LIBRARY

Michigan State
University

 

 

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

 

DATE DUE

DATE DUE

DATE DUE

 

94m 6 592004

 

N

{JV 2 2 2004

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1!” WM“

 

mum Em

Inn

WIRING HEAT TRANSFER FOR PARAMETER ESTMTION
in!irzausa:txrrrwnsrvunr:nxmumarumnurs

BY

Robert L. McMasters IV

 

A DISSERTATION
Submitted to
Michigan State University
in partial fulfillment of the requirements

for the degree of

DOCTOR OF PHILOSOPHY

Department of Mechanical Engineering

1997

 

m
A

mmm

mi"

... o i.
.9

u.‘
\
Ono.

t...

..o

 

..
L. as.
v . ~.
.. . .
«‘w
. ~ ~\~
‘4‘ .a.
A:
u .. a .
p u
n: C»
. . v a
a v .u.
w § .5 s
u a .. .
. x“

.
u‘ "

‘\

 

JUBSIEUUUT

WDELING HEAT TRANSFER FOR PARAMETER ESTMTION
ZEN ETJHNE DUHHFUSJNRPY EDUEBRUJBHNTS

BY

Robert L. McMasters IV

Since the early 19603, the laser flash method of
diffusivity measurement has been used on a large variety of
materials. Several parameter estimation methods have also
been used in analyzing such experiments, employing various
levels of sophistication.

This research investigates the penetration of the laser
flash beyond the surface of the material being heated.
Various heat transfer models are presented, each with
different assumptions about the initial temperature
distribution inside the material. Besides the mechanism of
thermal conduction, radiation transport is also considered,
.assuming a semi-transparent emitting and scattering medium.
An evaluation is made of the response of the methods to
factors which may enter into the experimental process. This
is done in quantitative terms so as to assess the adequacy
of the models in comparison to one another.

Estimation of thermal parameters, using the models
developed as part of this research, is performed on

experimental data from 15 laboratories around the world,

 

 

‘ o..,_
\ ' _c
‘..“'.'O...»

:53”!

I--...¢.

o
I

 

.0... ‘. -
.-
"-II ‘-.
u..-
v‘ '--.:I'c ‘7
.UI .'.' ‘.
u'.‘ -. ..
a... Q“ .:.~‘
A
'I“
."
.

-_"‘O

 

 

so
u .C"--.A
i...‘...:~
»
Q ..“ ‘
"I..‘ . -
. '5...

u-

(I)

 

 

5.
a.
I

o
‘r
n
a.
O

o.

‘M
‘-_
‘ o
A
‘v
4
‘..
‘-,.
O.”

o
‘O
*Q
-’.
.~‘
."
U.

 

 

:involving a variety of materials. Ambient temperatures in
the experiments range from ZOTIto ZOOOTZand estimated
diffusivities vary from 0.3 to nearly 80 square mm per
second.

When direct solutions which have closely correlated
parameters are used in parameter estimation, the equations
used in calculating the parameters can be very unstable. A
method of regularization is presented as part of this
research which imparts stability to the ordinary least
squares method of parameter estimation, where parameters may
be otherwise unobtainable. This method is presented

compactly in matrix format.

to the only God

who is, who was, and who is to come

iv

 

 

'-.' .‘b‘~
.
. - - I '
. _ .
‘.": In... (
U. h. --'.’ ‘ >
v. v- .I r ‘
h;- .;. “
... n" __.. .u
s
.. ~
5.: :’-=‘ : ~.
. . ~."'-.d. 4.
. .
.‘-- .u--- - .
~ ~ ‘
’- ~0.v¢~.‘ -
.
.- “-,.
-_ ... .
DR -“. ‘.
n
a..- :\‘ ‘
~‘.A
.
.__ .
‘zs.'~.“v-. ’ ‘
‘ Q
a.-‘, ~...
. Q
n . . -
‘Q.
5... ‘_~ -0 .. .
s -‘ - .-‘
‘...
.. '-~
.‘:‘.5 a...“ ‘
U -I.‘: ‘
. ‘
on V a? ‘4‘ .'
‘_‘ ‘-
‘v .I:‘
‘ s'
Q ..
a “
q.‘ ~.~
'. .
.3‘ a: ‘
o
g.
I‘. ~‘
.‘ a " ‘
A. “'
‘ I
Q
A
. ‘h u o
' Id‘og
I 9‘- v-'_‘
§
4

IUCKIKNNLIHNSMEHTTS

The input and help of my Ph.D. guidance committee,
James Beck, John Lloyd, Merle Potter, Indrek Wichman and
Norman Bell is acknowledged. To my advisor James Beck, I
owe special gratitude for helping me put my thoughts into
communicable terminology so as to be presentable to the
scientific world.

The assistance of the High Temperature Material
Laboratory, part of the Oak Ridge National Laboratory,
provided me with the opportunity to obtain most of the
measurements utilized in this research. Special thanks are
in order to Hsin Wang and Ralph Dinwiddie for the many
additional samples they tested for me.

Finally, I owe thanks to my family for their support.
To my wife MaryLynn, I am particularly grateful for her
exhortations which initiated my application to the Ph.D.

program.

 

.\~
.- nqu
—.~ nu.
._ . . .L
a; -..
E .. T
.0 R.- ”I.
“I. .I. \n—
. . . u m .
I ..:. w;
I 3. w...

u c
a; ..
.v o.
v. u.
u. L.
_.. v.

a .r.

Q ~‘

. ..

.. z a
o. .3 .. r.
a. u. ~ .3
C. .. .. a.
v. V; i. sxv
. . a. .u. L.
.q . a . nun

0..
‘u
- u...
U.
.u s '

. rt. «2 a
.a. .. .. s» a. Q.
.. v. a. a. s‘ —

v. a. a. a. .. ‘3

a. .3 pt. v. —.. v.

T. a. S .. .. .3

s a g . IF. ~ ~ t. s:
.s. (p. 7:

9. a]. «I. 9‘

q‘ a‘ .1. .15 q‘ «1- .1-

~

g

s

\P
«(a
. . 2 ‘3.

2.. . . .
“-W Q44 sflv \(v «1-

C
‘-
\h~k

I“: -\d 6

 

|i

13MBL£E<IF CKflTTFIIPS

LIST OF TABLES

LIST OF FIGURES

NOMENCLATURE

CHAPTER 1

CHAPTER 2

CHAPTER 3

CHAPTER 4

HiJF‘HiJP‘H
QChU1kithH

NNNNNNN

qmmpri-J

uwwwww

mmrhWNl-J

Background

Introduction

Previously Used Methods

The Need for Refined Models

Direct Problem Solution

Derivative Regularization

Optimizing the Analysis Method
Research Goal and Dissertation Outline

Flash Measurement of Diffusivity

Introduction

Description of Samples
Aspects of Measurement
Direct Solution Computation
Finite Difference Method
Parameter Estimation

Model Inadequacies

Accounting for Internal Radiation

Introduction

Direct Solution Development
Numerical Aspects of the Solution
Parameter Estimation Test Problem
Analysis of Laboratory Data
Integro-Differential Approach

Non-Radiative Alternative Models

4.1 Introduction
4.2 Non-Linear Temperature Dependent Models
4.3 Various Initial Condition Models

4.3.1 Initial Time Shift, Model 4
4.3.2 Exponential Initial Distribution,
Model 5
4.3.3 Various Initial Conditions, Models 6-16

vi

viii
xi

xvii

H

i-‘OKOQQUJH

Hi4

13
16
17
22
31
36
41

47

47
51
58
76
86
88

94

94
96
102
103

105
112

 

 

 

 

 

 

. . - . 1 p u . .I. . g n“
.u. I. .p. .3 a. .. .1 r... . . .1 Y4.
. . _ u. _ a. .n. ..
.t. .1. .q. .. o .v. .. ‘ .. .3 s: v.
. u . . . u . v . 4‘. b R r . .ua . .
0‘ "u I‘- s- o . p” .u. .\~ ‘ 4: . . xcd
T. u . . . . . i... a . .3 .2 V”
05¢
0‘ .~. . 4 ..s .u. !‘ .~. .5.
. . an“ . . . . ,
I, I, ... .~. .4. .~. as. us. .u.
.0
.-
‘1
.-.
.u

5*.

at.

. u” r.. . I
a. .. .. J .k.
p.. ~ .2 wg . \F
.. ». . . “3.4.
~ .~. Q. . 4 .. n-a
.. . . .4. .3 a: «.
. . a. .. .n. L. .2
.1 . . V.. \u. 2.. 4 .
4:. as. .s. ru. ~
.._. In. oh. PA. pu-

 

4.3.4 Two-Sided Flash with Penetration,

Model 17 118
4.3.5 Surface Transmissivity 123
4.3.6 Elimination of Excess Models 124
4.4 Estimating Extinction Coefficient 128
4.5 Utilizing the Models to Analyze Data 135
CHAPTER 5 Derivative Regularization 140
5.1 Introduction 140
5.2 Derivative Regularization Underlying

Assumptions 143
5.3 Cubic Spline Approximation 145
5.4 Parabolic Spline Approximation 150
5.5 Bias and Covariance 162
5.6 Matrix Condition Number 170
CHAPTER 6 Optimizing the Analysis Method 177
6.1 Introduction 177
6.2 Using Mollification 178
6.3 Eliminating the Heat Flux Parameter 185

6.4 Determination of Appropriateness of Competing
Models 193
6.5 Analyzing Sequential Experiments 201
6.6 Residual Frequency Analysis 210
6.7 CBCF Analysis Summary 220
CHAPTER 7 Summary and Recommendations 232
7.1 Summary 232
7.2 Recommendations 235
APPENDIX 236
REFERENCES 247

vii

 

r I . . C.
_..4 ~. .4. .u 1 ~ . «J . g «a; a . .44 .NA 5. 6 § h a. ‘ .3
y. .. v. a. v. .. v. .3 r. .. ya .5. r4 .f.. 3.. :4 \ s .r
i v . .1 _ . .1 v. .3 .u. .1 r a i . a .3 4 . K S a .1
up. a». a». .v u». A». ups \N. Eb. “wt nut ‘.§ “Dd A .v s: ‘h to u o'.‘ \b\
I‘- .s. c d «In 3 I“. .\J Ci 7 ad a: a s
. . . s . s ‘ s s s ~ \
.i. 41.. In; I? I, 15. 4 4 1.5. 1.1 4 .~v
an. a: a: Q. A. an. an. an. a. G. C» -\
. Q n Q ~ O ‘ Q - Q q Q n \ - Q q \ 9 ‘ Q \ u \
l as nfi I .u‘ .n nfi A ,n .t ,n ,-
.- 2‘ :a a. : .4 «u‘ u. uq .- ~< :
s . Q . Q a Q A s a 5 c D u s \ Q I h c s t

Table

Table

Table

Table

Table

Table

Table

Table

Table

Table

Table

Table

Table
Table
Table

Table

2-1

3-1

LIST OF TABLES

Comparison of Finite Difference Method
to the Exact Solution

Comparison of Direct Solutions Using Non-
Radiative Model and Center Node Temperature
to Define k2 in Radiative Model.

Comparison of Direct Solutions Using Non-
Radiative Model and Averaged Node
Temperatures to Define k.r in Radiative
Model.

Parameter Estimation with Tgtl

Parameter Estimation Using the Model 1
Procedure on Model 2 Data

Parameter Estimation Comparing Model 1 and
Model 2 on actual data taken at 700%:

Parameter Estimation Using the Model 1
Procedure on Model 3 Data

Parameter Estimation Using Model 1 to
Analyze Surface Radiation Model Data

Parameter Estimation Using the Model 1
Procedure on a Model 4 Direct Solution

Parameter Estimation Comparing Models
1,4 and 5 on actual data taken at 700 C

Purdue University Experiment C_R1

Oak Ridge National Laboratory CBCF Sample
at 700°C

A;R1 Palaiseau France
Using Temperature Compensation
Values of W

Parabolic Spline Matrix Terms

viii

35

61

67

80

81

88

98

101

105

110

125

125

126

129

132

154

 

 

. . s . ~ . . .. i ‘ u u . ~ 6 u 4 ~ . ~ wk. A «.4 w.
u v 0 U
‘ . . c.. p.. .IM v a . ‘ a: U M. \ \. h. \- \ d l a ~ ~40 A u . §
r: . _. . I. .3 3 a. Z. 2. 2. Z. Z. .. a. .2 S
n; V a .. a . a ..r 4 . .3. _ v A . n a a v a v a y «C A e N .
. v a 4 ~14 «is l‘ :4
n 6 «no 0‘ on. '5' Q Q A 1. «I. l‘ .\v o-v W nIv a, n s u s n § n s o \ c Q
. . . . s . . . . s . . . u s - . s ‘ ~
on. 9.. as. .u. 0.. .u. pp. an. :n. In. an. )5. on. thy PI- rh~ :v rhv Phd PL-
.uu a” J” .u.. .-.. .-.. 3.. 6.. .u.. :4. 3.. G.. A... 9.. 0.. an... RN Gs. Ans. an.
. . . . . . . . . . . . . . . ~ . . . 4
r ”a ”a“ W.“ .p. .p.“ .0.“ .fl.“ an. H.“ ”A“ on. g). gun .q-k .utc .nn. an. up. on.
. . . . . . . . . . . . ~ . a . . . . . a . ~ . s . g . g . . . .

.4
o o

 

 

Table
Table
Table

Table

Table

Table

Table

Table

Table

Table

Table

Table
Table
Table
Table
Table
Table
Table
Table

Table

Table

MJWE Matrix For 16 Point Example
MITM1 Matrix For 16 Point Example
MZTM2 Matrix For 16 Point Example

Monte Carlo Results Comparing Derivative
Regularization to Ordinary Least Squares

Parameter Estimation Using the Parabolic
Spline Method

Comparing Mollified to Non-Mollified
Experiment Results

Estimated Measurement Errors Using
Mollification

Contrived Test Case Comparing 3—Parameter
to 2-Parameter Method

Contrived Test Case Comparing 2-Parameter
to l-Parameter Method

ORNL Data at 700°C Using 2-Parameter Method

Various Sensitivity Coefficient
Calculations

Results of Calculation Method on ORNL Data
ORNL Data at 700°C Using First 100 Points
ORNL Data at 700°C Using First 200 Points
ORNL Data at 700°C Using First 300 Points
ORNL Data at 700°C Using First 400 Points
ORNL Data at 700°C Using All 463 Points
Comparison of Point Selection Schemes

Sequential vs. Individual Estimation

CBCF Samples Measured at ORNL 800%:
Using Model 1

CBCF Samples Measured at ORNL at 800%:
Using Model 5

ix

160

160

161

168

176

183

185

189

190

190

193

193

195

195

196

196

197

202

208

221

222

Table

Table

Table

Table

CBCF Samples Measured
Using Model 1

CBCF Samples Measured
Using Model 5

CBCF Samples Measured
Using Model 1

CBCF Samples Measured
Using Model S

at

at

at

at

ORN L

ORN L

ORNL

ORNL

at

at

at

at

1000°C

1000°C

1200°C

1200°C

223

224

225

226

 

 

ow. . . ‘ o . s « . s c 0.. ~ ~ 1.! 1‘4 u t «As . a . \ L. x x i. .
M T: ..: ‘E. on a.- $.. a. h. an a. : AN~ fi‘ n\w . ~40
E u. t e .. .1. 1. :.f e ; e e ... s 3
a. .n .3 u.. .2 «\V .‘I. .v .‘J .- PI. .. ~Q Aw .\ .\ A» ax» n- a”. v. aw. ‘fw «\s us
. ‘n ‘t ‘¢
And
. . . s .c. 0‘ .n. .u. a . . .1— xq. 1‘. :1. Cw 7 a A: . s
. . . . . . . . g g s \ ~ - g s s
. a . ‘ q s a L . t a 6 . 1- ... no. ‘1. 1: \‘u ace \(4 «(a fine ~‘d
:- gl. an. an. n: .n. «a. .c. a. any as an» as a. no a» a»
V. v. V. v. r. V. V. V. 7. fie V. V. Us Vs Vs Vs is
. . . . . . . . . g . a n .e .. a g
.n. v: ~: .c. *1. ‘3 s: s: s: ~u. s: ~q. s: \a. s: s: x:
. g . 0 ,s ‘~ . o . c . o . a . a . § . s . u . Q . \ . Q a \ ‘ s
no. .u. .0. up. nu. on. .u. n... ‘u. go. uu. ‘Ic II. ha. \u. \u. \I.

LIST OF FIGURES

Figure 2—1 Schematic Diagram of a Typical Flash
Diffusivity Measurement System 19

Figure 2-2 Photograph of the Anter System at the

Oak Ridge National Laboratory 19
Figure 2-3 Eigen Value Residuals as a function of

Biot Number for the X33 Case 28

' Figure 2—4 {Normalized Sensitivity Coefficients for

CBCF at 700°C 39
Figure 2-5 Residuals from Model 1 Used on CBCF Data

at 700°C 43
Figure 2-6 Residuals from Model 1 Used on CBCF Data

at 600°C 44
Figure 2-7 Sequential Estimates of CBCF Data at 700%: 46
Figure 3-1. Non-Dimensional Temperature as a Function

of Time km/k=1.0 T;=1OO 68
Figure 3-2 Non-Dimensional Temperature as a Function

of Time Bi=1.0 T;:1OO 69
Figure 3-3 Non-Dimensional Temperature as a Function

of Time Bi=1 km/k=1 71
Figure 3-4 Non-Dimensional Temperature as a Function

of Time km/k=1.0 Tgeioo 73
Figure 3-5 Non-Dimensional Temperature Bi=1 Tgeioo 74

Figure 3—6 Non-Dimensional Temperature as a Function
of Time Bi=1.0 km/k=1.0 75

Figure 3-7 Sensitivity Coefficients km/k=1 Tgelooo
Bi=0.1 77

Figure 3-8 Residuals from Analyzing Model 2 Using
Model 1 kro/k=l.0 Tm*=100 Bi=0.1 83

Figure 3-9 Residuals Using Model 1 to Analyze Model 2
km/k=o.1 T0021 Bi=1 84

Figure 3—11) Sequential Parameter Estimates Using Model 1
to Analyze Model 2 km/k=0.1 Tgel Bi=1 84

xi

 

 

 

.. . . ...I_N_ ... H. :. ... .4 ._ a. 2.. A. I. .
. I. v. : V: .3 .«c s. . ~ . s. _. .. s a a” pa. p... C. ... p. Q..
. L. . ~h .u. '5. .. ~: ..~ um; . . V n. :4 h. ‘ .e v. V. «a. as
:. .5 .A Z. V. ~ . .Q _-. .2 .2 ‘A. ”V V“ n. .. a. .0 .. s a. a: S p». p? s.
a o q C
. . . . .6 1. 0‘ .s. .u. a .I. .3 . s
c n n u o o g u g g Q
sad 4‘ I.- I‘ 1!. 0‘. IQ- 0‘ 0‘. I‘- 4
on. an. a. a. .u- Ah- 5. an. an. AM. n-
v. v. v. r. v. v. v. v. v. v. v.
. g . . . u a g u .n ‘
on. v.. o.- .q- .4. ’1. .1. ‘1. ‘1‘ \vu s.»
v. n. . O O .6 u. a Q O I Q Q. ~ Q
a... on. .g. on. ‘1. .C- no. .0. Qt. \U. slt

 

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Residuals from CBCF Material at 700°C 87
Sequential Estimates Using Model 1 to
Analyze Model 3 kl/ko=1, Bi=1 98
Residuals Using Model 1 to Analyze
Model 3 kl/ko=1, Bi=1 99
CBCF Sample from ORNL at 7OWTZAmalyzed
with Model 1 and Model 3 100
Residuals Using Model 1 to Analyze a
Surface Radiation Direct Solution
Sequential Estimates Using Model 1 to
Analyze a Surface Radiation Direct Solution 103
Modified Sensitivity Coefficients for
Model S (alpha=1, Bi=1, a=0.1) 109
Using Model 1 to Analyze Direct Solutions
of Models 4 and 5 (a=1, Bi=1) 111
CBCF Sample from ORNL at 700°C Analyzed
with Model 1 and Model 5 112
(Model S) Exponential distribution of
initial penetration of the flash in the
material with the fourth parameter measuring
the degree of penetration. In this and
subsequent models, the term To is not needed
since this temperature is indirectly
prescribed by the heat flux parameter. 114
(Model 6) Rectangularly distributed initial
penetration with the fourth parameter being
the distance of penetration. 114
(Model 7) Exponential squared distributed
initial penetration with the fourth

parameter a measurement of the degree of

penetration. 114

xii

 

 

. a ll 0 . ‘
. .1 .3 S a. 2.. 3

L. O .h § _ ‘u N‘ 5 ‘ N... o N 4 . Q —
.2 ... :. ... C. a. .. :4 .~ .. a. .5: v. .3 _.. .1 ...
r. : a. .. .7 ...£s Mics .2 : hf. —.. 51.13 .1 c. Q. a. a.
5 2.: 2.: a. C A. ~ .f. -2 a. u». we... pks A.
‘3 WI
as Q‘
s s
5%. 4
G. a.
b. v.
Q .Q
~.. 5:
u‘ .Q
s-.

