .
A 2— ------ (c1) —
Q 4 ----- (k2) non-opt (tufth) '
0
v - 2 -
+, ‘\ (C) _
.9 ‘ ‘
H5) _
O . -
L) -5- _
b a .1
IE ‘
8 -10— —
<1) - .
(f) .1 u
“14 ' l ' l I 1 ' r i I ' r '
0 100 200 500 400 500 600 700

Tune (sec)

Figure 6.2 Sensitivity Coefficients for the First Set of Optimal One-Dimensional
Experiments

121

 

 

 

 

 

' l ' I ' l ' l ' l ' I '
- —- (k3) opt (tm>th) -
A 2— ------ (c3) ~
Q) ~ ----- (k4) non—opt (t..p>th) -
SJ - ---- (c4) —
\
'E "‘ “
.02 -2* -
.9 ‘ ‘
E - _
O . _
O _5— ..
b s -
IE ‘ "
E? —10— —
(D 4 _
U) . -
—14 I I I I I I l I l 1 l I 77
0 100 200 300 400 500 600 700

Thne (sec)

Figure 6.3 Sensitivity Coefficients for the Second Set of Optimal One-Dimensional

Experiments

122

3. The same is true for the volumetric heat capacity curves and those of Figure 3.4.
To be able to compare all four curves ("kl", "c1", "k3", and "03") to those of
Figures 3.3 and 3.4, time is non-dimensionalized using the estimated thermal pr0perties,
and with respect to the thickness of the specimens. Furthermore, the sensitivity
coefficients of Figures 6.2 and 6.3 (which are really k(aT/ak) and pC(BT/apc) as
calculated by PROPlD) are non-dimensionalized the same way as the sensitivity
coefficients of Chapter 3 given by Equations (3.8a) and (3.9a). The resulting non-
dimensional sensitivity coefficients are (k/qoL)x(k(aT/ak)) and (k/qoL)x(pc(aT/3pc)). The

minimum of curve "k1" of Figure 6.2 is

13r_ 1

X’ =—— — — x(—8)=—0.915 (6.1)
mm qouk (8k) 337.5x5.62x10'3I0.217

 

which occurs at about

t =—— =2.
“m L2 1.35x10‘ (5.62x10'3)2

 

. at _ 0.217 550 s (6,2)

These two values X + and tum,+ are compared respectively with -1 and 3 obtained

k1,rnin

from Figure 3.3. Curve "c1" of Figure 6.2 shows a decrease up to a minimum, then goes

to zero. The minimum value is

 

 

. 1 ar 1
X. = pc[—) = x (—2) = -0.229 (6.3)
"mi“ qouk 69c 337.5x5.62x10'3/0.217

which occurs at about

. at_ 0.217 150

term" - =0.763 (6.4)
L2 1.35x10‘ (5.62x10'2)2

 

 

+andt

,1 mi, ’ are compared respectively with -0.3 and 0.5 obtained

these two values X c. m

from Figure 3.4.

123

The same is done for curves "k3" and "c3" of Figure 6.3 during the heating period
(which is the condition for which Figures 3.3 and 3.4 were generated). The minimum
value Xmmm+ was -0.921, and occurs at the non-dimensional time tk3mm*=2.82. The two

values XE“: and tkmm+ are compared with -l and 3 respectively. The minimum value

X arm; was —0.276, and occurs at the non-dimensional time tc3m*=0.719. These two

values are compared respectively with -0.3 and 0.5 obtained from Figure 3.4. The above
similarities indicate that the experimental results were consistent with the theory of
Chapter 3. This consistency was not achieved at an earlier stage of the research with the
setup described in Chapter 3.

The residuals of the thermocouple at the heated surface of the specimen were
examined next They are shown in Figure 6.4 for all four experiments. The maximum
value of each curve was compared with the maximum temperature rise. The largest value
of all four residual curves was about 015°C; the corresponding maximum temperature
rise was 17.13°C. Comparing these two quantities, the residuals are less than 1% of the
maximum temperature rise, as opposed to 2.9% using the setup of Chapter 2.

The similarity between the theoretical and experimental sensitivity coefficient
curves, as well as the low values of the residuals meant improved thermal properties
estimates. These estimates are shown as functions of time in Figures 6.5 and 6.6. Figure
6.5 shows the thermal conductivities corresponding to the four experiments. The
conductivity estimate was 0.218 W/m°C for the optimal experiments of Figure 6.5. For

the non-optimal experiments, the estimation process does not show any leveling off for

some time to a constant value. Figure 6.6 shows the transient volumetric heat capacities

Residuals (°C)

One-Dimensional Designs

124

 

 

 

 

 

 

1.0 r 1 1 1 r 1 i 1 ' 1 ' 1 '
r (1) opt (t..,=t,,) -
0.8“ ------ (2) non—opt (tup=th) -
" ----- (3) opt (tm>th) -
0-5‘ (4) non-opt (tw>th) ‘
0.4- —
0.2- r
0.0— , a
—0.2-* r
—0.4— —1
—0.5~ ~
, -
—0.8— -
—l.O 1 r 1 1 T i ' i ' l ' l ‘
0 100 200 300 400 500 600 700

Time (sec)

Figure 6.4 Residuals of the Four Experiments Performed to Investigate Optimal

CE}: 5.3.8328 65.2:

 

125

 

0.26 r I I I T I I I T I 7 I 1
‘ -"" (4) non—Opt (toxp>th) ...
"X ----- (3) Opt (texp>th)
0.25~ . x """ (2) non-opt (tuft) ‘

,’ “ — (1) opt (t..,=t..)
I_.'