Figure 4-12

Figure 4-13

Figure 4-14

Figure 4-15

Figure 4-16

Figure 4—17

(Model 8) Linear temperature distribution

over the first 10 percent of the incident

side of the material followed by exponentially
distributed initial penetration thereafter.

The fourth parameter is the magnitude of

the initial temperature at the heated

surface and the fifth parameter is a

measure of the degree of exponential
penetration. 115

(Model 9) Two linear and one exponential
penetration zones. The fourth and fifth
parameters are the temperatures at L=O and
L=O.1, respectively. The sixth parameter

is a measure of the degree of exponential
penetration. 115

(Model 10) Two constant and one linear
penetration zone as functions of

penetration. The fourth parameter measures

the depth of the first constant penetration
zone. The fifth parameter measures the

depth of the linear penetration zone and

the sixth parameter is the initial

temperature of the second constant

penetration zone. 115

(Model 11) One constant penetration zone

and one combined exponential plus constant
zone. The fourth parameter measures the

depth of penetration in the first constant
zone. The second parameter measures the

depth of the exponential penetration and

the sixth parameter measures the magnitude

of the second constant zone. 116

(Model 12) One surface flash on each side

of the sample. The fourth parameter

measures the magnitude of the flash at

x=L. 116

(Model 13) Parabolic distribution at the
incident side of the sample with a combined
exponential and constant zone. The fourth
parameter measures the depth of the

parabolic penetration. The fifth parameter
measures the depth of the exponential
penetration and the sixth parameter

measures the magnitude of the second

constant zone. 116

xiii

2:1

 

 

 

. . .. . . .. . .. .. Z. —.. ... c.
. . .. ~.. .. vs. 5 . b. .. a. fi. «.2 I ~Q .§ .Qu “ 9‘s
. . . . T .1 I :. .1 T. a. .. C .1 . S C. .t .. .. .2 J. . . . . a. . l a.
V .. .h r“ u. .H .3 U. .n. .u .1 .C S. u. a. an Ht. . . .1 a. .. a. ... 3. ~.. ».. : u p? r». .y w». .v
3.. .. _ .3 . :..: t. .1. .. . .L... .d C....:.:.. .. .. ... a: «CA. at A.
n-v .‘o . v Q Q n‘
a . . . . L a‘ «4 w. a‘ 3
o . ~ . g s - s
0.- 0.. I, 4 I“. .\v .\d -\.
8. .u. a: a: A. a: B. an.
r. v. .. v. v. v. v. >q
. . . a n .. ,¢ -
we. ‘4. ~¢~ -- 5: $‘- 5: 5‘.
.0 .0 .0 .Q .s .C .Q \.
ac. nv. cc. ‘3. ~-. ~-. ~.. In.

 

Figure 4-18

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

4-19

4-20

4-21

4-22

5-1

5-2

(Model 14) One linear and one constant

zone with surface heating at the x=L face.

The fourth parameter measures the depth of

the linear penetration, the fifth parameter
measures the magnitude of the constant zone

and the sixth parameter measures the

magnitude of the flash at x=L. 117

(Model 15) Surface heating at x=O and a
penetration zone with combined exponential

and constant distribution. The fourth
parameter is the maximum temperature

associated with the exponential component

of the penetration. The fifth parameter
measures the depth of the exponential
penetration. The sixth parameter measures

the magnitude of the constant penetration. 117

(Model 16) Surface heating at x=0 and x=L

with exponential penetration. The fourth
parameter is the magnitude of the maximum
temperature associated with the exponential
distribution. The fifth parameter measures the
depth of the exponential penetration.

The sixth parameter measures the magnitude

of the surface heating at x=L. 117

The First 100 Points of File A;R1 Showing
Initial Temperature Decay 119

Residuals from File A;R1 Comparing Models
1,4,5 and 17 123

Spline Approximation of Residual Curve 148

Spline Approximations of Sensitivity
Coefficients 148

Spline Approximations of Sensitivity
Coefficients 149

Monte Carlo Results Using Ordinary Least
Squares 168

Monte Carlo Results Using Derivative
Regularization 169

Condition Number Ratio for Cubic Case
Using 16 Points 173

Condition Number Ratio for Cubic Case
Using 100 Points 174

xiv

 

ph-

:.

.s
5‘.

.u.

 

. on
. ”J on .un v. & ‘q-.—;
.3 pm .H. . .u. q .65 [_ x‘
L” . . uh .3 .2 .3 ... .H «.2
.c- l, .s- ’5. 5.
§ . o u .
;bu r.. as. uh. ph-
a. a. a. a. A.
v. v. v. v. v.
. . . a .
s: v: 4.. ~: ‘:
. o .c . Q . Q .A
.0. an. §-. :3. hit

.3. . .1 L.
y . 3 v.
‘3. . . L. pm
. Q

.r C. 2. . _
file a,
s .

:- an.

3. a.

v. v.

. .

‘u. s.-

. Q - §

Sn. su.

~..

1 \
rh—
A»
‘s.
s

‘6.

3....n. .. Z
a. .. . T.
v. .u ab ..a
8.. «\~ H-M has
2 «J
«a .
s ‘
C). C.
a» Q.
’3 .¥\
5.. \y-
.C ‘Q
‘0. us.

. s
3 .3
a. 5%
sun. a
4.1.
o s
s
th
a»
Vs
-
\a.
o \

 

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

5-9

6-1

6-2

6-3

6-4

Number Ratio for Internal
Case Using 16 Points

Condition
Radiation

Number Ratio for Internal
Case Using 100 Points

Condition
Radiation

A Comparison of Raw Data to Mollified
Data, Blurring Radius = 2

A Comparison of Raw Data to Mollified
Data, Blurring Radius = 3

A Comparison of Raw Data to Mollified
Data, Blurring Radius = 5

A Comparison of Residuals from Mollified
and Non-Mollified Data

Sensitivity Coefficients Comparing
3-Parameter to 2-Parameter Methods

Percent Difference of Sensitivity
Coefficients Between Perturbations
of .001 and .0001

Sequential Estimates of Diffusivity
(CBCF 700°C)

Parabolic Fit for Parameters Using
Simultaneous Experiment Analysis

Residuals Using Model 1 to Analyze Three
Unrelated Experiments

Residuals Using Model 1 to Analyze Three
Unrelated Experiments

Frequency Distribution of Residual Graph
Shown in Figure 6-9

Frequency Distribution of Residual Graph
Shown in Figure 6—10

Estimated Parameters, Using Model 1, as a

Function of Flash Penetration

Residuals Comparing Direct Solutions With
and Without Heat Loss

XV

175

176

181

181

182

182

188

192

198

209

211

212

213

214

215

216

o ’ -
.‘Iv-va -. 5 ~~
..

u!

p .I a- -'
I'.": h- D .
'0'..- v -9 u. h

-
x.“
6....
n p .q n
.q. . .
V.’ .- '
3.1.5.- . o v.-

 

Figure 6-15

Figure 6-16

Figure 6-17

Residuals Comparing Various Model 1 Direct
Solutions to a Model 5 Case with a=l, Bi=1
and Penetration=0.1 219

Diffusivity vs Temperature for CBCF
Samples at ORNL Using Model 1 229

Diffusivity vs Temperature for CBCF
Samples at ORNL Using Model 5 230

xvi

.

‘i

... .:

.~.a Ni.

.v.. 3

.\

 

. . . o
a . .

«O .Q
. ¢ . s
I ~ 2
.1‘ ‘nm
«1. .u.
I... (m,

EKIQENCEJHFURI:

Absorption Coefficient
Estimated Parameter Vector
Biot Number (hL/k)

Specific Heat

Residual

Gmn(x,x',t,t) Green's Function for the X33 case

h

I(T:9,¢)

n

L
P(G,¢)
P(I)

qo

qr

Convective Heat Transfer Coefficient
Radiation Intensity

Thermal Conductivity

Radiative Conductivity

Radiative Conductivity at Ambient Temperature
Mean Free Path of a Photon

Spline Conversion Coefficient

Spline Conversion Coefficient Matrix
Composite Spline Conversion Coefficient Matrix
Number of Measurements

Sample Thickness

Phase Function

Weighting Function for Mollification Method

Magnitude of the Laser Pulse
Square mm)

(Joules per

Internal Radiation Flux

xvii

R

S(Ilel¢)

5(t)

Pre-Conditioning Matrix

Radiation Source Function

Time

Non-Dimensional Time (at/LO

Half-Rise Time

Temperature

Calculated Temperature Vector

Transformed Calculated Temperature Vector
Experiment Ambient Absolute Temperature
Non-Dimensional Absolute Temperature TLkL/qxx
Temperature Calculated at node i

Variance

Spacial Dimension

Sensitivity Matrix

Transformed Sensitivity Matrix
Temperature Measured at Time Step i
Spline Approximation of Temperature at Time Step i
Temperature Meaurement Vector

Transformed Temperature Meaurement Vector
Thermal Diffusivity

Parameter Vector - True but Unknown

Eigen Value

Blurring Radius in Mollification Method

Dirac Delta Function

xviii

 

III . . . .3 i I

. 1 .. .. .-. 3 1 L. z.

z. . . . C. . . .6. p“ .. .~_

. .2 . .u .3 G. .3 v. ..

v .2 H” L. .1 . . . . .3 ...

p.“ z. ._ . .C 3C :5 no .
\o- .h o v ad. a». v

. l...
.3
4C 9‘
o. qn‘
c . . g
u». . .
u up.
n
A!

Emissivity

Residuals Yi-Ti

Extinction Coefficient (a + o)

Cosine of ¢

Density

Scattering Coefficient

Stefan-Boltzmann Constant (5.729 x 10‘8 W/mfiC)
Standard Deviation of Residuals

Variable of Integration for Time in Convolution
Integral

Optical Thickness

Polar Angle of Radiation

Constant Expressed as a Function of Albedo
Azimuth Angle

Albedo (O/K)

xix

CfiflAPHEHR 1

EHMNKGEUUUNE)

1x1_INIBQDD§IIQE

Many new materials are being synthesized for various
high temperature uses in the aerospace industry. Knowledge
of the thermal diffusivity of these materials is extremely
important in evaluating their appropriateness for particular
applications. Because of the extreme high-temperature
environments in which many of these materials are expected
to perform, parameter estimation experiments must also be
conducted at these temperatures in order to establish valid
estimates of the thermal parameters of the materials in
their intended working environments. At these extremely
high temperatures, traditional methods of contact heating
and temperature measurement become impossible. The flash
method of diffusivity measurement has proven extremely
useful for this application. The procedure for conducting
flash diffusivity experiments is described in detail in
Chapter 2.

Some of the objectives of the research conducted here

are

2
1. To determine the thermal properties of the
materials tested, specifically thermal
diffusivity, from transient temperature

measurements .

2. To investigate the possibility of internal
radiation as an ancillary means of heat transfer
to Fourier conduction and to investigate
penetration of the laser flash beyond the surface

of the specimen.

3. To investigate non-radiative effects which
could be responsible for systematic disparities

between measured data and the mathematical model.

The common underlying motivation behind each of the
above objectives is to develop and utilize a heat transfer
model for these experiments which will more accurately
conform to the physical phenomenon observed, thereby giving
'greater confidence in the parameter values reported. In
many cases, changing the mathematical model can change the
estimated parameter values by as much as 20 percent. This
in turn can have an impact in the design phase of the

utilization of the material. For example, changes in the

ha
Iv-pzybou‘
5...-.vouu
o .
DA chap...‘
" w‘
'5 U.-.‘. d-..
- u
o - _ .
,_ V~~
ec-o ._‘.-
.
-- ~
.
‘
0..-.
‘ .
\;~o...
s.--._.. -.
.-
‘ ‘ "a 1~
.“‘ u... d-
.
O
.' 'eo.,_
oi. ‘-
‘--..-
.5...
. "Z“-...
'-
~Q.‘..~..
‘- .
-' 'vg.‘
.‘Is I! VI“
1. .~
v
"Q._ .
.:“‘ .A.

P.
‘5 Q..
'U..‘ "I..
‘Ql
-
. O
u.
.'c. n'-‘
' “-C
Q
A .
:3‘ "r o
§
“"-|
.0
4.‘

\" _
-.~:QF.“ ‘
‘\ I!

‘n

 

"F '.

‘
vnnuu

U'
(I)

'
c..
”
o.‘
-_‘
.

.
t

’l!
(1'
"
D"

 

 

3
properties of materials used in space vehicles can translate
to changes in space vehicle weight, which in turn can affect
fuel requirements and mission capability.

This chapter addresses Previously Used Methods in
Section 1.2 as well as a brief description of the experiment
and the direct problem solution in Section 1.3. The need
for refined models is discussed in Section 1.4 which is the
motivation for this research. Section 1.5 introduces
derivative regularization as a tool necessary for extracting
radiation parameters. Further work in exploring the
sensitivities of the experiment are introduced in Section
1.6. Finally, the goal of the research is summarized in
Section 1.7, including a summary of the remainder of the

dissertation.

1lZ.28IIIQQBII;HBID;MIIIQDS

Prior to the widespread availability of high speed
computers for use in parameter estimation, simplified
methods were utilized to facilitate more rapid parameter
estimate computation. When the procedure of laser flash
diffusivity measurement was introduced in 1961, the primary
means by which diffusivity was calculated involved the
principle of “half rise time”. This pioneering work is

described in Parker et. al. [46]. Early models assumed a

 

. . ‘
. on .;;b :o‘
‘0‘ n.-.o. ““
hue-v
.
““ ‘.“ “a. ‘.
‘ .I‘ ‘— ..‘.
.‘--‘.-.—‘
-~‘~ .' .-- ~-..
F b u fl...
.34... -.. -‘I- _
.
... A .A'
‘ ~ = a
U... ‘ ' ---
o. .
:- DA--4‘ -;'-:
‘H‘ .O.” '5‘: u-
.
'52:”vA1 .’._ -
- .- -.
“~‘." u.-- -
.
I - _
. C a. .‘Q ~-..
.._ d
.. ds ._~ ‘..~-
-‘V\~~.c" A“‘
My“ ‘-.~ ‘
v-.~-~'_‘-
’0 _ .
.-
.:‘: .:.‘- _ _
.v-a~‘.-~ .4 -
‘5
.‘A
' ~‘a 7 .
' s
‘.
‘- ~ ‘Q .“

".
~. b.‘
'A
vv“::‘A—‘_.’
‘\
v..“-o; ‘ ‘
‘~
I"
n.‘ .
K‘ 2‘ “0 c.
‘ U
~.-b “ .;
“.‘

 

4
pulse heat addition to the incident side of the sample, and
insulated conditions otherwise. An additional assumption
made in the development of this model is the approximation
sin(e) = e for small values of e. Using these assumptions,
the non-dimensional time corresponding to the point where
measured surface temperature on the non-heated side reaches
half of its maximum, or equilibrium, temperature is 1.38
non-dimensional time units. Based on this correlation, the

relationship was established

:1.38L2
n2 t1/2

a (1-1)

where L is the sample thickness and tug is the time
corresponding to that at which measured temperature reaches
half of the final, or equilibrium, temperature.

Numerous revisions to this method have been presented
in the literature in subsequent years. Cowan [43] provided
accommodation for radiative heat losses from the sample
surfaces. This modification required the analyst to enter
correction charts with parameters such as ambient
temperature, maximum sample temperature, and surface
emissivity. The correction factors are then applied to the
half-rise-time method described above. A similar method

accounting for heat losses from the sample circumference is

 

 

. o .
I
. n u ‘. nfia MN“ .'~ -. § 0 o
v. o
u A
a Q. . «v 9. . . flu .- .~¢ u Av
. .. . .c a. .. .. . s. a. C .3 .
n.. .3 .I a. e .3 a 2‘ as s 1“ s x . ~
. . . v . .. . q ~. ~ ~ .
to h . . ~¥ N. 5 n o . ‘
v. v. sa . ... an a. ‘1. u. .n . ‘1‘
a? a. J. a. Z . .. ”a. a; e . t a u.. .3 Av 14 a». .»
.u r.. . . a. 3 . V _. 2‘ a. 3 ~ .. s s e ; . G. u .. Q
.. 2. .3 a: .. .. NM ” . .. a. a ... r. u. . . ‘5
v. u h pu- Q s q~v V .“ gum Q-a ‘ «Rd g-‘ u— .~ \
:4 f. u . ask A , . . s‘ . . s . NU a‘v . s
95. v . .: .\. .ow :— g... "g .g» ~ » ~\~ .«n .. s
.w :u 4... .:.. ...‘.. .... ...... ; .. . .. . .. . .. . .. a. . .
n~ .n v .u. . . I a a Q .. .N .s- . . . q . c . n \ a . u

 

\

N

.‘x

5
examined by Clark and Taylor [44]. Several test cases
comparing this method with the Cowan method are examined by
Clark and Taylor as well.

Later work incorporated the minimization of least
squares error as a means of estimating diffusivity and heat
loss coefficient simultaneously. Early work in this area
was performed by Koski [42] and includes discussion on the
conformance of the mathematical model to the laboratory
measured data. Taylor [47] also examines conformance of the
mathematical model to the measured data for, not only
surface heat loss, but for finite pulse length as well.
This effect has to do with the time duration of the flash,
including considerations for the flash being non-
instantaneous. This paper by Taylor examines the residuals
in various experiments, that is, the difference between the
measured data and the mathematical model.

Input from many of the above contributions has been
utilized by the American Society for Testing and Materials
(ASTM) as put-forth in 1992 [40]. This ASTM publication
provides a standardized procedure for accounting for both
heat loss factors and finite pulse duration factors. The
standardized method is based on the time of half-temperature
rise principle.

Further investigation into aspects of this experiment

 

a. vv;pon..c -. ‘.l
' "--Viv . . _.
~"--"“ -. n‘ .
-'- ‘ -
-h* ."".u.. ‘
A..--..~ ‘ H
U H: A: .-
.‘V.A‘.~' u- .-
-'3 “ 's .= ".
CO. .‘fl-o‘ .~ ‘.
.
'3‘;0. .. ‘g ”‘3
.‘hollh-., .0.” . .l
-~. .~ ~ -
.-‘ ‘ 4' .
sv-q uic‘.:s ‘-=
.
£\>‘-"- ' m
u-I, - 9 .. K.' -
.h.".”h >1...-
. ~-.. ~ . “-
I
.-
:‘_:-~;.= -: -
c._ g
.‘t- '. ‘--
”i
v-._
.. c ”.A-A
.‘ 0.. . ‘
.9‘5 5...:‘. h .
‘ v. #‘-.
.-
v...
hz‘n C‘CD.‘
’ G... ‘
‘3‘...
-
.
§
“‘5 5 z. .. h
a“..‘:‘
“‘
“I. .v.‘ I‘
2' .Ma ,. A.
.-..‘3 . ';
.‘V“‘
'5
F I
~._ ‘I‘. .
1' ‘ ‘.'--
H. "~" 2*
‘ ‘~“ 0:.
a ’9‘
I. h‘ .
.. . .
~~~§Ied Q-.“‘
Q..: b..-
“a.
..-
N v‘A
.‘~~.::‘~_‘ .
~cs~s :“.A
.'-..-
a
F.
\'._ v
i ‘.
\.
n“:I-a" ‘
..~v.S A- . A
b
3“
Q
."‘.;,‘\‘ ‘
5 CV.“ >
‘ y
\“b ‘ ‘
“‘v Q'
~‘
.-
~.
J‘s...‘
‘5‘ g *
e.“ ‘
5‘~3 A?
‘v‘. C
V
‘
‘:..
.~,.~D~‘ .
“3.9“ c
*1 ‘V‘..~
“0. ‘§
§..
‘;‘.C
‘5 ~S.
§
‘
‘4 ~ ‘P
A ”"
b..‘~f\.“‘n
v
"1
d”.
§
.Q
\n‘ ’
'n
s D‘.
V ~‘V.’
§\‘ “
“. Q.‘
\“ h‘.
I ~1
\‘.
\‘ ‘A
n\‘~A ‘.
~a .\
“-a -.
‘ a A
H.‘ t.
.
I‘D‘. ‘
- a
‘b-ih.
t.‘ ‘
\_.c-‘
-‘. Q
‘U

 

 

6
is made by Raynaud, et. al. [48] in regard to the adequacy
of various models. This discussion not only includes an
examination of the residuals, but of the sequential
estimates as well. A discussion of how sequential estimates
are found is presented in detail in Chapter 2. More
recently, the method of non-linear regression using ordinary
least squares has been applied to the flash diffusivity
experiment by Beck and Dinwiddie [37]. This method has the
advantage of allowing model flexibility to account for
various phenomena by simultaneously calculating multiple
parameters.

The standard deviation of the residuals in many of the
experiments noted above is considerably higher than that of
the anticipated measurement errors, pointing to some
unresolved inadequacy in the models. Although large
improvements have been made in model conformance through
simultaneous calculation of heat loss and diffusivity, a
characteristic signature is evident with the analysis of
experiments for some materials. The shape of the signature
generated from these models suggests that the “wave” of heat
diffusing through the solid material arrives earlier than
that predicted by conventional kinetic conduction. This
implies that another mechanism is at work in these

experiments, such as internal radiation or a penetration of

 

 

I II.
I I I c . m. a:
.. I . I . r r : . . t. .x. .. .
. . . L. L. . g .3 rs .» Q. .. s «a .c Q» Qu
a 5.. N L1 g. u .. .5 .. 0‘ .H.. a». Q. is d4 {s Lu
.: a“ .2 -. . ... «4 .3 .t s. 0. ~. ;. um ..u u.. fix n» D» a»
.3 .w. o . s. C. A. J. B‘w \ y 8 s 5 Q» .. s in vhk .~ \
_ . .. 3 z. . . . a. ... ... c . 3 . . ~ . .c ... a. c. T. a t
.9. .. .. .3 C. 5.. s e .3. a» 4‘ s: .3 rs av on Q. .1 u. 3‘ ‘.s «V.»
.._ S 3 v. a. . . 2. ~. ... s» .c .... :4 .. Q. ~w. w 4 ..a
r. 3. T .. a. .. ... a. a. ... r. .. 1.. Q. T« .1 r
.7 ... .. ‘3 n- .. sq A. an .. m3 a» ... is Q.» Q» :s v... Q.
.3. up. .. .. C. . . . . ..u. u a . . .3 .2 .q. . an . v .n. he .2 a .
.3 .U ax; .: v. .2 s . s 2. ,u . \n. u . .o. a e a 2.. a.» s ~ 53.. a»
. . 3. an .3 v . . a .C i a .3 r h . n». «a e» . . . . .. ..s. «C:- v:\ M:
as. . v u .. . v y g u. u u . q A v y . -.h “in u x ‘i .nn\
1. . . . . q . .... . . 2.. .... “a u... .:.. : a u... . . .. c . .. . . r.
p .0“ n.“ .o u A._.. or 5.. .0» us- . k . .. . a v .p s 5' . a, a e ._ ~ »

7
the laser flash beyond the surface of the material. Both of
these phenomena are investigated as part of this research,
which consists of a further examination of heat transfer

mechanisms at work in various flash diffusivity experiments.

1i3__IEI_NIID_EQB_BIIIHID;MDDIL§

Several means are used as part of this research in
order to determine the adequacy of the models used and to
evaluate the need for further refinement of the models. The
primary method of evaluation is by comparison of the
standard deviation of the residuals to the standard
deviation of the expected measurement errors. Even more
important is the emergence of a characteristic signature in
the residuals, indicating that the model does not conform to
the physical measurements.

A characteristic signature is exhibited by long periods
of continuous positive or negative residuals. This type of
behavior in the residuals can also be evidence of a problem
with the measurements, specifically, correlated errors.
Another key indication of model inadequacy is the stability
of the sequential estimates. These phenomena are discussed
in detail in Chapter 2. In the following chapters, various
models will be examined in an effort to account for the

characteristic signature in the residuals and the lack of

o
a

.~-~~
_

a
‘ \
vv..v--"..- '

g..."

 

' e.- 9..
. 2 .

. i. .9.

.3 .\. .2 ..
.. .. ... ..
.. p“ a: .3

e.. .: ... .c

.: .u . . ...

.. .. Z .1

p. .. ..

.t A. -

‘. .. i.

.-¢. av ‘5.

.u. .- -\.

~
-nu‘
o

\

... ~.
.3 s: a» ».
... a. w. v.
s. .3 .n C.
C» s .
w. .c .3 f.
a. . . r. S
su. a» ...
w. .. 2. a.
.3 .2 s
. . z. a a.
.h.. s y s ..
in s .
.e I a.
n. n u .h u .s ‘
.k A.~ y s a v

. « w.
4 g a. s .
. . . . .Q
mg .3
. . s
Q» .. «
u. e
a. . .2
'u. .F A. “V
s . n. .
3 rs
. s .3 .an
Q. .s. . .
flu usv D s
Fa use .—1
s... h . Ah.

.2 z. §~
r. . .
k ‘ .H‘ n .o
S S E
v . s.
I‘ ‘ Q“ h‘ ‘
c . L .. r.
a .. c.
as up.
.. .. . u v...
. . . : a.
nu ‘
a c a ..
... Q.
h . .ru... 5

8

consistency in the sequential parameter estimates.

1x5__DIBIQI_EBQBLIMLEQLQIIQE

Typically, specimen heating is accomplished in these
experiments by laser flash. Temperature measurement is
accomplished by infrared photometry. This research
specifically investigates the heat transfer through a disk
shaped object in one dimension from a flash heat source and
the subsequent heat loss to the ambient surroundings
following the flash. Since the diameter of the sample is
much larger than the thickness, normally by a factor of 15-
20, heat losses on the perimeter of the disk are neglected.
The temperature measurements are made on the opposite side
of the disk from the side exposed to the flash.

In order to estimate properties for a sample in an
experiment, a direct solution must be obtained. Several
models were utilized for direct solutions as part of this
research. Later models are generally more rigorous than
early models at the cost of increased complexity.

The first and most simple model assumes conduction
through the sample and convection from the exposed surfaces.
When experiments such as these are performed at very high
temperatures, the materials may be likely to become more

transparent to radiation of the frequencies emitted from

 

-a..;, .,.;-V
‘Juvoo oat-.-
-: 9- .-..c:.
.' _V - .b-
"§--...._‘ .
‘ ‘ _ ‘
.“O¢- -_

.

- a .v..‘ ’u
. - fl ‘ -
o... ‘0‘--“
n."-"' ..-

‘ I.

—.~.-~--‘--

. C

'~q....-- ‘
p-
c¢¢-u‘--v.. ~

-_ ‘
c... :‘..- ‘
“§-“‘.: .
c
b. g
c‘.:- ‘

‘Q.‘~ -‘

o ~.
.

._.

A ‘..
~ .-
uu.‘ _ -._.
a. 8"

‘ in ‘

§..‘- ‘
‘..~

s.- .

6": ‘~

..,“ ~ ‘

v Q .—
~ -

‘v
. -‘-

- c

g‘-:.‘ -Q‘

"\I.
‘9... y
‘3“- ‘-
u“
‘..
.D.

.

A
*_ h.“

.-‘.

“C.
h

§ .‘

‘ ..:-‘ 5
‘\ ‘
Ԥ
...
‘n~.“.
n‘ _ ‘-
“~ ‘
V‘.'
A
.‘
u. fig
u..‘-~“‘
‘Np a.
n
‘
.- .
U ‘>.
-.‘¢ ‘8
C \. ‘
5 .Q
I

\

.A
u...

‘ 52-...
\_.‘.‘
fi“

\.‘

‘

.“Q “

V
\.._ «_‘
‘V‘V
\

5
Fa
‘._~ A
‘ \

L‘h‘
“\y~~
‘c.C
‘

‘b
q
s‘~;‘

‘
-

9
matter internally. Part of the objective of this research
is to investigate the effects of combined conductive and
radiative heat transfer inside the material. Several models
are available to address this phenomenon. The most
appropriate model for optically thick material utilizes a
radiation coefficient which models the radiation as a
diffusive phenomenon. This model is discussed in detail in
Chapter 3. Other factors which are to be investigated
include the penetration of the laser flash into the material
at the instant the flash heating occurs. Additionally,
there is an investigation into the reflection of the laser
flash inside the furnace. These factors are discussed in

detail in Chapter 4.

1x5__DIBIEIII!I_BIGHLLBIZBIIQN

When studying the subtle effect of a small contribution
of heat transfer from a mechanism such as internal
radiation, in addition to the dominant mechanism of
conduction, the extraction of a parameter such as “radiative
conductivity”, becomes extremely difficult. This is because
the sensitivity coefficients are closely correlated. When
this occurs, the final set of equations used in solving for
the parameters becomes nearly singular. Without being

treated with some type of regularization or stabilizing

 

 

a . 3
T. C
at
i . c x
I C
N-
2.
\.‘
~ \
. ts
.q‘

 

lO
influence, the parameters cannot be found. Occasionally in
solving singular sets of equations, the use of double
precision in the computing method can improve results. With
measurement errors in the data however, this is of no use;
each iteration of the non-linear regression gives
increasingly wild answers. A part of this research includes
the development of a method of regularization which allows
parameters to be calculated even with inaccurate initial
assumptions as to the parameter values. The stabilization
is achieved by incorporating information related to the
first and second time-derivatives of the sensitivity
coefficients and the measured data. The Derivatives are
approximated by fitting parabolic splines through the
respective curves. This method allows estimation of the
“radiative conductivity” term in the internal radiation

model. This procedure is explained in detail in Chapter 5.

1x5__QEIIIIZINECIII_LNILISIEHMIIIQD

Investigating methods which may simplify the
calculations in this problem and provide better accuracy is
an important part of this research. Variations of the
method of calculation include the elimination of heat flux
as a parameter by normalizing the solution to a temperature

measured at a specific time in the experiment. Further

 

 

 

Fl . .. . .

.3 — .3 a.
on O u o.- n. C»

. v as. ..
.. . . C . ‘4.
O o O u Q A‘ Q“ Q.-
.aa .H. H; .h— 4.
an. .s. 2. up. i. ..
. g 3‘ u . a» :u .3
.. -. C. L. o. .
2. 3. . v . .a
.t . . ~.. ... a.
u .x. L. 2. a 4 .
. - M Q a up ~n
. s n.‘ c Q i

O
ViA
-~-‘:-

C.

yr.

hhu

a»
.3
.r! C»
a . . .
.. s .2
... 2.
. . _. .
3 . .
.. u a:
c a l.
C ‘ ...
.~ ‘ g ..
4‘ ..

u H w .
a. Q.
Q ...
.. \
3 a .
a. x C
. . a.
s s .s s
x: : s
. . 1.

E

s :
1.x» S
\ ~
FV.

.1

\.~.

ll
investigation into the optimum duration of the experiment is
also made. Various criteria are investigated pursuant to
the establishment of a systematic way of evaluating the
adequacy of various competing models. An analysis is
undertaken to determine the validity of the method used in
calculating sensitivity coefficients. Finally, the
parameters are estimated over sequential experiments in
order to establish the functional form of the temperature
dependence of the parameters. The results of the

investigation into these issues is presented in detail in

Chapter 6.

 

The main goal of the research is to obtain a greater
measure of confidence in the results of the flash
diffusivity experiments. Various physical phenomena are
investigated as possible causes for model non-compatibility
with the experimental measurements. This is a critical area
of investigation since small changes in estimated parameter
values can have a large impact on the cost and ultimate
effectiveness of the devices in which the material is
intended for use.

As an outline of the remainder of the dissertation,

Chapter 2 describes the fundamentals of the experiment and

I...‘
§
...u.._;‘
' -
.."’-
"' -
“ --. .--
e...

I... .‘
‘

~'~u _.

‘vn

:- - ‘2:

-.- 9".

fl
.4...“-~:
su. ‘.-‘

I .1
H" ‘c. .
';. ‘vi
..--.:“

'!
(I!
it

 

'7‘

'1

in

LJ

(3'

L)

 

 

12
aspects of parameter estimation for a three-parameter model,
solving for diffusivity, heat flux and Biot Number.
Internal radiation as a diffusive mechanism for optically
thick materials is discussed in Chapter 3 with analysis of
both simulated and real measurements. Non-radiative models
are described in Chapter 4, involving an effectively
instantaneous penetration of the flash into the material at
the time of heating. The concept of derivative
regularization is discussed in detail in Chapter 5,
examining the benefits and costs of the method. Various
aspects of sensitivity analysis and method refinements are
presented in Chapter 6 and the work is summarized in

Chapter 7.

F'

.-
..~

= ..-.‘
U--.

Q~

~
5-

n.os‘

‘. e.
a»
3 1
F». c .
o..
.t ;.
... ..
o. ’y.
.-
3. . .
a. Z.
.N u-
.c :-
h“ ‘-

‘ flu

‘a‘

2.

c\ Q.
s .
.l. 1‘
. v . ‘
3s «v
a s C.
. . :«
u~ ».
2 ‘ s e
s. .r..
e . G.
. .. .
I . s
u.‘

 

CEHMPTEHR.2

FTJUSHIBHEASCHNEMEQEP CW’IDIFTHJSINKFTI'

2x1_IflIBQDQCIIQN

The historical discussion of the flash method of
diffusivity measurement, featured in Chapter 1, highlights
some of the milestones and significant refinements made
since the procedure originated. During the course of this
development, the flash method of thermal diffusivity
measurement has been used many times in dozens of
laboratories around the world.

A.sample list of countries having made contributions in
this area is as follows: Reference [54] presents an
investigation of the thermal diffusivity of thin coatings of
tungsten and molybdenum which are applied by plasma spray.
This type of material application is in use in the nuclear
industry. This work was performed by Canadian researchers.
Reference [52] discusses experimental measurements of semi-
transparent materials in the temperature range of 300 to
800K. This work was performed by researchers in France. A
contribution from Germany was made in reference [36] where
an extrapolated radiation heat transfer solution was

superimposed on a kinetic conduction solution in order to

13

 

Q"-
‘
.-‘_ -...‘.

~
“

C.“
V‘-

~.
‘.

v I-q

'f“

& s
a.
sq.
‘:

l4

obtain more accurate estimates of the thermal diffusivity of
alumina. A contribution from Ireland is presented in
reference [57] which explores thermal diffusivity
measurements taken on alumina based ceramics used as
packaging substrates. A contribution from Japan in
reference [53] discusses measurements taken on casting
powders used in the steel making industry. This method
examines measurements of material consisting of multiple
layers. Reference [56] is a paper written by Scottish
researchers regarding flash diffusivity measurements taken
on dielectric crystals which exhibit non-linear optical
characteristics. The work specifically looks into crystal
damage from absorbed laser radiation as a result of the
measurement process. Many contributions have been made by
the USA. Reference [40] presents a standardized method for
measurement and calculation of thermal diffusivity by laser
flash by the American Society for Testing and Materials.

The flash method of diffusivity measurement employs
non-contact sample heating and temperature measurement.
Contact methods of heating and temperature measurement of
solids typically produce non-uniformities which come about
from such factors as imperfect contact, serpentine heater
construction and the additional thermal mass inherent in the

heaters, thermocouples, adhesives and lubricants. The

.3
3..
v u.

.a
.‘b

:v

Q-

..\‘-‘

s.
.. a”
m. w.
. ¢
.. s.
:.

.. In
... L.
.c

. -\~
—.. .s
.. u"
.H .-

... T. ~ C s .
3. . . at
z. 3
.7. c . T. 3.
by ...
T. a. .C I
a. . . ..
e s . .1‘ .3
Q ~ § . sh» :—
u'.‘ u v
2 . . C ...~.. \. ~. .

q
~

 

L .L
T.
Q~
C P.
,7. 2
AV 4‘
s\
...
. _\
a»
a:

4 Q
S S
a .. t .
At .2
a. . ..
.. \ V‘ \
.3 .1
‘s
a. . .
. .. . .
a e

15
effect of these factors can be minimized by employing
samples of physically large size. One of the prime
advantages of the laser flash method is that the non—contact
nature of the heating and temperature measurement eliminates
non-uniformities which allows the use of small samples.
This in turn allows small furnaces and other sample
enclosures to be employed. The small size of the samples
also allows the ambient temperature of the experiment to be
changed quickly, so that successive experiments may be
performed at varying ambient temperatures over a relatively
short amount of time. The flash method can also be used in
extreme high-temperature environments due to the non-contact
nature of the heating and temperature measurement. There
need be no concern for the degradation of the heating and
measuring equipment from exposure to high-temperature
environments. The small sample size typically allows the
duration of the experiments to be quite short, with most
experiments lasting less than one minute.

Once laboratory measurements have been taken, a direct
solution must be computed for the problem in order to obtain
the parameters. The model utilized in this chapter assumes
one dimensional Fourier conduction in a rectangular
coordinate system with convective heat loss on both sides of

the sample, each side having the same heat transfer

   

‘-
s-‘

" 5's >—
Ov-n -_,- ugv

..

.2 ...
.»

5. ..

. . o.

I. . .

_ C.

.. «\u

.u»

2.

n ..

D u

n —.
T. ..

‘. .. ~
.H .. .*
A: . . ~\~
- .qu «K»
w& s s a:
r. ..
o _ as. I
:v . IQ.
on . . n
. g a w .is

. . ~ - . u . r \ er r . Q». .
.. S .. a. I E c . . . E
a .. a”. «\g ‘ ‘ ' g$ \ «\H 8.. .N -§ \ ch c \ z
4‘ .s ‘ . § . .. s3 ~= .. \ Q» .\ \ $ ~ A&
x. C C C a r C. -2 .2 r
.3 :. f. .. a . . 4‘ Q» .2 s . ~3 .2
a. 3 . . a. . . .: a. C . .. . a - s. a. Q.
‘ u . . *1 ‘ . Q. a C as al. a: 2‘ n... a; u .
... . c a. . . ..v. C. \. .. .C .
no a. N a ' h u . s ,f. s ‘ s C s \
. C u s. 2. - . s. .: .= s: a: p s. u s . ..
.. t. ... n .. ... .... .‘ .... us. .. s.
- L. 2. . g . L a\~ 2. s . ~ q s . .a. \\s .s \

l6
coefficient. Due to the large sample diameter in comparison
with the thickness, heat losses from the perimeter edge of
the sample are neglected.

Section 2.2 provides a brief description of the types
of samples used as part of this research. The measurement
instrument used is described in Section 2.3. The
Computation of the direct solution is presented in Section
2.4, assuming simple one dimensional conduction and
convection. A finite difference version of this solution is
provided in Section 2.5 with the parameter estimation
aspects of the problem discussed in Section 2.6. The
inadequacies of this simple model are discussed in Section
2.7, demonstrating a need for further research into more

sophisticated modeling techniques.

2x2__DIEQBIEIIQN_QILSBMRLIS

Data from fifteen laboratories around the world were
analyzed as part of this research. Ambient temperatures for
the experiments performed ranged from 20°C to 2000°C. The
disk shaped samples ranged from 1mm to 18mm in thickness.
Thermal diffusivity estimated from the data ranged from
approximately 0.3 to 80 nmfi/sec. Heat loss from the
surfaces of the samples following the pulse heating ranged

from near zero to a Biot number of approximately 10.

17

The primary area of study for this research is centered
on the Carbon Bonded Carbon Fiber (CBCF) material developed