 

 

Thermal Conductivity (W/mK)

 

I I I I

1
600 700

 

 

1 r r 1
0 100 200 300 400 500

Time (sec)

Figure 6.5 Transient Thermal Conductivity Corresponding to the First Set of
Experiments Performed to Investigate Optimal One-Dimensional Designs

126

 

13999990 I 1 1 1 ' 1 1 1 ' 1 1 1

12999990-

 

119999904 . -

Volumetric Heat Capacity (J/m3°C)

 

 

 

E": "J", I I”
-: II] \‘ III d
:1 ‘\ I
1099999.0~ 1; a (4) non-opt (t...>t..) r
1' ~ ----- (3) opt (t“p>th)
. ------ (2) non—opt (tm=th)
— (1) opt (t“p=th)
1000OO0.0 I I I I 1 r 1 1 1 I 1 r r
0 100 200 300 400 500 600 700

Time (sec)

Figure 6.6 Transient Volumetric Heat Capacity Corresponding to the Second Set
of Experiments Performed to Investigate Optimal One-Dimensional Designs

 

113

3C(

127

corresponding to the four experiments, and the same convergence/non-convergence
behavior noted in Figure 6.5 was true for Figtue 6.6.

The estimated thermal properties corresponding to the four experiments of Section
6.2 are summarized in Table 6.2 along with their confidence intervals. The confidence
regions of the four experiments are shown in Figure 6.7. The small squares formed by
the open circles (labeled 5) and triangles (labeled 3) (there are in fact four triangles: two
on two of two) correspond to the optimal experiments while the large squares formed by
the closed rectangles (labeled 6) and diamonds (labeled 7) correspond to the non-optimal
experiments. The areas of the square confidence regions corresponding to the four
experiments were compared. The area of the confidence region corresponding to the first
optimal experiment (exp 1) was 16.32 (W/m°C)(J/m3°C) as opposed to the non-optimal
one (exp 2) of 2883.58. The area of the confidence region corresponding to the second
optimal experiment (exp 3) was 11.43 as opposed to the non-optimal one (exp 4) 2291.53.
Much better experiments (reflected by smaller confidence region areas) were therefore
achieved when the dimensionless heating time was about 2.5; these experiments were
further improved by taking measurements after the end of the heating period for a
dimensionless time of 0.73.

In conclusion, the setup developed by Garnier, Delaunay, and Beck (1991)
improved the quality of the experimental results to the point where these results and the
theory of Chapter 3 on the design of optimal experiments are the same. Furthermore, the
transient experiments demonstrated that optimal experiments produce substantially more

accurate parameter estimates than non-optimal experiments. This is exactly what the

128

theoretical investigation of Chapter 3 showed.

Table 6.2 Comparison of the Results of the Four Optimal One-Dimensional
Experiments

 

exp. time time time time thermal volumetric rms
step heat heat exp. cond. heat capacity
on off off
(sec) (sec) (sec) (sec) (W/m°C) (J/m3°C)

 

1 2.2 22 550 550 0.217 :1: (1.35i0.01) 0.021

 

 

0.0004 x106

2 4.4 4.4 110 110 0.2388 :1: (l.19:l:0.07) 0.041
0.0101 x106

3 2.7 27 558.9 675 0.223 i (1.35:0.01) 0.028
0.0003 x106

 

4 5.3 5.3 109.7 132.5 0.2365 :1: (l.19:l:0.06) 0.061
0.0089 x106

 

 

 

 

 

 

 

 

 

 

 

129

 

 

 

 

A 1400000. 1 1 1 1 I
g) 2 E3 opt
"’ non—opt
E COM A (.3) opt 4
3 (ILA O (4) non—opt
>\
T’: 1300000- r
t)
O
O.
O O. O I
O r .
6
CD
I 12000004 2
.9
'23
§ 0 .
‘5 I I
> 1100000. 1 r 1 1

0.21 0.22 0.23 0.24 0.25

Thermal Conductivity (W/m°C)

Figure 6.7 Confidence Regions Corresponding to the Experimental Investigation
of Optimal One-Dimensional Experiments

CHAPTER 7

RESULTS AND CONCLUSIONS

The goal of this research was to find and investigate new ways of improving the
estimation of the thermal properties of composite materials. To reach this goal, analytical
and experimental approaches were followed.