at Oak Ridge National Laboratory. This material is
primarily intended as insulation for objects exposed to
atmosphere re-entry conditions in space applications. The
material is manufactured by Oak Ridge National Laboratory.
CBCF insulation is vacuum molded from a slurry of chopped
amorphous carbon fibers in a mixture of water and phenolic
resin. The material is then dried and the resin, which
accumulates where the fibers touch, is carbonized. This
process results in an open structure where both the density
of the solid fibers and the overall porosity are continuous
throughout the material. One specific use of the material
is as an insulator in radioisotope thermoelectric generators
such as the General Purpose Heat Source used to supply power

to deep space probes like Galileo.

2‘3__LBEIGI8.9lLMlhflnBlMINI

Since the flash method of thermal diffusivity
measurement has been in use for several decades, equipment
designed for conducting flash diffusivity experiments is
available for purchase in the form of pre-manufactured

systems. Two companies which sell such equipment are

18

Holometrix Inc.

25 Wiggins Ave.

Bedford, Mass. 01730—2323

(800) 688-6738

Anter Corp.

1700 Universal Rd., Dept. 10

Pittsburgh Pa. 15235-3998

(412) 795—6410

www.anter.com
Measurements were made as part of this research by equipment
made from both these manufacturers. A majority of the
sample measurements, however, were made using the Anter
system at Oak Ridge National Laboratory. A schematic
diagram of this system is shown in Figure 2-1. In this
system, the sample is held horizontally and the laser flash
is introduced vertically, incident on the top of the sample.

Figure 2-2 is a photograph of the Anter system at Oak
Ridge National Laboratory. This system happens to be
configured for four modules, but the systems are available
with as few as one module, depending on the range of
intended temperature applications. The four modules
provided in the Oak Ridge machine include a low temperature
aluminum.block furnace for temperatures from -150%: to
500°C, a room temperature to 1700°C furnace, a 500°C to
2500°C graphite furnace and a room temperature to 1200°C

quench furnace. The quench furnace is cooled by a blast of

helium gas once the heat source is turned off. The

l9

DISK
SHAPED INCIDENT LASER FLASH

SAMPLE

1

HOLDER\A

FIBER OPTIC
CABLE

  
   
     
   

INFRARED TEMPERATURE
SENSOR

PC DISPLAY
AND DATA
STORAGE

  

INFRARED
TEMPERATURE
DETECTOR

  
     

 

DATA
ACQUISITION
SYSTEM

 

 

Figure 2-1 Schematic Diagram of a Typical Flash Diffusivity Measurement System

 

Figure 2-2 Photograph of the Anter System at the Oak Ridge National Laboratory

i

,,.c- -
— .9‘ V".
o¢.A-"'." -
: . v

my
«‘-

a. a.

v . . .
fl “‘5

. . .
.3 .—
v. a:
.u.

9..

..

._.

h '

xx.

.0.

...
a. U.
c . ._t
I c.
c.
xx. a“.
.3 s.
.. ..
~.
... . .
.. I.
‘s ‘§.

is

a:

‘3

~\~

-~

-.
n. I
E ...
S a C
‘~‘
8 . .
..~ a.
. v ‘ .
as \\ ‘6“.
~-- \ a

20
quenching process is capable of reducing the sample
temperature by approximately'ZOOWC per second.

Samples held at high ambient temperatures in a furnace
are typically in a vacuum or an inert gas in order to
minimize chemical interaction with the surroundings. The
laser flash is generated on the outside of the furnace and
passes through a spectrally neutral neodymium-glass window
in order to reach the sample. Since the sample is small,
the laser flash is able to cover the entire surface to
promote even heating. The laser has a maximum pulse energy
of 35 joules. This pulse magnitude is adjustable and is
established based on an initial operator-assigned maximum.
Subsequent laser flashes for a given experiment are adjusted
through a trial and error process, facilitated by the
computer control system, after the initial flash.

The infrared temperature sensor used in the experiment
is also physically located outside the sample furnace and
takes readings through a spectrally non-interactive glass
window at the bottom of the furnace. The measured signal is
passed through a fiber optic cable to the actual detector
unit. The detector unit includes a pre-amplifier, which
allows several furnaces to be served by the same infrared
temperature detector without moving the detector itself.

Additionally, the sensitive electronic components of the

QA.AQOA
‘- u— '

".-~"‘

'o-gq
.-

~4....-_

‘Rbg.
" no"

a

O

‘
. 4
' . -‘v.,

‘c. . -.

an.

“‘0 b...

b O
a

.'q -.
~—.—.~'.-.‘-u :

a.
‘

-0
5-

a-

V .
h
x

f

 

I
(I;
(I)

21
detector are protected from the high temperatures of the
furnace by the physical separation. The Oak Ridge machine
uses two detectors: a silicon photodiode detector for high
temperature applications and cryogenically cooled Indium—
Antimony detector for low temperature applications.

The detector is calibrated using simulated black bodies
at different temperatures. From this calibration procedure,
a look-up table is generated which provides a mapping of
intensity versus radiation wavelength. The lookup table is
stored in Read-Only-Memory (ROM) in the detector's
controller unit. Measurements taken using the instrument in
actual practice are arrived at by interpolation of the
various values which are stored in the lookup table during
calibration.

The output of the detector is usually expressed in
terms of volts, which are directly proportional to
temperature. The detector is normally calibrated to read
zero for each experiment at furnace temperature. The final
signal recorded in the data acquisition equipment is an
expression of a signal proportional to temperature rise
above ambient temperature.

The temperature measuring instruments tend to perform
better in higher temperature experiments due to the more

distinctive spectral signal emitted at high temperatures.

L.

O

.Q“
.-
‘a-O¢"

a‘~ A. ‘
._ A.
.3 >. w.
as c.. Va.
2. ,...
.u. a. ..
.. z. .3
. . v. u.
.. .
a. :— A—v
an .—o D.
.. L. . .
.. .. ..

III-
up-

A..
‘CQU. F

.-

;. a.
.. .
.. .
gum .. A
a. a)
n.. A:
Q “
. .
M.
..
AL.

. . c
.. . . .
. . . . =9
.3 a.
3 .. . ...
.s s s . s .
u . .2 .3
. T . ..
.s ~ . u
. s s y Ah.
s.» . n I .4
uni m .

22

An additional requirement for low temperature experiments is
that liquid nitrogen must be used to provide a cryogenic
environment for the detector. This is necessary so as to
minimize background radiation not related to the temperature
measurement of the sample. In spite of this precaution,
more background noise typically exists in measurements taken
below 600 degrees centigrade than is exhibited above that

temperature.

2.i__DIBIQI_§QIQIIQE_QQMEQIBIIQN

The Anter system described in the previous section
provides immediate analysis of diffusivity by evaluating the
half-rise time as described previously. The system
automatically computes several diffusivities, each based on
the methods of Parker [46], Cowan [43], Clark and Taylor
[44], Koski [42], and Heckman [59]. The computer displays
this information in tabular form and automatically stores
the information in an individual file for each experiment.

The method of analysis used in the present research is
based on non-linear regression, as described in reference
[25]. This method requires the formulation of a direct
solution. The most basic model, which assumes conventional
kinetic conduction of heat through the material, can be

solved analytically. In order to accomplish this, the

'<

a
- ‘V‘

-co"'

.3

—
‘.

,--.
.

-c :.A--.
-.~~-..,.

. .s.
g.
is.

.1

v.
-.§
K.

sne

23
Green's function is used for the X33 case, that is one
dimensional conduction in the Cartesian coordinate system
with convection at both boundaries, as put forth in
reference [51]. Assuming no internal radiation and constant
conductivity with respect to temperature, the one

dimensional differential equation for this problem is

 

M 621'
QCpB-tg:kaxz (2’1)
The boundary conditions are
kg] q06<t> min-T...) (2-2)
6T
«[532] =hiTx.,-T.) (2-3)
x=L

In these equations, p is the density, cg is the specific
heat, T is temperature, t is time, k is thermal
conductivity, on is the magnitude of the heat pulse,
typically expressed in joules per square mm, 6(t) is the
Delta Dirac function, h is the heat transfer coefficient, L
is the sample thickness and x is the spacial dimension.
Prior to the initiation of the flash at t=0, the sample is
assumed to be at ambient temperature, T.. The Green's

Function Solution Equation given in reference [51] is.

L
L.

3.

\\\

24
T(x,t)=Tm+%L:OGX33(x,x’|t,I)qoo(I)dI (2-4)

where the Green's function for X33 is given as

 

0° ‘ 20 'T 2
6x33(x,x’lt,t)=%§e ”"1 “ ”L Amm nm(X’> (2-5)
where
Bmcos (em—2‘.) +Bilsin (swig)
A (X) = , , (2-6)
a 2 2 812 .
(Bm+B-il) 1+-2—'—; +8.11
Bm+BiZ

I /
n.<X’) =Bncos (6,33) +Bilsin (BMLL) (2-7)

In these equations, Bi1 and Bi2 refer to the Biot numbers on
the right and left hand sides of the specimen, respectively.
Since the temperature measuring instrument used in the flash
diffusivity experiments reports voltage proportional to
temperature rise above the experimental ambient temperature,
it is convenient to set T;=O in the solution. Since the
same medium exists on both sides of the sample at
approximately the same temperature, it is also reasonable to

set Bi1=Bi2. The solution now becomes

a “ -2 q
T(L,t)=__qfl ‘ 2.2 e mm: ”L2 A 6(I)d't (2-8)
k (=0 Lmzl m

av
z.
.

.

~I~

« .

K. ‘
Q

'-
.-..‘.3-

~. I
... t.
A. ,

c. ‘N
.. ;
s. “H.
«q .:
... v.
Q.

.. y...

.. ..
w . p
— .. u ..
d .. r .. r
C . r . c
km 2. .2
a . . .. a
a. 3. 3.
tr 2. :.

.5

ca

H5
.3
3
8 s
Q v

C,-
\.\

25

where

B,[B,cos(Bm)+Bisin(Bm)l

A = _
" Bfi+512+231 (2 9)

 

Integrating and non-dimensionalizing, we have

T(L___,__ t) kL 4:8 ~Bmcxt/L2Am
qu ”:1

T (L, t)= (2-10)

In order to compute the eigenvalues, the following eigen

condition must be satisfied

t (B ) zsmsi
Bi-Biz ( )

A rapid procedure for computing eigenvalues is Newton's
method, which involves finding the roots of the following
equation, which is simply a re—arrangement of the above

equation, expressed as a function set equal to zero.

f(Bn)=tan(Bm)(Bi-BiZ)-ZBmBi=O (2-12)

Using this:method, an initial guess is made for Bm which, in
the case of this research, was 2.5 for Biot numbers less
than 6 and 3.5 for Biot numbers of 6 and greater. This
method proved effective for Biot numbers ranging from near

zero to approximately 30, well above any experimentally

o..- .‘O
V. r;.
.3

a.
p. D.
.v .o
n. . .
.3
‘5 un
.u
3.
«a.
..
u:

. ‘ u. ‘
a. r i. s a L
r. 1 .4
.. :
C. .q.
ax
— .‘
g ‘u u-
on L.
~\~ a» 09‘
o u . 1 ts
. v a:
.. en.» 3 .
.u. n. ..I..

in.

26
observed values.
A more refined method for arriving at an initial
eigenvalue approximation is given in reference [51]. For

Biot numbers less than 1.0, the following approximation may

 

be used
_ _ . . 1/2 1/2
[31" 45 3031+jr<3n1 (2_13)
481
where
f(Bi)=225(3+231)2+36031(312+231) (2-14)

For cases where Biot number is between 1 and 4.85 the

following relation may be used

U2
.121+[(_IZI.11/2+(231+1)J
Bl: (2-15)

(ZBi+1)
Bi

For values of Biot number greater than 4.85, the following

approximation may be used

z Bin
1 2+Bi

 

(2-16)

As a more general approximation of the initial eigenvalue
for this problem, the following relation was arrived at as

part of this research

 

‘na
'-
t-Q‘QO—
v v.
A.

a. n

2

1.

a.»

27

1'1

8 =
1 1+ H 2.151 /2.15]. (2-17)
J2Bi

The above expression is valid for Biot Numbers from 0.2 and

 

 

higher with an error of less than one percent when compared
against the converged value. Figure 2-3 shows a graph of
the residuals of the approximate eigenvalues arrived at
using this method compared to the true converged values.
This graph shows the very close approximations achieved
using this method. Equation 2-17 was developed as a
continuation of work initiated by Yovanovich [61] who used
an equation of the same form for computing the first
eigenvalue for the X23 case. The constant 2.151 in the
exponent in this equation was arrived at as part of this
research by ordinary least squares parameter estimation,
similar to that used in estimating parameters in the heat
conduction equations.

After computing the first approximation of the first
eigenvalue, subsequent iterations are made using the Newton

method by adding a correction value, AB, computed as follows

 

rm) fan‘s“?
AB= " = . 2-18
f'm) ___1___+.§.£ ( )

coszw) ‘3

m,

i. .u. .-.-.-\o
OWWWO m

I-lauuzlc I:~G)c0-!W

2.. .< . s.
r. a c . .. _
1 S
_ F . a»
1‘» s ~ .1
.L ‘3
.. s
a» .. e .9.»
L .. x c . c.
a r . . a
‘2
.. .. a 5 pa.
. . e
. Q A“. ANN
5‘ .h.~ .m..

4. ~ 2
.. ‘
I e
.
r
«o a t
-t t
.3 Q.
c s Na.
«1 .2
h. .. a
\nu .u‘.‘
\\\

28

 

 

 

 

 

Eigenvalue Residuals
i
on

 

 

 

O 2 4 6 8 10
Bknlmlnbor

 

 

 

Figu'ez-s EigenvalueResiduahasaFuncfionofBiotNumberformanCase

With the ABm term.calculated, a revised value for the fig can

be calculated as follows

Bfi’lwfigefi. (2-19)

where the i superscript designates the iteration from 1 to
convergence. For the work conducted as part of this
research, eigenvalues were computed to six significant
figures.

Once the first eigenvalue has been computed to a
satisfactory degree of accuracy, the initial value for
subsequent eigenvalues is normally obtained by adding n to

the previously calculated eigenvalue. Newton's method is

 

I h—

. g r-

.ifi

I a.

.g‘ r.
. v

.1 ..

an.

. . .no

. o

a». p.

n.. . .
p...

L. e
we.
va a..
.3
. . . .
u
v. \m
AU
.0 s : Q»
I p.
v“ ...
.n. u.

.3

ah
..‘

\
H1»
v“
. ..
and

v a
a. .—
Fhs v~
C» h a.
a»
E .. .
_ .. s .
e t . e
u .s
. .. A »
2‘s

29
applied again in each case until convergence is obtained.
For a more refined initial estimate of subsequent
eigenvalues, the following equation is provided in reference

[51] for small Biot numbers

 

 

 

28'
B"=(m-1)n+-(_m—-11TI; (2‘20)
and for large Biot numbers
B =(m-l)n + ———3—— (m-1)n - Biz B
m 2(3+28i) B’ a (2’2“
I
where
. . uz
33‘: l + 881(3+281) -l
.2 _
“w- w, ] «2
II

and Bm' is arrived at from.Equation (2-20).

The number of eigenvalues necessary in order to achieve
convergence of the infinite series temperature solution
varies depending on the value of diffusivity, sample
thickness and the time associated with the first time step
in the experiment. Since the term in the infinite series
solution which drives the series to convergence is the
exponential, a convenient convergence criterion for six

’significant figure results is

 

3.

v...
a. .
. a FL. .1 I
s .IH .
Q» . ‘
....._ ..s e
u.. e s NS
2. -hy . C s .. x
s .
v. a e ‘ \ h|v Q‘ ya»
2.. x
. . ‘am «we. 9
h .. AV .h so f
.. s \ £ 5 C
§\s . K
. .. Q. he
s\ C»
u .. nV
~ \ \ ,
‘\.v ... ‘
n ..

. .
:—

30

 

= 106 (2-23)

 

 

Taking the log of both sides we have approximately

_Bit++B:t+ : 14 (2'24)
where t+ is the non dimensional time defined as

t’=-- (2-25)

Since a typical value for fin is approximately nn, the
eigenvalue to be calculated in order to achieve convergence

can be expressed approximately as

2 4
Bn:= ll4+n t (2-26)
t’.

Since t+ is typically on the order of 0.01, we have the

number of eigenvalues necessary for convergence expressed as

n = 1 fl <2-27)
II

1'

Since the eigenvalues are very close to nn in value, the
number of required eigenvalues for convergence for a typical

initial time step measurement of .01 unit of non-dimensional

9...- .'~
0" \

.“' -~ -b.
p .
u:"~ -
Vb-..‘ :

.:"l... . .-a -

a»- .‘_-' u

.A

"o...
'7 -" "'5
‘ H.-.»'

“a
g- ‘
r
.
." W:
.. ’“‘
.e ‘
a§~ ..V ‘
‘
9“
.. A
'n ... ‘
V n.“~
'-
n o ,
‘~. ‘fiuk
..~A.‘C.v-c‘
"'\.
‘-A
“
\ IA
~‘ ~F ‘
E‘ss “ “
\‘~
§
A
-~“
e». ‘QF“I
s‘s "na
5‘-»
.
"»
r‘:=:‘e. .
V-..~ “~A
‘ N
‘1..—
I,‘
si‘
‘
A
k.
..
$5-
‘c
:u‘.
‘
‘__ ‘
." h
“'9 r:
«‘5».

31

time is 12.

W

Using a backward difference scheme for the time
derivative and a central difference scheme for the spacial

derivative, the energy equation becomes

Tij ' T11.1 _ le+1 '2 T11 + Tij-l
oc —k (2-28)
p At sz

 

 

In this notation, the i subscript represents the spacial
node to which the temperature applies. The spacial nodes
are numbered from right to left from zero to n. The j
superscript represents the time step to which the
temperature refers, where time j is considered to be the

present time. The left boundary condition is approximated
by

(xxnx
2At

+h(T -T)+A{T°-T1]-0
° ” Ax —

and the right boundary condition by

 

()
(2-29)

 

 

 

 

 

 

. .. . .n -. .. . . . . 2‘ . . 2. v. .. .- .. .. u. c.
. .\‘ J‘ ”y. s.. awn“ u.§ RV .‘n an «an Q~ Q‘ s» 5‘ C» u. :‘ k.» u..
.. z.s . ... C S C .2 C T. «C 5 .C T. T. C
as o . “O ~ .~ . s _ s a. Q» .‘s “We .5 § .~\ ‘ ~ ' Q.‘ .s \ e. \ J ‘ F4
.c C. t I . . t v. I .x ... L1,. : L. .C at
a. a. ... . . 9 a r E a.c . . C. r . .. s . u. 1‘ s: l
.- . ~‘c \» a» 9 ~ .hu A» gfl‘ ~¢. “H '\ We HIV e «V \ .H. N ‘
. .. I ..... S T. . . i C. S x. C e 2 .i 3. 3. a c . . ~.1 .d
. Q AH.» Ah» A.» F \ ahh he. in. § » NB Q» l. 0 Q5
. . v . a.» .Q. .. . KL. 3 .t .. ‘ K-..
. . a... g e. 5.. ~ . .. v . Q.» C. 2. a: J s > x i s s . - O
..._.. ..... u... .... . . ..... v: .... . .. .. .. .. .. .. .. s .. a . .. . c . v .
. ~ . a. .5 e. se :— ‘Q ..I ... ex.

32

ocpr(Tj_
"_'__—' n
2At

T -T
+h(T,,'T..) +k{ .igxi’i) :0

The flash heating is simulated in this problem by

Tj'1+Tj 4'“)

n-1 n-1

(2-30)

setting all initial temperatures at zero, i.e. ambient, with
the exception of To0 which is set at ZqO/Axpcp. This
represents the temperature rise resulting from absorption of
the entire energy of the flash at the surface node, To.

The solution to these equations, gives the direct
numerical solution to the problem. Although the solution to
this particular problem can be found exactly by the infinite
series expansion mentioned above, it is necessary to build
the numerical solution in preparation for the addition of
the non-linear radiation terms to be utilized in the next
chapter. Since this numerical method serves as a basis for
the computation of direct solutions for subsequent models in
this work, it is very important to observe accurate
agreement between this and the exact solution prior to
adding the non-linear terms.

Combining the two boundary condition equations above
with n-1 interior equations, such as equation (2-28), leaves
n+1 equations and n+1 unknowns. At the initial condition,

t=0, all n+1 node temperatures are assumed to be known. The

33
n+1 unknowns in the first set of simultaneous equations to
be solved are the n+1 node temperatures at the first time
step. The first of the n+1 equations, the equation
generated by the left hand boundary condition, has two
unknowns, T3 and T1. The second equation has three
unknowns, Tm.15 and T2. ‘This overlap continues until the
n+1 equation which, like equation 1, has two unknowns.
These are qu and Tn. In matrix form, this set of equations
takes the form of [A][T]=[D] where the [A] matrix is n+1 by
n+1, and the [T] and [D] vectors are n+1 by 1. The [A]
matrix is completely zero except for the main diagonal and
one place either side of the main diagonal. This is known
as a tri-diagonal matrix. The [T] vector contains the
unknown temperatures for each node and the [D] vector
contains the temperatures from the previous time step.

The method of solving the tri-diagonal matrix generated
by these equations is very important and can significantly
affect the results of the calculations. Although
algebraically correct, solving the equations sequentially by
substitution can generate roundoff error through successive
close subtractions, significantly degrading the accuracy of
the solution. In contrast to this method of evaluating the
tri-diagonal matrix, the equation transformation method as

outlined in reference [24] eliminates this accumulation of

 

 

. .. e. 5 ‘ ~ I
I _ Av K§ \» ‘3 O C O fist
.. . . e . .
.. .. z. 3.. .. .. .. .. se ‘t fiv g o .‘
.. .. -lc- -[ln.-i.-- . a v a. T.
A; a a . ~ 3 . 0 V .t
s ‘ \ N/s
f 9 u. : p t w r
S T. e. .A c: 2 . .. C 1 .. 2
e 0
.~. ~ d ' Q» ‘1 s ’ t
.2 .3 6. ~ . . . . . x a e D.
r. ~ \\ .3 AV R v . \ AU
A.» u .a. as F‘ v.‘ c \\ h~ ow \
v. 7. 3 3 ...‘ a r .5 k.
v. A. 3.. gnu us .. a. v. s. .a
L. .o . ..l «\e .N “H. .u... {see 43.. ”in

34
errors. Given a tri-diagonal matrix expressed in expanded

form as follows

 

 

 

 

 

 

bocoO 000... o o 0 To do
a1 b1 01 0 00 0 O 0 1 1
0 a2 b2 c200 0 0 0 T2 d2
0 O 0 O O 0 ‘°' an-l bn-l Cn-l Tn-l dn‘l
LOOOOOO'” an "damn;
(2-31)
we define
a C0 ‘ do
c:.__ d=_ ..
0 b0 0 b0 (2 32)

Subsequent terms are defined through the recursion relation

C;=--J;7- d; -—-—-:—' (2-33)
br-atcr.1 br-arc

c __dr-ard,._1

r-l

for r=1,2,3, ... n. When dg'is finally calculated, this is
equivalent to Tn. The other temperatures are calculated by

the following expression

Tr=d;-C'T (2-34)

1' rd.

for r=0,1,2,...,n-1. The results of this method are shown
in Table 2-1 in comparison to the results from the exact

solution.

 

_ . q . c g ‘1. A s D. . A u. .
l‘|ul. >- .. v c . N. h—.- aP- . Q A... l‘ a”. ... -‘. u‘. .‘o .4. .4. .4. —<. i
3. .... . . . . .1 .>. ... ... ... 3. .4. .3 .3 3. 3. .3 3. ... ..
p” A‘s Can u. . e o e e o e e e e I e 0 e e .e 0
...~. “I A»-.......»q.~.-v....-...a.-\.auaeqv
a: .. 7* «x» .
n
w. . '9. ... ... a... .h. ..L ... 11 a». .l .n. ls. phv
... O a u u g a o a s fi‘v ‘1' LIV. I‘- -\v n" the‘ n ... AJJ 3‘4
. . . . . . . . . . . . . . . . . .
o . s a . n n u s a . n u . o 9 n q n g - a n . u . h a v N b A . A .-

n‘b

35

Table 2-1
Comparison of Finite Difference Method to the Exact Solution
Values Shown are in Percent Error From Exact Solution
d=L=1 At=.0005 Bi=0

EXACT SPLIT GRID*
TIME SOLUTION Ax=1/30 Ax=1/60 Ax=1/30
0.06 0.07142 1.093531 0.449454 1.503115
0.12 0.405587 -0.11292 -0.04635 -0.l4088
0.18 0.663191 -0.15863 -0.10148 -0.18787
0.24 0.81295 -0.12215 -0.08525 -0.14183
0.3 0.896468 -0.08511 -0.06169 -0.09785
0.36 0.942727 -0.05696 -0.04211 -0.06519
0.42 0.968321 -0.03738 -0.02799 -0.04256
0.48 0.982477 -0.02392 -0.01812 -0.02718
0.54 0.990308 -0.01515 -0.01151 -0.01717
0.6 0.994639 -0.00945 -0.00724 -0.01067
0.66 0.997035 -0.00582 -0.00451 -0.00659
0.72 0.99836 -0.00351 -0.0027 -0.00402
0.78 0.999093 -0.0022 -0.0017 -0.00244
0.84 0.999498 -0.0013 -0.001 -0.00143
0.9 0.999722 -0.0007 -0.0006 -0.00082
0.96 0.999846 ‘-0.0004 -0.0003 -0.00047

* The first five nodes of the 30 total were in the first two
percent of the material.

The example shown in Table 2-1 compares the finite
difference solution, using the tri-diagonal matrix method,
to the exact solution. These errors are approximately 1/10
of the magnitude of the errors generated when using a simple
sequential elimination algebraic scheme in solving the tri-
diagonal matrix. More importantly, the time grid can be
refined without instability when using the tri-diagonal
solution scheme given by reference [24]. The algebraic

elimination method becomes completely unstable for very fine

av;
'4“
O

0“-
bw’e

al.

. .
.3

<~

Vu.

Q...

h‘.

u.

L.
V.

Ne
-
‘
‘IIQ

fl.

4.. I

~2
«2 .
..u n
z 2.

Av
s. Cs
3 !.
.... ...
e. .u

\n.

N
5“
‘ 'Nek

‘_‘I
wC

 

. c .2
\Q Q»
a s .. ..
.. .. ‘
x c a
w .4 ... .
«2. ... w

36
time grids.

As an additional area of investigation, pursuant to a
greater level of accuracy, a split spacial grid was
attempted in the finite difference solution. In order to
treat the flash problem, it was suspected that a finer grid
might be required at the left hand edge of the material
where the temperature derivatives are extremely large. In
the example shown in Table 2-1, the refined grid scheme is
actually shown to be somewhat inferior to the uniform grid
spacing, using 30 nodes in each case. It was discovered
that the 30 node split grid was superior to the 20 node
uniform grid, but the best utilization of nodes seemed to be

a uniform distribution.

215.233Allllflulflllllllgfl

With the appropriate methods in place for computing the
direct solution, the parameter estimation aspect of the
problem can be undertaken. It is desirable to solve for the
minimum number of parameters necessary. This facilitates
the greatest degree of stability in the parameter estimation
procedure and the greatest degree of confidence in the
calculated parameters. The three parameters shown below are
obtained by dividing the differential equation and boundary

conditions by k. Estimation using two parameters instead of

 

... .J v.
t .3 . . —.
... . C. C. .. ..
... .1; ~. C. .« .3
a» N‘ 4‘ «y ...
a“ v. a: «v
.(. ... e. S S
. .u .... v. .3 a.
... s . a: r. . we
2. 2 .u .as 6» fig .3
. v . ..m 8.. Q. .~ ~ . AC .. .
.3 u . . . ... . v. c» 51.
o . n . . . o .. a . . a =~

 

37
three is investigated in detail in Chapter 6, however the
parameter estimation algorithm was found to be less robust
than the method which used the three parameters given below.

The unknown parameters are for this model are

 

Bz=— 5.5—:- (2-35)

The third parameter can be used as the more familiar Biot
number simply by multiplying by the sample thickness, L. In
order to find the parameters, the method of least squares is
used as outlined in reference [25]. The method of least
squares was chosen because it is a simple method and the
results are the same as those obtained by maximum likelihood
and Gauss-Markov, assuming that the following statistical
assumptions are valid:

1. The measurement errors are additive in nature to

the true (but unknown) temperatures.

2. The measurement errors, considered over the

duration of the experiment, have a zero mean value.

3. The measurement errors have a constant variance

over the duration of the experiment.

4. The magnitude of each measurement error is

unrelated to it’s predecessors or successors. In

 

. Fs .3;
. .l

A. A»

a». .3 2. ‘§
2 ... .... .1 ...

. . a
o A. n. V:
A v .. a a A . u U . . .‘ .
s . -~ v §
AH» ~\» § .

u M ... s.“
an r . a“

. ?

... W... ...
. 2. . .

q 4 ...
2
o 5‘“
e ....
u. ..a
2‘ s.
y a a
... ...
«y {s
s v s
«an -.
... i.

A‘s

38
other words, the errors are uncorrelated.
5. The measurement errors, considered over the
duration of the experiment, fall in a normal, or
Gaussian, distribution pattern.
The method of least squares minimizes of the following

expression
-T1)2 (2-36)

In matrix form this becomes

s=[r‘-r‘]’[r‘—r‘] (2-37)

In order to minimize this expression, aside from using trial
and error, the sensitivity coefficients must first be
calculated. This is accomplished by taking partial
derivatives of the direct temperature solution with respect
to each of the parameters, one at a time. The sensitivity
coefficients are then normalized by dividing by the
respective parameter. In this way, the units of the
sensitivity coefficients are always in temperature and the
magnitudes of the coefficients are directly comparable. For
example, the sensitivity coefficient for Bl,'the first

parameter in the model discussed above, is

6T

Xlzalg'é’
1

(2-38)

 

\
9|.\0
.oe4s

.... .. i 28 >3...
lap-0‘0" In...“

as

6..

5
-Vct

§I§.¥

.2 E

..§
2» «y
>.. 4‘

c e
vs C.
a y A.
.c s -

: .... .
A. .u
. . .L
.2 «E
I .. .
C c i
.3 c .

.. .
.q- .1:

39

 

—’{F——A"*IL -‘*F—‘tiNNFkxz ‘-*F—‘Bklthbor

 

 

 

 

J—
T

 

 

Sensitivity Confident:
60 «L K: t: n: «D a» «a
i

 

02 0.4 05 0.8 1 12 1.4 1.6
Time (wounds)

0

 

 

 

Figum2-4 NormallzedSonthyCoetfidemsforCBCFanOO’C

A graph of the sensitivity coefficients for this model, as a
function of time, is shown in Figure 2-4. The parameters
for the direct solution from which these sensitivity
coefficients were taken correspond to those of an actual
laboratory experiment with a Biot number of 0.142 and a
diffusivity of 0.33 an/sec. The nature of the flash
experiments is such that the heat flux parameter and the
Biot number are somewhat correlated, an undesirable
condition. This is evidenced by the similar shape of the
two sensitivity coefficient curves. The diffusivity
sensitivity coefficient curve, however, has a different
shape than the others which makes it a more salient

parameter. This characteristic suggests a good experiment

 

 

I I .I
|IH . . . yt ..
. . . k p» i: 5.. ,
_ . f y. 2 .C .c L. ..— «\ vs C. M4
l .
.b.‘ o 5 . 2‘ NJ .I. rL. \~. be 0 %.¢ a k s 5‘
. . . c c. .3 S r ..J E ..n I .3 C. a. -4
I .2 I S a. ... e 3 a . . .u . 5 . . ... .3 9 U .k
... I .. ... S I. .... ... e .3 ... 2 a. C N.“ e . t z . T
a: a: . v .3 .. . > . a: . . a. .. ‘ .C R» .. s s y s \Q a
3 . . n . .. . a. S S E I . t C. t . o. ..c . a I
3. .u .. .h S ... a. .. .. ... .... z. .. 3 ... at
... .. z. .3 a. r 2 .. 3 .... .R v. .. a s» u».
.3 . . 8.. ... ~ 3 . . xi .... x c A.» a... a. NC» ....a a?
. . . n 1 . 5 . .c
3. a. . w.” .. .. Q“ g h. v_.. «I... ‘3‘. no ...... M: ..a s p . A: u a a .

40
design for estimating diffusivity, since it is the parameter
of interest.

Using these sensitivity coefficients, a set of
equations can be developed to be solved by least squares.
The measured value of temperature for a given time step, i,
is assigned the symbol Y; and the calculated solution is
given the symbol T1. In order to perform the non-linear
regression procedure, an initial estimate for the
parameters, designated b1, b2, and b3 in this case, is
required. Assuming a locally linear approximation to the
sensitivity coefficients, a revised or improved value for
each parameter can be found by the partial Taylor Series
shown below. .A superscript designates the number of
iterations, using the letter k.

The term Ab corresponds to the adjustment necessary in the
estimated parameter to affect a change from the initial
calculated temperature to the refined calculated
temperature. This can also be expressed as

. 61‘ +1 k

41
Using this principle, an appropriate change in the estimated
parameter can be found by knowing the desired shift in
temperature necessary to make the calculated temperature of
the model match those of the measurements. Substituting Y
as the symbol for the measured temperature, and using the
method of ordinary least squares to solve for the revised
parameters, we have, in matrix form, as given by reference

[25]

b(k*1)=b(k)+ (xtk)fx(k) ) -1x(k)f(r_r(k’) (2_4l)

This process is repeated until successive iterations result
in a change of less than 0.1 percent in any parameter
between iterations. At this point, convergence is
considered to have been obtained. The residuals, expressed

as

01::(Y1-f1)2=): ef (2-42)

are considered minimized where the “hat” designation on the
calculated temperature signifies the calculated temperature

based on the converged parameter values.

W
The primary means of determining the adequacy of the

direct solution is to examine the residuals, ei. .Assuming

.3

.u.

a.
D a
.‘d

on
.n‘

.—
‘CG .
nurQirv -"“‘

.

.

“A

U---

Lg‘

.. v.
a»
a»
I a.
,.. ...
._ ...
:u ..
... ..
~.. .1
.. ..
a‘. It.

‘6

‘-
but

Qw
3 i .
.. ‘ p ..
c.c
$ 5 u ..
.-.. .. a
c. u k
a: 9.
i. n 9
p a ..r..
n M. Q»

42
an appropriate model is used, the calculated direct solution
and the measured data should match identically once the
parameters have been found. The unavoidable presence of
measurement errors should be limited to small background
noise which causes the residuals to oscillate randomly from
positive to negative about zero.

A likely indication of an inadequate model is a
characteristic signature in the residuals. A characteristic
signature is exhibited by long periods of continuous
positive or negative residuals. Another problem of which a
characteristic signature in the residuals can be an
indication, is a problem with the measurements,
specifically, correlated errors.

It is important to make a distinction between a
characteristic signature, which suggests an ill-applied
model, and correlated measurement errors, which suggest a
problem with the measuring instrument. Both of these
phenomena may look very similar if only one experiment is
conducted. One way of discerning which of these problems is
present is to plot the residuals from two similar
experiments over top of each other. If the basic shape of
the curves is the same, this suggests a characteristic
signature since correlated errors are not expected to be

repeatable from one experiment to another. If correlated

 

01°

0
8
\!J

".9

.... .n-uu. . .
3 m. n. M...
flab 0 n0 Alfie
«Ignaz Oiafiailfiu

s

4
Au

0.]

w

_ 5..
. x «.9 : 1 7
E 8 «nu . § ”‘4‘
Fly O V ‘04
A.. «\u 3“ ~.~
In an I . .

A . ...

v. R»

v. .u ..

.5. «\~

V ‘ \ t

as .c s e

3 C. . i
A. )5 w \
a... ‘

.5

a. A
. .. s s

. Q5

43

 

0.12 -
0.1
0.00
0.06 I u
0.04 it $.31
0.02 4- “

 

IA‘
Q? —a——Run1 ——-—-Run2 ——-—-Run3
. 1‘

u:
.
.

   
 

 

-ODZ~~
-Ofl4‘~
-006-
-008

Residuals (Volts)

 

 

 

0 02 04 06 08 1 12 L4 L6
Time (seconds)

 

 

 

Fmez-s ResidualsfmmModeH usedonCBCFDatsat‘lOO’C

errors are observed, the “signature” from each experiment
should look different than the others.

Examples of characteristic signatures are shown in
Figures 2-5 and 2-6. As a basis of comparison, the maximum
measurement value in these experiments was approximately 6.8
making the maximum residual value roughly 5 percent of the
maximum measurement. This appears to be a classical case of
a phenomenon in the experiment which is not accounted for in
the model.

Another key indication of model inadequacy is the
stability of the sequential estimates. The sequential

estimates are each calculated for their respective times,

 

0.12 ._-

I
1.
0

304.1. . a . .--.r_o
0

meme wmmm

«080a 10:8..0‘

bud)

Q ' .1 t l E

s i R . \ Q,» n.-

‘3 fix !
. ‘ .... . i 5

at a. Q.
a... re A .. x I
Q; . u s t s .
w ‘ aw 3 n
51 3‘ in .. \
Q~ Cw . .. G» ....v
«\s \ . -u .niU -. U

44

 

 

0.12

—‘3— Run + Rmz + Rm3

op
888
’

004)

 

 

 

Residuals (Volts)
é>éa «a
secs

6
3

 

 

 

a
8

0.6 0.8 1 12 1.4 1.6
Time (seconds)

c:
:3
h)
c:
h

 

 

 

Fbm326 FMflmmhfluandfl1uumonCBCFDmaaflmUC

each assuming no subsequent information is available. As
each additional data point is added, the value of each
parameter should ideally be unchanged from the parameter
estimate generated at the previous time. For example, in

the equation

b13+1) =b (I) + (x(k)fx‘3’ ) -1x(k)!(r_r(k) ) (2-43)

with three unknown parameters for this model, the first
sequential estimate may be calculated after say 10
measurements. In this case, the x matrix would be 10 x 3
and the Y—T vector would be 10 x 1. The final product would
be 3 x 1 to match the b‘“ and the law“ vectors. At this
point, a new b”‘” vector is calculated and becomes the b””

vector for the next iteration. Now another time step is

45

added to the equations so that the x matrix becomes 11 x 3
and the Y-T vector becomes 11 x 1. A new b””“ vector is
now calculated and the process continues. The sequential
estimates can be plotted as functions of time and observed
for stability.

In order to avoid the computational expense of
calculating the matrix inverse in the explicit form shown

above, the equation can be re-written in implicit form as

x1312x13, (bu+1)_bm) =xt8)!(r_r(k)) (2-44)

where the vector b””“ is the only unknown. In the present
example of the estimation of three parameters, the 1"!
matrix is 3 x 3, the b””“-b”“ vector is 3 x l and the
XVI-'1') vector is also 3 x 1. Using P—L-U decomposition as
set forth in reference [26], the b”"“ vector can be found
by reducing the fix matrix to a lower diagonal with zeros in
the upper half using Gaussian elimination. After scaling
the equations to minimize numerical rounding errors, the
b”’“ vector can be found by simple substitution. Another
alternative to calculating the matrix inverse is to use the
matrix inversion lemma as described in reference [25].
Figure 2—7 shows a graph of the sequential estimates
for the Oak Ridge National Laboratory experiment from which

the residual curve was generated in Figure 2-5. This figure

Kw

. s s s s AU a. C_ V q
C ...
Q» ..

at .

a: s a

..n ... at

A. as .1“

n .. s s A»

.1 .l ...

o

1 5
0
06.653. W iallafiflm

.

«saunas

46

 

 

 

 

 

 

2
15~
a 1 .. ”..."
g 05»
5 . 1
g .05 1 + mummy
a -14~ +HestFlut
lg-45‘* ‘—'—’Em1hhmbflr
-2
25

 

0 02 0.4 0.6 0.8 1 12 1.4 1.6
Time (seconds)

 

 

 

Figure2-7 SequentialEstimetssotCBCFDeteetNO'C

shows some incremental change in the parameter estimates
with time which is an indication of some inadequacy in the
model. In the following chapters, various models will be
examined in an effort to account for the characteristic
signature in the residuals and the lack of consistency in

the sequential parameter estimates.

CfiflKPTIEI 3

EKMIBLJEHS IfiflflERNUUL RIEIUBTEIni

3i1_IHIBQDDCIIQH

As discussed in Chapter 2, one indication of a
disparity between the mathematical model and the physical
mechanism in an experiment, is a signature in the residuals.
The residual curves of the experiments which were conducted
at the Oak Ridge National Laboratory, as exhibited in
Chapter 2, serve as examples of characteristic signatures.
As discussed in Chapter 2, a characteristic signature is a
pattern which repeats itself when comparing residual curves
of successive experiments and indicates a disparity between
the model and the measured data. Also, it is important to
make a distinction between a characteristic signature, which
suggests an ill-applied model, and correlated measurement
errors, which suggest a problem with the measuring
instrument.

Both of these phenomena may look very similar if only
one experiment is conducted. As evidenced by the residual
graphs from the six experiments plotted in Chapter 2, a
characteristic signature is present in the CBCF experimental

data. These curves point to a characteristic signature

47

0'"° 1“"
S-Hib ‘19.-

A.

‘v'un A9: .—

..yl-i V“- -.

Ann. "A‘i .
~..'

~;::C’O" .

““5‘—an‘ -

(“v-c- :. ~.

~'.‘ u Qt. --

esew F‘- lib“

.l.9‘.ta \
n

.::‘_A'
‘VC..H 3--‘~-
:U‘l‘e ‘ ..v-
~¢1I~S_s :-

'l~

A." ‘
-N‘e‘.
A.
Q “A f
VO.~ kws:

 

h- ‘.
““‘e . N‘Kp .
‘V . '.
yin. “
4 .
e0 s

“‘9
VI. 3 lflh“ .
”Ca V 7
“sc.‘
“
h
r.
p
“a. :c .
”‘93:~.
K

48
since correlated errors are not expected to be repeatable
from one experiment to another. If correlated errors are
observed, the “signature” from each experiment should look