The first way of improving the estimation of the thermal properties related to a
new experimental paradigm. This new paradigm consisted of two series of thermal
properties estimation experiments. Each series consisted of four independent experiments.
In the first series, the imposed heat flux was made large to cover the whole temperature
range of interest (20°C to 100°C) in each experiment. In the second series, the same
temperature range was covered but each of the four experiments covered a small portion
of it. It was found that the temperature residuals were large, and led to large confidence
regions. The large size of the confidence regions masked the outcome of the paradigm,
and it was necessary to look for other ways of improving the transient experiments. One.
way was to design the experiments so that they are Optimal; the other way was to
quantify the thermocouple measurement error that led to the large residuals.

The second approach of improving the estimation of thermal properties was to

130

131

design the experiments so that they are optimal. This design of optimal experiments was
done analytically, then verified experimentally. It was based on maximizing the
determinant of the product of the sensitivity matrix and its transpose, which is the well
known D optimality criterion. Different geometries were exarrrined, and different
parameters were of interest. First, a one-dimensional geometry was considered. It was
found to have an optimal dimensionless heating time of 2.25, and a dimensionless
"cooling" time of 0.73, or a total dimensionless optimal experimental duration of about
3. The choice of the Optimal dimensionless heating time was found to be flexible
(between 1.5 and 3.5). The optimal location of a sensor for this one-dimensional
geometry was found to be at the heated surface. The temperature boundary condition
used with this geometry proved to be superior to an insulated one. Next, a semi-infinite
geometry was considered. The optimal dimensionless heating time was 1.5, the "cooling"
time was 0.22, and the total optimal experimental duration was about 1.7. This geometry,
unlike the one-dimensional one, showed less flexibility in the choice of the heating
duration. The finite duration of the heat flux (with measurements lasting a dimensionless
time 0.22 after the end of the heating) for this semi-infinite geometry was found better
than a heat flux lasting through the whole experiment. ’

A two-dimensional geometry was also analyzed in the same context of designing
Optimal experiments. For this configuration, the Optimal experiment was found to occur
at steady state instead of a finite time. The optimal placement of two thermocouples was
found to be at the two edges of the heat flux surface. Finally, some global design curves

were obtained for different ratios of the area of the heated potion of the heat flux surface

132

to that of the whole heat flux surface. The recommended curve corresponded to a,/a=0.75
for (a/b)sqrt(ky/k,)=2.25.

The third and last technique used to improve the estimation of thermal properties
was to characterize the error due to the embedded thermocouples. This error was
quantified using the USE method, which was compared to the two popular numerical
methods: the FE method and the FD method. The USE method gave more insight into
the problem, required considerably less computational time for the estimation of the small
quantity thermocouple disturbance, and gave accurate results. An experiment was
implemented to verify the results of the USE method. The experiment showed higher
dependence of the temperature disturbance on errors in the thermocouple location than
errors in the thermal properties.

Finally, four experiments were run using a technique developed by Garnier,
Delaunay, and Beck in 1991; the setup uses only surface thermocouples and resistance
thermometers. The goal of the experiments was to verify the one-dimensional finite
geometry optimal experiments’ results. The experimental results were found similar to
the analytical ones.

The following results and conclusions are drawn from this research:

1. The use of embedded thermocouples to investigate a new experimental paradigm led
to large residuals and masked any possible improvements expected by the paradigm.

2. For the finite one-dimensional X21B50T0 case, the optimal heating time was 2.25, and
the Optimal total experimental time was 3.

3. The choice of the optimal heating time for the X21B50T0 case was flexible, between

 

133
1.5 and 3.5.

4. The heated surface was the optimal location to place a sensor at for the X21B50T0
case.

5. For the finite one-dimensional case, a constant temperature on one boundary with a
finite duration constant heat flux on the opposite boundary led to better experiments than
an insulated boundary with a the same finite duration constant heat flux on the opposite
boundary.

6. For the semi-infinite X20B5T0 case, the optimal heating time was 1.5, and the Optimal
total experimental time was 1.7.

7. The choice of the Optimal heating time for the X20B5T0 case was less flexible than
for the X21B50T0 case.

8. For the two-dimensional X22BO0Y12B0x5T0 case, the optimal experiment was
attained at steady state.

9. The optimal placement of two thermocouples for the two-dimensional
X22BO0Y12B0x5T0 case was at the two edges of the heat flux surface.

10. The optimal experiment for the two-dimensional X22BO0Y12B0x5T0 case occurred
for a,/a=0.75 and (a/b)sqrt(k,/k,)=2.25.

11. The USE method permitted more insight than the FE or FD method in calculating
the thermocouple-induced errors.

12. The USE method used less computational time than the FE or FD methods in
calculating the thermocouple-induced errors.

13. The USE method gave more accurate results than the FE or FD methods.

 

\

134

14. The thermocouple location was the experimental parameter with most effect on the

thermocouple-induced disturbance.

15. An alternative technique avoiding embedded thermocouples made it possible to check

the analytical results of the design of Optimal experiments.

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