different than the others. The characteristic signature
shown in Chapter 2 is not only present in successive
experiments on the same material, but is exhibited in
experiments from different laboratories around the world
using different materials. The shape of this residual curve
suggests an initial rate of heat transfer through the
material which is more rapid than that which is
mathematically predicted using the conventional conduction
model.

One possible explanation for the more rapid initial
heat flow is that a mechanism other than conventional
kinetic conduction may be at work, superimposed on the
dominant heat transfer mechanism of conduction. Although
the CBCF material is considered opaque, radiation could be
such a mechanism, using the material as a participative
medium. The principle which drives this mechanism is based
on a localized area of heated material in the continuum
transferring heat by radiation to a lower temperature part
of the continuum which is close enough to allow direct
radiant exchange. The process continues, spreading the heat

by radiation from the warmer parts of the material to the

~A"

\A'

vvnnu

C.

\-

-..

t A. ~.. we
as u? s 2 a»
3‘ ~. .3 be a?
.. a. .3 .1 ....
an ... ... ... Q.

49
cooler parts in a diffusion-like process. All of this would
take place in addition to and simultaneously with
conventional kinetic conduction. Modeling the combined
conduction and internal radiation mechanisms is more complex
mathematically and the differential equation cannot be
solved explicitly. The solution must be obtained
numerically.

Internal radiation has been shown to be evident in
other materials such as alumina powders as shown by Hahn et.
al., in reference [36]. In this alumina work, the thermal
diffusivity was found in a de-coupled way from the radiative
component of heat transfer. The radiation portion of the.
heat transfer was assumed by eXtrapolating known radiation
parameters at temperatures other than that at which the
experiment was conducted. The radiation was then treated as
a parallel heat transfer mechanism superimposed on the
conduction mechanism. The radiation contribution to the
overall heat transfer was assumed as a known quantity
throughout the duration of the experiment. The diffusivity
was then found using conventional conduction to model the
remainder of the heat transfer.

In contrast to this method, the objective of the
present research is to compute radiative parameters

sinmltaneously with the kinetic conductivity or diffusivity

50
parameters. The remainder of this chapter discusses the
various aspects of computing parameters related to combined
conduction and internal radiation. A model is developed to
accomplish this and is referred to as “Model 2". The non-
radiative model discussed in Chapter 2 is referred to
hereafter as “Model 1". Section 3.2 provides a development
of the direct solution and experimental design
considerations. Numerical aspects of computing the direct
solution are addressed in Section 3.3 and examples are shown
of various direct solutions over a wide range of selected
parameters. Section 3.4 presents the parameter estimation
aspects and sensitivity coefficients for this model. This
section includes a test problem in an attempt to correctly
extract parameters from an exact solution with known
parameters. A reduction in the number of unknown parameters
is attempted in an effort to stabilize the parameter
estimation procedure. An investigation into the residual
signature is shown for a case where an exact solution with
internal radiation is analyzed by a parameter estimation
rnodel assuming no internal radiation. Analysis of actual
.Laboratory data is performed in Section 3.5 with results
shown for a number of different approaches. Finally, an
ithergo-differential model is discussed in Section 3.6 which

is more applicable to materials which are optically thin.

a
~ \

flu.-.
u..' ‘

... I
-w.

'u
5 U
“V:

“y- ‘. v
“.
s I
We
-
‘

 

A,
uC-t u ,
V. v.
I P
N
‘>
“ha 9-.
“ a
\f: t
\-
\
\ A.
‘Vds
.
V'L
‘n.Q‘
“o SA 1
V1

51
3i2__DIBIQI_EQIQIIQNLDIYIHQEMENI

The type of model applied to a particular experiment
where combined conduction and internal radiation are both
suspected as operative mechanisms, depends on the optical
characteristics of the participative medium. The primary
consideration in selecting a model is based on the optical
thickness of the material. A material is considered
optically thick if the mean free path of a photon in the
medium is relatively short in comparison to the sample
thickness. These materials appear opaque upon visual
inspection. Conversely, in an optically thin material, many
photons are able to pass through the medium without being
absorbed. Even though a photon may undergo many scattering
interactions as it passes through a material, the material
is considered optically thin if absorption of the photon is
unlikely.

The method of modeling the combined conduction and
internal radiation as diffusive processes is most
appropriate for materials which exhibit optically thick
behavior. This model is best suited to the physical
characteristics of the CBCF material being studied and
allows a solution to be generated at a relatively rapid rate
when solving the equations numerically. Optically thin

Inaterials are more appropriately modeled using an integro-

kt,

v..

9.. ...

52
differential approach discussed in more detail in Section
3.6. The diffusive model, as given in references
[l],[8],[9] and [10], is used in evaluating optically thick
materials.

The radiative heat flux vector for this model is

lGobT3
q =-——————VT (3-1)
‘ 3K
which can also be expressed as
lGobT3
qrz-krVT Where kr=-—3-I-(—— (3-2)

In this context, k1 can be thought of as the “radiative
conductivity”. The K term is known as the “extinction
coefficient” which is the sum of the absorption coefficient
and the scattering coefficient, each having units of inverse
length. The reciprocal of the extinction coefficient
represents the mean free path of a photon in the material,
or the mean distance traveled without an absorption or
scattering interaction. When, K tends toward infinity, this
corresponds to negligible radiation transfer due to a near-
zero mean free path of the photons. Conversely, radiant

heat transfer is maximized when x is small, corresponding to

fi‘izo.

()

f)

f

53
a long photon mean free path and a very rapid rate of heat
transfer through the material.

The cm term is the Stefan-Boltzmann constant which is
5.729 x 10‘8 W/mfiC. Additionally, all temperatures in this
equation must be expressed as absolute temperatures. The
overall heat flux, including conductive and radiative

components, then becomes
q =q,+qc=-<k,+k>VT <3-3)

where k once again corresponds to the conventional kinetic
thermal conductivity. The energy equation in the x

dimension, with no internal heat generation, becomes

6T

8T 3
= -V. =—— -_ -
q 6x[(k’+k) (3 4)

C—
996::

6x

In order to solve for a temperature solution, the proper

boundary conditions must be applied. They are

41.91.)ng =q.6<t)+h<T.-T.=o> (3-5)
x=0
6T

-(kr+k)l—L = MT“ 4;) 3-6

ax % L ( )

.Additionally, the initial temperature throughout the

 

WU-bO‘O Q

n-‘
~‘:~ :H...
‘O-b‘uvv‘
'Vfl ‘hp:‘ ‘
18: ....--_,
.. D--. .

a N;-
--.- u.... u
' t

A . ’_'
‘--. ..-. ‘
“‘Qnu..-

—‘ P,
...~."~ru .-
PA..":"’ a
~unu4... "

0
DA
~’y::a'.:
...-...»-..5
9
s
‘ e. .
I‘ ‘q..A'
I

.1!

‘ "2‘5 ‘A

vfi.‘~~ V
q
I“ .‘ '-
~a' “Q!

(n

54

material is assumed to be T.. .As in the non-radiative model
discussed in Chapter 2, the parameters in these equations
are considered to be constant throughout each experiment and
the heat transfer is assumed to be one dimensional. The
left hand side of each boundary condition represents the
combined heat flux into or out of the solid side of the
boundary due to conduction and radiation. The q°6(t) term
represents the heat flux at the boundary due to the flash as
a function of time. The heat losses from the material
surface to the surroundings is modeled as convection.

By dividing through these equations by k, the number of

unknown parameters can be reduced to the four groups

 

az=kx B3=— 55-23 <3-7)

Groups 1, 3 and 4 are the same as the three parameters
utilized in the model presented in Chapter 2. Because the
groups are obtained the same way as those presented in
Chapter 2, which is by dividing the differential equation
and the boundary conditions by k, each of the parameter
groups contains the kinetic thermal conductivity term. The
fourth parameter can be expressed as the more familiar Biot
number by multiplying by the sample thickness, L. The

Parameter, 62. represents the additional degree of freedom

‘:‘*r454 u
4
iotbodv“ ‘

-.
'--\ u..."
\ ~
°"‘V uvgo-a

' .9. -.
.- ,_r‘ ‘
' I..."-‘n
.

.IQFA'.;‘.
b~vb~g .-V.
. I

.3 I

_-"‘r;‘ ‘

... .~“_‘-
.

1h
(\-

‘. ‘ n
4.
-~-.c 1»

55
afforded by the additional mechanism of internal radiation.
This term is expressed as shown rather than k} since k} is a
temperature dependent term. The szparameter is
proportional to k/kg‘with the known constants and
temperature variabilities divided out. This is an easier
way of dealing with the second parameter, the only parameter
related to the radiative transfer.

In an effort to optimize experiment design factors, an
order of magnitude analysis for each term in the
differential equation and the boundary conditions can be
performed in order to determine optimum magnitudes of
parameter groups. It is desirable to have a radiative
conductivity on par with the thermal conductivity, or larger
than thermal conductivity, in order for the radiative
parameter to be adequately estimated. This is evidenced by
the magnitudes of the sensitivity coefficient curves
presented later in this chapter. Additionally, the
following experiment geometry should be established in order
to give approximately equal weighting to each of the
parameters. This provides the proper combination of

diffusivity, sample thickness and experiment time duration.

-——=2d (3-8)

~ .

~-..-

.—

“‘

P.
r»-
r 7
.4.

v' I o
. c . .
. . a:
. s

a: .3

:—
v~.

e~
‘-

A

s..~"‘
hr

5

L

 

56
Since the experiment time duration is normally of a non-
dimensional time of 0.5, the optimum thickness will be
approximately the square root of the diffusivity.

Although it is essential to simultaneously calculate
all parameters, the heat loss coefficient and the magnitude
of the heat flux are not normally parameters of interest.
When this is the case, the optimum experiment design would
allow heat loss to be assumed as zero which would result in
a simpler model with fewer parameters to estimate. In
experiments performed at high temperatures, however, even in
a vacuum, surface heat losses are quite high and far from
negligible. In cases such as these, it is desirable to use
a heat flux that is of large enough magnitude to allow
temperature readings to be two orders of magnitude above the
ambient noise.

As with the linear differential equation outlined in
Chapter 2, the non-linear conduction-radiation model can be
non-dimensionalized. With the addition of one non-
dimensional group to the three developed in Chapter 2, the

non-dimensional form of the equation is

61" a . 1"3 61"
__.=——— 1+km———- -—— (3-9)

as aw Ty ax*

where T. is the experiment ambient absolute temperature and

§.
5.

3..-.

(
0
' a
(n

A'-
.va.

AN’
\
V
1‘ AFA,‘
\ n
V L.
A
‘v I
.u‘

57

 

 

+ T + TakL x
T : Too-‘- X =— (3‘10)
qo/(Locp) qoa L
, 160 E?
tag km=——L— (3-11)
L2 3kK

This differential equation is obtained by dividing
Equation (3-4) by k and substituting the non-dimensional
terms listed in equations (3-10) and (3-11). The boundary

conditions can be non-dimensionalized in the same way as

+3

 

 

 

 

__ * x’=0 6T? _L26(t) . +_ +
1+km T? aw x.:o—-——E——+BJ(T, T Ix.:o) (3-12)
, T’3 .- + ,
l+km 3;“ aT+=Bi(T.—T*Ix.=1) (3-13)
6x

In Equations (3-9) to (3-13), the four dimensionless unknown

groups are

2 . . , k 160 T3
3.933., Bi, T. and km where km: ’°= b

(3-14)
a k 3kx

with x? and t? as independent variables. By varying these

four groups in the direct problem, some insight can be

gained as to how these factors affect the direct solution.

~FIIA'.‘ F‘

‘b a ‘

vii-.‘. b.
.

“‘ON-h.

"'
vs..¢-_-'”‘
O.» ‘I"r'

,-
...-v "“5

.
U
’ s
§..

.

c‘VA'.‘

‘5. ~N ‘- -.
. "u

58
Several plots of these direct solutions under various
conditions are shown in Figures 3-1 through 3-6 later in

this chapter.

.2a3_EnHIBIQBL.ABEIQI§_QE_IEI_§QLEIIQH

In the first attempt at computing the direct solution
for this model, the temperature dependent radiant
conductivity, kg, was calculated for each node using the
temperature at the applicable node. For example, from the

differential equation

8T_ 6 6T
pcpE—Elmgk) 5;] (3-15)

the spacial derivative can be approximated as

.1 j .1 J'
T, -T T -T_
8T= 11 1 or 6T: 1 11 (3-16)
6x Ax 6x Ax

and the time derivative can be approximated as

1 1‘1
6T: T1 "T1

.__ (3-17)
at At

Substituting this into equation (3-15) we have

2‘
\\

 

59

1 1‘1
DC T1 ‘T1 ___ 6
P At 6x AX

 

j _ ;
W321i] (3-18)

This can in turn be approximated as

 

TI: 1 ' T1) T1] ’ TIJ- 1
-1 k +k -
DC T11”T11 ___ ( r )[ AX AX

9 [At Ax

(3-19)

 

 

A typical equation in the tri-diagonal matrix using this

scheme then becomes

 

—T}_1 +(2+B) T11 -:rf,1=m;"1 (3—20)
where
2
‘8: pC$Ax
l6obe (3-21)
kA l+——————
3kK

As defined in Chapter 2, the temperature subscripts refer to
the position within the one-dimensional geometric grid and
the superscripts refer to the time step to which the
temperature applies.

This method was found to be inadequate in its accuracy
when used to analyze an insulated case, i.e. Biot number
equal to zero. A value of km/k=l was used in each test

case using this method shown in Table 3-1. In each case,

...»

A..-
\
-4

UV"

...
:»
2,-
. .

 

‘A

 

60
the temperature rise should be one unit of non-dimensional
temperature. This is because the contrived sample is
considered insulated and, in order to satisfy the First Law
of Thermodynamics, the temperature rise should be the same
in each case since the flash magnitudes are the same in each
case.

The reason for the inadequacy of this method was that
temperature differences at the boundaries of the control
volume surrounding each node were not taken into account in
computing the radiant heat flux at those boundaries.
Instead, the temperature of the node at the center of the
control volume was used in computing the radiant
conductivity at both of the corresponding control volume
boundaries. This resulted in a disparity between radiant
conductivity calculated at common boundaries of adjacent
control volumes based on differing center node temperatures.
The resulting error generated using this method is shown in
Table 3-1. The “No Radiation” case differs from the Tgfi)
case in that one simulation accounts for kinetic
conductivity only and the other is performed at an ambient
temperature of absolute zero with internal radiation present
as a heat transfer mechanism. In the latter case, km=0 but
lg is non-zero once heat is added to the sample from the

flash.

\CA\‘

.v-.\‘

-."-
.----

(s

.... a». .h. .4 .u. 4‘. .F. «.L .-. 4V. 1« 1. ..
. ... .HJ .3 a o a, . . a «I; ..s .2 . .\ _.‘
0 I a n o I o I ..- s ‘ ~§s “ \\~ -A\‘
..«s...~..... ‘ .a :s .. ...\ {\

Au. .2 .4

A 5 «iv ... a? uh. C. pt ’F. ~

. O . I . C O . I
a a . . a y A a . . . . a . a c A u A n a u A

 

61

Table 3-1

Comparison of Direct Solutions Using Non-Radiative
Model and Center Node Temperature to Define kr in
Radiative Model.

NON-DIM NO

TIME RADIATION T.*= T,*=10 T.*=100
0.06 0.075954 0.075827 10.0757 100.087
0.12 0.402901 0.402222 10.4015 100.417
0.18 0.651622 0.650522 10 6493 100.648
0.24 0.800543 0.799191 10 7977 100.781
0.3 0.886319 0.884822 10 8832 100.855
0.36 0.935278 0.933697 10.9319 100.896
0.42 0.963161 0.961534 10.9597 100.918
0.48 0.979033 0.977379 10.9755 100.93

0.54 0.988067 0.986397 10.9845 100.937
0.6 0.993209 0.99153 10 9896 100.941
0.66 0.996135 0.994451 10 9926 100.943
0.72 0.9978 0.996114 10.9942 100.944
0.78 0.998748 0.99706 10.9952 100.945
0.84 0.999287 0.997599 10.9957 100.945
0.9 0.999594 0.997905 10.996 100.945
0.96 0.999769 0.99808 10.9962 100.945
1.02 0.999869 0.998179 10.9963 100.945
1.08 0.999925 0.998236 10.9963 100.945
1.14 0.999957 0.998268 10.9964 100.945
1.2 0.999976 0.998286 10.9964 100.945

As in the numerical solution outlined in Chapter 2, the
initial condition for this problem attempts to simulate the
flash by setting all temperatures at I; with the exception
of the temperature at the i=0 node, which is set at
zkk/Axocp. This represents the temperature rise resulting
from absorption of the entire energy of the flash at the
surface node. As the ambient temperature increases, the

effect of the radiant conductivity increases due to the

n~p.. ‘

.
2'.
v¢~-. “:,.
"‘ ‘9 .
a 'A a I
nu“. “-- ~
-
.. _ '
.‘_ .
5. "

7"."Ao...

.
hVnti‘y‘- .

I. M-
u. ,.
..‘. ~
-. A..-‘
..
::e~~;‘;~.
“1“"§4

b
N ‘ . .
...: Vs .

'a
4‘” 6"“
‘V
I
-o~.
ny‘y
‘~
5
:0 :m‘.‘
“ 4
H
. a_‘h‘
V.
.1
‘1 I
‘1
v
.‘e A:
V n

62
cubic dependence on temperature. This can be observed in
the rate at which the final temperature is attained in each
case. The higher ambient temperature cases result in a more
rapid attainment of the final temperature.

In a second approach to solving the direct problem, an
average temperature was used to determine the radiant
conductivity between qu and Ti.as well as between T3 and
T91. This technique was implemented in order to more
adequately balance the radiant heat transfer between nodes.
In other words, without averaging the temperatures used in
computing the local radiant conductivity, the radiant
conductivity assumed at a particular node in one equation in
the tri-diagonal matrix is not necessarily the same as the
radiant conductivity assumed at that same node in an
adjacent equation. This factor led to the inaccuracies
illustrated in Table 3-1. With regard to time, no average
values for radiant conductivity were required to be
calculated between time steps. This is because radiant
conductivity changes between time steps at each node in
general, and all radiant conductivity values are consistent
within each time step, without mis-matched overlap.

Applying this method to the differential equation shown

as Equation (3-15), the discretization of the right hand

side of node 1 becomes

o . A
an" “H
“41" b.‘-

Can‘a " ‘

a." ‘

...Q r
'..- o
v

's

63

.1 1‘1 .1 .1
T1 “T1 6 a Titl‘Ti
c -———=.._ k +k ___.. (3-22)
0 P At dx[( ’ ) Ax

and the corresponding equation for the left hand side of

 

 

node i is
j 14 j j
Ti-Ti 6 L T1 ‘T1-1
C ————=— k +k __ (3-23)
0 P At 8x[( ‘ ) Ax
where
2 Tj_ T1 3
kf: 0‘" 1 1+ 1) (3-24)
3K
2 1" I", 3
k3: 0“ 1+ 1 1) (3-25)
3K

In this notation, the superscript "R" or "L" refer to the
right or left temperature used in the T’3 term. The "L"
superscript indicates the average temperature between Ti.l
.and T1 is used and the "R” superscript indicates the average
temperature between T1 and TM is used. Completing the
discretization by approximating the second partial

derivative of temperature with respect to x we have

V)

. ‘
,
" qu

~ ‘~
‘5.

4
"‘Vuu.
d

'A..‘ . ~C
Ar... J...

A I
\ A‘A
V"‘:..,:, '|
n‘ '
l'. ‘ne A1
V H V
I
cf: .H‘.”
v n.‘.‘~c
‘
Lu QII‘ I-
§‘-‘¥ ‘

7‘.
he ,“

sql‘e'

64

1 .1'1 J' J' .1 1
T1 “T1 1 a T1+1'T1 L T1 ‘T1-1
p 9 At Ax( ’ ) Ax (I ) Ax

Re-arranging this equation so as to be used in the tri-

diagonal matrix of the finite difference scheme we have the

non-dimensional form

 

(3-27)

In deriving the left hand boundary condition using this
scheme, we consider a control volume which encompasses the
volume of material bounded by the two left hand nodes, which
are numbered 0 and l. The time rate of change of energy

content of this element can be expressed as

pch . ._ _
zit (To’+T1’-To’ l-Tf 1) (3-28)

 

The rate of heat transfer from the surface at node 0 is

Mini-T.) (3-29)

The rate of conduction into the solid material is

65

Tj-Tf

(k+kr) (3-30)

 

Combining these three factors in accordance with the first

law of thermodynamics we have

CMZAX . _ -
22c (To’+T1’-To’ 1_le ‘) mug-T.)

 

j 1 (3-31)

T0 1
+ k+k =0
Ax ( Jr)

 

Re-arranging this equation so as to be used in the tri-

diagonal matrix of the finite difference scheme we have the

non-dimensional form

 

 

2 (3-32)
AX
ZaAt

 

(TJ“+T3“) +35%

In this equation, N refers to the number of spacial nodes.
This becomes the first equation in the tri-diagonal matrix.
Likewise, the right hand boundary condition becomes the last

equation in the tri-diagonal matrix as

 

 

Ax2 k, Bi j Ax2 k, ,
+1+_+— T + -l-— T
( J " [ 2aAt k: "4

(3-33)

 

66

In these equations, the radiant conductivity term is

2 Tj'lmj" 3
kt: Ob( °3 1 ) (3-34)
K

 

or

: 20b( T3-1+T13:11) 3
’ 3x

(3-35)

 

as applicable.

The results of implementing this scheme are shown in
Table 3-2, again using an imposed condition of zero surface
heat loss in order to compare the radiative model accurately
with the non-radiative model. As with the previous attempt
at this solution, a value of km/k=1 is used in each case.

In each solution, the final temperature is very close to one
non-dimensional degree above the ambient temperature which
is what is expected from the physical parameters of the
problem. As in the previous case, the heat transfer takes
place much more quickly at the higher ambient temperatures
due to the greater effect of radiant conductivity. The non-
dimensional time in this table is based on the kinetic
conductivity only and is not affected by variations in
radiant conductivity. The solutions are all generated by

finite difference using 20 spacial nodes and a time step of

I.

ah

u
.4

A‘s

 

67

Table 3-2

Comparison of Direct Solutions Using Non-Radiative
Model and Averaged Node Temperatures to Define krill

NON-DIM NO

Radiative Model.

TIME RADIATION T;:10 TJ=100 TJ=400
0.06 0.075954 10.075979 100.09284 400.88195
0.12 0.402901 10.402959 100.44162 400.99301
0.18 0.651622 10.651674 100.68633 400.99934
0.24 0.800543 10.800582 100.82657 400.99993
0.3 0.886319 10.886346 100.90445 400.99999
0.36 0.935278 10.935296 100.94739 400.99999
0.42 0.963161 10.963173 100.97104 401

0.48 0.979033 10.979041 100.98406 401

0.54 0.988067 10.988072 100.99122 401

0.6 0.993209 10.993212 100.99517 401

0.66 0.996135 10.996137 100.99734 401

0.72 0.9978 10.997801 100.99853 401

0.78 0.998748 10.998749 100.99919 401

0.84 0.999287 10.999288 100.99955 401

0.9 0.999594 10.999595 100.99975 401

0.96 0.999769 10.999769 100.99986 401

1.02 0.999869 10.999869 100.99992 401

1.08 0.999925 10.999925 100.99995 401

1.14 0.999957 10.999957 100.99997 401

1.2 0.999976 10.999976 100.99998 401

0.01 units of non-dimensional time. .A comparison of the
non-radiative model using finite difference and exact
methods is shown in Chapter 2.

Figures 3-1 through 3-3 show examples of the direct
solution for various cases. There are four basic parameters
that can be varied for this model which have an effect on
the direct solution. They are a, TJ, Bi and km/k. The
three that offer the most insight are T:, Bi and km/k.

Varying a simply changes the time scale of the temperature

I. o. . ..
.Cfiooinun-EQOJCOZ u 9 ....
a

N .. Ayn“

68

 

——'——'BFO ‘_*}-‘EW411 ‘_'—“BF¢D ‘—{*_‘BF40

 

...-.. JLflaiir‘vf'I“ rugs-lgqflg...‘

C)
k)
I

L
Y

Non-Dimensional

Tempemtura Rise

c>c:c:¢: c>cac3

o :.. N u 'o- 8 u :4 ho
\

L
fi

 

.-.'-'I'-
.. .
I...
II.
A‘s-Eunnlllllllulltannin-uIII-Illa.-

 

 

0 0.1 02 0.3 0.4 0.5 0.6
hkuromnonflonafTflno

 

 

 

Figuna 3-1 Non-Dimensional Temperature as a Function offlme W13 T,’=1OO

plot of the non-heated side of the sample and little else.
The effects of variations of the other parameters are
examined throughout the remainder of this section with a
held at unity.

Figure 3-1 shows the effect of varying Biot Number with
the other parameters held constant. The non-dimensional
temperature history at xEfl.is shown in each case. A Biot
Number of zero corresponds to an insulated condition, and
the specimen is heated to a non-dimensional temperature of 1
due to the heat addition of the flash, which also has a non-
dimensional magnitude of 1. None of the other cases ever

reaches a temperature of 1 because of heat loss via

I 0.”; 0.3-CsOQEIP
.fl-‘O...‘.E.uolc°z

Qu-
HAuQr~'
vgg
d

‘ .
/\H~\
V "'I‘

ANN

V

0‘ ‘
hit

C
x.

‘ ~~
\ ‘0‘
§

H
Lz"

69

 

+ klik-U + krlk 80.1 + krfk-LU _‘D— krfk-10

d

 

O
CD‘S

  
  

A
I'

Non-Dimensional
Tompomtura Rico
6 p p c: o o o p .

     

 

 

ohmwlhin'm‘u

0 DJ 02 03 0A 05 as
Non-Dimensional Time

 

 

 

Figure 3-2 Non—Dimensional Tompomtum as a Function ofTimo Bi=1.0 T.‘=100

convection. The most extreme case shown is with a Biot
number of 10 where the maximum temperature reached is
approximately one order of magnitude smaller than the
insulated case. Due to the greater rate of heat loss,
especially from the heated side of the specimen, most of the
energy is transferred away from the material before it can
be conducted to the non-heated side.

Figure 3-2 shows the results of allowing the
conductivity ratio, km/k, to vary while the other
parameters remain constant. The limiting case with a
conductivity ratio of zero corresponds to a condition of

pure kinetic conduction with no radiant heat transfer. As

70
the conductivity ratio increases, the peak temperature
becomes larger and occurs at an earlier time. The rate of
heat transfer through the solid increases with an increasing
conductivity ratio because, the addition of radiant heat
transfer has the effect of adding a temperature dependent
conductivity to the existing kinetic conductivity which is
assumed to be independent of temperature. This effective
increase in conductivity causes the faster response seen in
the curves. The peak temperature becomes higher with
increasing conductivity ratio because the dominant mechanism
of heat transfer tends to be internal, by means of
conduction and radiation, with comparatively less external
convection. This causes the non-heated side of the specimen
to receive more energy than in a case where convection is
dominant and much of the heat is carried away from the
heated surface before it can be transferred to the non-
heated surface. .An additional feature of the model which
influences this phenomenon is that the Biot Number is based
only on kinetic conductivity and not on the radiant
conductivity. The internal heat transfer tends to dominate
very quickly then, whenever the conductivity ratio is
increased. In the limiting case of an extremely large
conductivity ratio, the sample behaves much like a lumped

system where the heating at the time of the flash is uniform

"IN. -.IIC'."III' F
~060.ICIE.Q -COZ

-‘

A

Q»
". J ‘v‘ ‘
v ~ a \
‘ «a s:
7.4 .Jo

 

71

 

’“4F—‘Tb=1 ‘“**“'RF40 ——'_‘”hwfl00 “—{*—’Tb¢KDD

 

 

Non-Dimensional
Temperature Rise
:3 c>c3

o 8 N 8 :2 8 a. u 8 8 _.

 

 

L

1

O 0.1 02 0.3 0.4 0.5 0.6

hknrDMnonfionflFfinur

 

 

 

ngaaalWxHJmamkmflTbmnmmuaasaFummxuflfimn8:1thfl

throughout the material, causing the non-heated temperature
to go to a non-dimensional value of one at the time of the
flash and decaying due to convective losses thereafter. In
such a case, it would not be possible to measure the
diffusivity or radiative conductivity. In order to
accomplish parameter estimation in such a case, it would be
necessary to use a much thicker sample.

Figure 3-3 examines the case where T;'is allowed to
vary and the other parameters are held constant. With a
value of Tfi'at 10 or above, the results are all very
similar since the temperature rise in the material is

relatively small in these cases compared to the ambient

 

d

n»

I“

{D

'I’

72

temperature. With T;=l on the other hand, the effects of
an increased effective conductivity can be seen, similar to
that observed in Figure 3-2. Since the temperature rise in
the material is approximately on a par with the ambient
temperature, the effect of the '1‘3 term in the equation for
lq/k causes this ratio to rise locally to extremely large
values, even though the ratio based on ambient temperature
is held the same in this example, as in the other curves.
Although the condition of Tj=l is not very realistic from
an experimental point of View, it is a situation that makes
extraction of the radiative conductivity parameter somewhat
easier because its temperature dependence is much more
pronounced. This fact highlights the necessity of utilizing
very large heat pulse magnitudes at the higher temperatures
in order to accomplish adequate parameter estimation.

Figures 3-4 through 3-6 are identical in all respects
to Figures 3-1 through 3-3 except that the non-dimensional
time is based on the sum of the radiant conductivity and the
kinetic conductivity. In other words

k+kr

och (3-36)
L2

 

The most obvious difference that can be detected in Figures

73

 

‘—4**—EN43 “—fib_'8h01 ‘—4F—‘Bhflll"‘D‘—'Bhflfl

 

 

0 z
041- %g”
034* A?
0.2 4-

,
...-.......l"."lIIIIIIII
1 '.-'.....'.

Non-Dimensional
Temperature

04- '7'

0 01 02 03 04 05 08
Non-Dimensional Time [based on lulu]

 

 

T

 

 

 

Fbue34lkmifinuubmflTuwmmmueasaFunflmmlfimenJeflflTr=flm

3-4 through 3-6 is that the time scale is essentially
compressed by a factor of 2 because the effective
conductivity in determining the non-dimensional time is
twice the conductivity used to define non-dimensional time
in Figures 3-1 through 3-3. That is, since km/k is equal
to one throughout both figures, the effective conductivity
at ambient temperature is 2k.

Figure 3-4 is nearly identical in all respects to its
corresponding Figure 3-1, with the exception of the
different time scales noted above. As expected, the
limiting case of Bi=0 corresponds to a rising temperature

which becomes asymptotic on a non-dimensional value of 1.

74

 

 

 

 

 

 

——'——iqkfii '—4}—'MMGOJ ——'—‘iqk#U3'—4}——kuhflo

05
'3
.5 g 04"
{£5 as
.§ 3 .
2 5 °2
ol'lfl1‘
12

0‘ ‘ c t . : i
0 DJ 02 03 04 05 05
Non-Dimmionel Time (based on k‘I-kr)

 

 

 

Figure 3-5 Non-dimensional Temperature Bi=1 T.‘=100

The larger values of Biot number generate temperature curves
which reach a peak point followed by a temperature decline
due to heat loss.

Figure 3-5 is noticeably different from Figure 3-2
because the curves are virtually on top of one another in
Figure 3-5. The only changing parameter between the family
of curves in both of these figures is km/k. Since the time
scale is effectively shifted along with the effective change
in conductivity in Figure 3-5, there is almost no difference
between the four curves shown on this graph. The only
detectable difference between the curves is that the higher
km/k curves show slightly higher temperatures than the

lower km/k curves. This is due to the higher rate of heat

75

 

+ To-1 + To-10 + Tia-100 —13— TO-1000

0.9
0.8 i
0.7 i

 

.09:
010)

0.4 ~~
0.3 +

Non—Dimensional
'mepenflnn:

P
N

 

 

 

 

0 DJ 02 03 04 05 06
Non-Dimensional Time (based on k+kr)

 

 

 

FbueaalflmifimmMuurnmmemmnuuaFamflwutfimeahtonfisto
transfer in the localized areas of elevated temperature near

the heated surface brought about by the temperature
sensitive nature of the radiant heat transfer.

The curves in this figure would be slightly more varied
between each other if a value of TJ=1 were chosen rather
than Tfisloo. The at the lower non-dimensional temperature,
the temperature sensitive radiant conductivity would be more
significantly affected by the proportionally larger pulse,
thereby causing more of a distinction between curves of
varying radiant conductivity.

This concept is illustrated somewhat by Figure 3-6
Which again matches very closely its counterpart, Figure 3—

3. With a small value of T;, the magnitude of the pulse

76
becomes more significant in the radiative transfer
mechanism. For example, with T;=l, the magnitude of the
flash effectively doubles the absolute temperature of the
material in low heat loss cases, making large temperature
gradients inside the material. These large temperature
gradients cause a significant impact on local value of the
1'3 dependent radiative conductivity. The proportionally
larger contribution from radiative conductivity causes heat

transfer to be more rapid than in the higher TJ'cases where

the pulse heating is less significant.

 

As with Model 1 discussed in Chapter 2, which did not
account for internal radiation, the sensitivity coefficients
are plotted below for Model 2. Figure 3-7 shows a plot of
the modified sensitivity coefficients for a simulated
experiment where the ratio between radiant conductivity and
kinetic conductivity (km/k) is 1.0, Biot number (Bi) is
also 1.0 and non-dimensional ambient temperature (Tgi is
1000. In this plot of sensitivity coefficients, the non-
dimensional time is based on the kinetic conductivity only.
With a small radiant conductivity, the sensitivity
coefficients for this parameter are predictably small. The

best experiment design pursuant to extraction of the unknown

 

‘A

b
“'6

W-

 

77

 

—“'_‘dem3 ‘—D—“'qk ——I——-Bi “—{y—‘RKB

1 .

 

u 3“" “I" 1 fl rl'fltw

 

  

 

Non-Dimensional
Temperature

7 ' I'....‘IILIJ‘IIIIiI...-IIIIIIIIIIIIl-lal ea
fi"' ' "F‘*""'--IIIIIII*II

 

 

Non-Dimensional Time

 

 

 

Figure 3-7 Sensitivity Coeflidente w1 T:=1ooo Bi=0.1

parameters, seems to be with the radiant conductivity
approximately on a par with the kinetic conductivity. Of
course, this is generally outside the experimentalist's
control since it is a material property.

Following the calculation of the sensitivity
coefficients, a program was developed to extract parameters
from experimental data using the radiative model. The
equations used in estimating parameters in the case with
internal radiation, Model 2, are not as stable as when
attempting to extract the parameters using the non-radiative
Model 1. In the non-radiative test case, the parameter

estimation method converged with initial parameter estimates

78

larger than actual parameter values by a factor of up to 3.
Initial attempts at extracting parameters using the
radiative model would not converge for initial estimates
ranging only two percent above the actual values. Part of
the reason for this is that solving 4 sets of simultaneous
equations for 4 unknowns is inherently more unstable than
solving 3 equations for 3 unknowns. More significantly,
however, is the fact that the radiative conductivity is
essentially a temperature dependent conductivity and is very
difficult to separate from the kinetic conductivity when
there is no appreciable change in temperature throughout the
sample over the course of the experiment. This situation is
known as correlation among the parameters. Evidence of this
can be clearly seen in Figure 3-7 where the sensitivity
coefficient curves for diffusivity and km/k appear to be
mirror images of each other. This is an undesirable
condition in an experiment design because the cause of the
unique shape of the measured data curve can be almost
equally attributable to either of the two correlated
parameters. This phenomenon is manifested in the
sensitivity matrix in the final set of parameter estimation
equations, making the equations very ill-conditioned.

As a quantitative comparison of the stability of Model

1 and Model 2, with regard to parameter estimation, it is

I.

”a

u .5
sum

 

79
useful to examine the number of iterations required for
convergence. The program developed as part of this research
for this application, entitled “flash.exe”, requires 9
iterations to reduce the difference between the actual
parameter values and the estimated parameter values from 1
percent to 0.1 percent. Comparing this performance to that
of the program using Model 1 with no radiation, a 100
percent deviation of initial parameter values may be reduced
to 0.1 percent deviation in as few a 6 iterations.

In an effort to accentuate the effect of the radiant
conductivity's temperature dependence, another test was
performed at a non-dimensional ambient temperature Tfi'equal
to one. Physically, this means that the average rise in the
temperature of the material is approximately equal to the
absolute ambient temperature. This situation is somewhat
unrealistic in an experimental situation, particularly at
high ambient temperatures. For example, a test run at room
temperature, 273K, would have to be heated to 546K in order
to provide a unity value for ng The results of this test
were that the program was able to reduce a 10 percent
initial deviation in all 4 parameters down to 0.1 percent in
approximately 6 iterations. The actual values used in this

test are shown in Table 3-3.

 

1"

e“

S

...:
\

80

Table 3-3

Parameter Estimation with'L;=1

Diffusivity Heat Flux Biot Number' lgO/k

Actual

Values 1 l 0.1 0.1
Initial

Values 1.1 1.1 0.11 0.11
Estimated

Values 0.9999977 1.0000008 0.1000018 0.1000016

As a further attempt to refine the process, a reduction
was made in the number of parameters estimated in order to
minimize the singular nature of the simultaneous equation
used in finding the solution. The Model 1 parameter
estimation procedure was used to analyze data which was
generated using Model 2. It was discovered that the Model 1
parameter estimation procedure interpreted the radiant
conductivity as kinetic conductivity, but that the heat flux
value was properly estimated. The example shown in Table 3-
4 illustrates this result.

Even with the initial parameter values set at 3 times
the actual values, the solution converged within 5 or 6
iterations using Model 1 to estimate parameters from the
fictitious data generated using the Model 2 direct solution.
Since the radiant conductivity is not acknowledged by the
Model 1 parameter estimation program, the kinetic and

radiative conductivities are simply seen as one conductivity

.:
n~e

81

Table 3-4
Parameter Estimation Using the Model 1 Procedure
on Model 2 Data

Diffusixiix. W3. Miami}:

Actual

Values 1 1 1 1
Initial

Values 3.0 3.0 3.0 NA
Estimated

Values 1.7265 1.05293 0.58275 NA

which is equal to the sum of k and km. For this reason,
the value of diffusivity is returned as twice the actual
value and the Biot number is k of the actual value since
diffusivity is directly proportional to conductivity and
Biot number is inversely proportional to conductivity. The
value of the heat flux, however, is preserved as
approximately the correct value due to conservation of
energy.

With this fact in mind, an attempt was made to recover
3 parameters using the Model 2 analysis instead of 4
parameters, assuming that the heat flux is known from the
Model 1 analysis. The three unknown parameters in this case

were

 

 

82: 83:8]: (3’37)

 

«\V

«dd

u .
«Hm

82

This method of analysis, however, provided no improvement
over the previous method of attempting to simultaneously
analyze four parameters. With T;=l, the method would not
converge using exact data unless the initial seed parameter
values were within 10 percent of the actual values. At
T;=100, the method would not converge using exact data
unless the initial guess for the parameter values was within
1 percent of the actual values. In these test cases, the
simulated experiment was errorless. Other combinations of
conditions were tried with Biot number and km/k equal to
1.0 and 0.1 with similar results. In no case was
convergence obtained with an initial parameter guess of more
than 10 percent above the actual parameter values. As
discussed earlier in this section, the reason for this is
the close correlation between the parameters of diffusivity
and radiant conductivity, causing the final parameter
estimation equations to be ill-conditioned. Without
extremely accurate seed values for the parameters, even
using errorless data, the parameter estimation problem is
unstable.

Extending the investigation of the use of Model 1 to
analyze simulated measurements generated using Model 2,
several four-parameter direct solutions were generated.

This was done in order establish the magnitude of the

83

 

W—l

 

g.

Non-Dimensional
Temperature
c:

 

 

 

0 .1 .2 .3 .4 .5 .6
hkurDhnenflonafTflne

 

 

 

Figure 3-8 Residuals from Anaiyflng Model 2 Using Mode” W10 T_’=100 Bi=0.1

residual signature using a three-parameter estimating scheme
in analyzing a four-parameter problem. Part of this
analysis included an examination of how close the estimated
parameters came to the known values from the direct
solution.

The graph of the residuals, as shown in Figure 3-8,
shows the extremely small residual signature resulting from
this test problem. A small ripple, which shows up in the
first 0.1 unit of time, is the only indication that the
chosen model may not be perfectly suited to the experiment.
At the maximum value, this ripple reaches a magnitude of
0.000245 with the nominal temperature measurements reaching

a magnitude on the order of one. For this problem, the

. Illa .lwflcaaoew

 

84

 

 

0.01

 

0.005 ~~

 

 

-0.005 4

T

-0.01 <~

-0.015 «-

Non-Dimensional Temperature

 

 

-0.02

 

0 0.1 0.2 0.3 0.4 0.5 0.6
Non-Dimensional Time

 

Figure 3-9 Residuals Using Model 1 toAnelyze Model 2 W.1T,’=1Bi=1

 

 

—-'— Alpha —+— Heat Flux + Biot Number

 

1.8
1.6 1F
1.4 1,

d
_. N
1 L
T Y

"-I ;.'- 55,528 V

. 08 4..

E
9.0
no

Sequential Parameter
strmates

9
re

 

 

O

 

f r

.2 3 04 .5 ,6
Non-Dimensional Time (based on k+kr)

O
_a

 

 

 

Figwe3-1OSequentielPerameterEstimatesUsingModel 1 toAnelyzeModeiZ
lgk=0.1. T.’=1.Bi=1

actual versus estimated parameters were shown in Table 3-4

85
previously. Had the direct solution been unknown, this set
of estimates would have given the impression that the model
chosen was perfectly appropriate and that accurate
parameters had been obtained.

A drastic change is noted when repeating the above
procedure using the same parameters, but with TJ=1 instead
of 100. At this level, the magnitude of the temperature
rise due to the pulse heating is approximately equivalent to
the ambient temperature. This causes the effects of
internal radiation to be much more significant, even with
the same km/k ratio. Figures 3-9 and 3-10 show the
residuals and the sequential estimates for this problem,
respectively.

Although the residuals are much larger than in the case
with TJ=100, they are still only 1.5 percent at the
greatest and could still be masked by errors in a real
experiment. In this example using exact data, there is an
unmistakable signature, however this signature does not
match the one shown in the residual curve from the actual
data as shown in Chapter 2. In fact, the signature shown in
the residual curve in figure 3-9 is nearly a perfect mirror
image of the signature from actual data shown in Chapter 2.
This seems to imply that internal radiation transport is not

responsible for the signature in the residuals from the Oak

86

Ridge National Laboratory data.

3l5__ANILIEIE_QI_LBEQBLIDBX_DLIL

As stated in the previous section, the parameter
estimation equations were found to be unstable even for
errorless data and unrealistically accurate previous
knowledge of the parameter values. Using derivative
regularization, an attempt was made to analyze actual
laboratory data measured at Oak Ridge National Laboratory.
The sample measured in this experiment was Carbon Bonded
Carbon Fiber (CBCF) as described in Chapter 2. The
measurements were taken at an ambient temperature of 700%:.
Analysis of this experiment indicated a poor fit of the
internal radiation model to the laboratory data. The
convergence criteria of 0.1 percent change or less for every
parameter between iterations used in the three parameter
model was not obtainable for the km/k parameter. Although
the other three parameters Ud,<%,and Bi) all converged, the
km/k parameter tended to hover between iterations from -.O7
to -.05 without converging in one particular area. Worse
yet, the negative value arrived at for km/k is physically
impossible since there can be no negative radiation.
Further confirmation of this model incompatibility was

evidenced by the fact that, forcing km/k to be positive

87

 

—+— Hodell + HodelZ

 

D
'D
Shit.
I
'5'

i115 .. \

u-L
l
‘r
1’
Far
7!
E
....
i
a:

 

Temperature (volts)
9
o 8

c:
d
or

 

 

 

do
as

0.2 0.4 0.6 08 1 1.2 1.4 1.6
Time [in seconds)

C3

 

 

 

Figure 3-11 ResiduelsfromCBCF Metenalet700’C

resulted in non-converging parameters with km/k driving to
zero. Additionally, model non-comparability is shown by
Figure 3—11. This figure compares the residual curves from
the Oak Ridge National Laboratory data using Model 1, which
assumes no internal radiation, and Model 2, assuming
internal radiation. The internal radiation model provides
no real improvement in the conformance of the calculated
solution to the lab data. Additionally, the fact that the
residual curves are mirror images of the known direct
solution residual curve, casts serious doubt as to the
responsibility of radiation for the residuals in the lab

data. Table 3-5 shows the parameters arrived at by each

88

Table 3-5

Parameter Estimation Comparing Model 1 and Model 2

on actual data taken at 700%:

Diffusivity Heat Flux: Biot Number kp/k
Model 1 0.3703 16.131 0.9646 NA

Model 2 0.4201 15.206 0.8245 -.06

method. These factors seem to confirm that internal

radiation is not evident in the experiment.

3A§__IflIIGBflzflllllfllflllhiahflflflfllgl

A more rigorous model for the combined radiation and
conduction problem, accounting for scattering absorption and
emission in a gray medium, involves solving an integro-
differential equation with appropriate boundary conditions.

Considering the one dimensional energy equation,

pc ar_ 5(k21’-

P'a‘E’F; ax qr) (3’38)

the spacial derivative of the radiant heat flux can be
written as given by ref [1] as follows, assuming non-

spectrally dependent parameters with isotropic scattering

89

 

i:’=—2nxflai (0, u)exp(--——)du
-2nx[§ (L,-u) exp(3£E%:£L)du (3_39)
-2nxfo" I(x ’)E1(x-x’)dx’
-2nKIx‘I(x’)E1(x-x’)dx’ +4nI(x)
The symbols in the above equations are as follows
K = extinction coefficient (a+oJ
L = thickness of the sample.
x' = Dummy variable of integration.
i*(0,u) = Radiation intensity from the x=0 face
into the material.
if(L,-u) = Radiation intensity from the x=L

face into the material.
u = cos(6) as measured from the normal to the

surface.

Eg(x) is defined in the following equation

Eaixi =L1u"'2exp(--§) du (3-40)

Finally, I(x), which is known as the “source equation", can

be expressed as follows for isotropic scattering.

I(x)=(1-Q)i(x) +B—o-f‘"i(x m)dw (3-41)
0 4n 0 '

90
In this expression, O0 is the albedo which is the ratio
between the scattering coefficient and the sum of the

absorption and scattering coefficients or

0'

Q0 =a+; (3‘42)

 

The denominator term, a+og, is sometimes also known as the
“extinction coefficient” as noted above. Using 6 as the
emissivity, the intensities shown in the above equations can

be expressed as

i’(x)=—T‘(x) (3-43)

and

[mi/(de =2€OT‘(X) (3-44)
0

Using this method of modeling the local intensity, the

source function can be simplified as

I’(x)=€oT‘(x) (1-&) (3-45)
2

Using this term in the expression for radiant heat flux

yields the following expression in T(x,t)

91

 

 

qr _ Q0 4 1
_ 4oxel1 —2— T (x, t) ~20Kefo(n1—n2) udu

dx
Q (3-46)
_ 2 ___3 L 1 /
Zoxe(1 2’10 Ion3dudx
where
nl=T‘(O,t)exp(—-§§-)
u
n2=T‘(L,t)exp(-:(—1(x-L)) (3-47)

n3=T‘ (X’. t:)il'1e><p(-:L-1IS Ix-x’l)

If non-scattering conditions were to be considered, the
scattering coefficient, 0, is zero which causes no to be
zero. This causes the equation to match the non-scattering
forms in references [8],[9] and [10]. With the radiant heat
flux gradient now expressed as a function of temperature,
the energy equation can now be expressed as a non-linear

second order integro-differential equation in T(x,t).

 

oc (3-48)

9 at -5; 6x 6x

6T(x, t) _ 6 (deU-c, t) ) _ (39'r
This equation requires two boundary conditions and one
initial condition in order to be solved. The initial
condition is taken as T(x,0)=T..and the boundary conditions

for the parameter estimation problem are

92

l-k%+qrj. =q06(t) +h[Tm-TX:L] +€C[T:—T::L] (3‘49)
x w
61" _ 4 4
[dc—”I’l- - “TM—1;] +eo[T,,=L-T..] (3-50)
8x =L

In a similar manner to the way the spacial derivative of qr

is written, qritself can be written as

q: =203af01{T‘(0, t)exp(;S§—)

—T‘<L. t)eXp[-3- (x-L) 1 ludu

Q (3'51)
+ZOea(l-——°)f"f1T‘(x’, t)exp[3 (x-x’) ]dudx’
2 o o p

Q
-203a(1--3)f‘f1T‘(x’, t)exp[3 (x-x’) Jdudx’
2 x o p

This model requires that 6 parameters be evaluated, which
makes it considerably more complex than Model 2 in that

regard. The parameters are as follows

 

(3-52)

With the difficulty experienced in estimating four
parameters in the previous model, it is unlikely that the
more complicated model would be practical in estimating

additional parameters. Moreover, since the diffusive model

93
is more appropriate for optically thick materials, the added
complexity of the integro-differential solution is not

warranted for this experiment.

CEUKPTIEI 4

IUUTIEUIAIUNHB EKMIEIéBIN1150301!IDNTlflleJhIKADflJHPICEI

1l1_INIBQDDCIIQB

As shown in Chapter 3, the inadequacies of the three
parameter model, which includes only diffusivity, heat flux
and Biot Number, do not seem to be rectified by the
inclusion of internal radiation as a mode of heat transfer
in the model. In past research, other phenomena have been
studied in order to rectify disparities between measured
data and conventional kinetic conduction models.

One of these secondary mechanisms, which has been
studied as non-Fourier conduction, is the concept of
electron interaction. This mechanism, as put forth by Lin,
Hwang and Chang [49], has to do with sub-micron regions of
high thermal gradients over very short periods of time, on
the order of picoseconds. This model attempts to account
for heat transfer by electron interaction within a lattice
and is most appropriately applied to thin films of highly
conductive material, such as gold.

Since the Carbon Bonded Carbon Fiber (CBCF) material
studied here was originally developed as a thermal

insulator, other models are investigated as part of this

94

95
research in order to find alternatives to the conventional
model presented in Chapter 2. The research presented here
involves the development of other physical phenomenon which
may explain the discrepancy between the calculated direct
solution and the measured data from Oak Ridge National
Laboratory. None of the models described in Chapters 2 and
3 has been able to adequately explain this disparity.

The remainder of this chapter explores various models
in an attempt to more adequately describe the mechanisms of
heat transfer taking place in the samples, thereby achieving
greater assurance in the parameters estimated. Section 4.2
discusses a linearly temperature dependent conductivity
model and a separate model in which the surface heat
transfer is purely radiative instead of conductive. Various
linear problems which assume constant parameters and some
type of flash penetration into the material are discussed in
Section 4.3.

In one such model, the assumed time of the flash is
considered to have occurred prior to time zero. The time
interval between the flash and time zero in this model is
treated as a parameter. Also investigated are various forms
of internal deposition of the flash which are treated as
initial condition temperature distributions. Also discussed

is the concept of surface transmissivity which addresses the

96
disproportionate amounts of heat deposited on the surface of
non-homogeneous or coated materials. Section 4.3 also
briefly identifies several models used in the development of
the more preeminent models ultimately used in describing the
observed phenomenon. The exact mechanisms of the flash
penetration are dealt with in Section 4.4 wherein the
extinction coefficient is shown to be related to the
parameters calculated by the salient models. Finally, the
models deemed most appropriate are utilized in Section 4.5

to analyze data from several laboratories around the world,

comparing the various thermal parameters.

 

Pursuant to the notion that the signature in the
residuals may be due to a simple temperature dependent
conductivity rather than an internal radiation phenomenon, a
direct model was developed which assumed a linear dependence
of conductivity on temperature with no internal radiation.
The differential equation for this model is

g

6T_ 6
oc - 6x

p5? ax(ko+k1(T"T~))

(4-1)

where kg and k4 are coefficients of a linearly temperature-

dependent conductivity with units of W/mK and W/mKfi

97

respectively. The boundary conditions for this model are as

follows
-(ko+k1(Tx:o—Tm)){%§J =q06(t)+h(Tm-Tx:0) (4-2)
x=0
aT
-kkT-T-—=hT-T -
(0* 1‘ H 016an ( x=L .) (4 3)

The initial condition for this problem is assumed to be
T=T.. This problem was solved in a similar manner to the
model for internal radiation using a finite difference
method.

The direct solution generated by this temperature
sensitive conductivity model was analyzed using the three-
parameter Model 1 in order to estimate parameters. The
direct parameters used and the parameter estimates are shown
in Table 4-1. Figures 4-1 and 4-2 show graphs of the
sequential parameters and the residuals, respectively, for
this parameter estimation problem. Like the internal
radiation case discussed in Chapter 3, the sequential
parameter graph shows that the parameter estimation model is
not well suited for the simulated laboratory data. The
sequential parameters are not constant toward the end of the
experiment indicating a mismatch between the direct solution
and the model used in parameter estimation. Moreover, the

graph of the residuals does not at all

98

 

+ alpha —9—— Heat Flux + Biot Number

15

 

9"
won

'4
“’13.?“ , r
”'ditmfismxusasiaxuxns_
a,

 

Esfiuunes
inm'm

—fi

Sequential Parameter
no
-‘
+ L
I!
E!
l
.
is
K
I:
it.
t
B-
I!

051’

 

 

 

0 DJ 02 03 0A 05 06
Non-Dimensional Time

 

 

 

Figure 4-1 Sequential Estimates Using Model1 to Analyze Model 3 lolly-'1. Bi=1

Table 4-1
Parameter Estimation Using the Model 1 Procedure on
Model 3 Data

Diffusivity Heat Input Biot Number kzl/kO

Actual

Values 1 1 1.0 1.0
Estimated

Values 1.4936 1.9006 1.5064 NA

resemble the graph of residuals shown in Chapter 2 for the
actual experimental data. This set of results seems to
indicate that the linearly temperature dependent
conductivity is not a reasonable explanation for the
behavior observed in the Oak Ridge National Laboratory

experiments .

99

 

 

001

osmsv

 

 

s5
53

{unse-

Residuals (non-dimensional
temperature)
«5
8
0'1

 

 

-002

 

0 01 02 03 DA 05 05
Non-Dimensional Time

 

 

 

Figure 4.2 Residuals Using Model 1 to Analyze Model 3 woe-1, Bi=1

Figure 4-3 shows a comparison of residuals from actual
data as measured at Oak Ridge National Laboratory with the
CBCF sample at 7008:. The two sets of residuals are
generated from Model 1, which assumes conventional kinetic
conduction with constant conductivity, and Model 3 which
assumes a linearly temperature dependent conductivity. Once
again, there is no real improvement in the fit of Model 3
over Model 1. This is to be expected since the residual
curve of the exact solution of Model 3, when analyzed with
Model 1, does not match the residual curve signature from
the laboratory data when analyzed by Model 1.

Another possible explanation for the residual behavior

100

 

“**_“hkxkil ‘“**_‘lkxkfl3

0.3
025‘r
0.2 t
0.15 ‘-

 

 

Residuals (Volts)
c:
{3

.b
. 3

 

-015"
-02

 

 

 

0 02 0.4 0.6 0.8 1 12 1.4 1.6
Time (in seconds)

 

 

Flgtl'e4-3 CBCFSUnplofromORNLat700'CAnelyzedwlfllModel1 andModel3

exhibited in the Oak Ridge experiments is that the dominant
mode of heat transfer at the surfaces is radiation as
opposed to convection. At the higher temperatures of 700%;
it may be likely that the model used in estimating
parameters was not properly accounting for this. A direct
solution was generated utilizing a constant kinetic
conductivity in the material continuum and pure radiation at

the surfaces. The differential equation for this problem is

6T 621'
OCpat 5X2 (4 4)

and the boundary conditions are

 

101

Table 4-2
Parameter Estimation Using Model 1 to Analyze Surface
Radiation Model Data

Diffusivity Heat Input. .Biot Number e

Actual

Values 1 1 NA 1.0
Estimated

Values 0.938 1.607 10.87 NA

6T 4 4
«[5],; q06(t) +€ob(T..-T,,.o) (4-5)

6T _ 4 4
-kl-é-;Jx:z- €Ob(Tx=L—Tee) (4'6)

where e represents the surface emissivity of the sample.
The initial condition for this problem is assumed to be
T=T.. .A finite difference scheme was used which was similar
to the internal radiation model in order to generate the
direct solution. As in the cases above, this direct
solution was analyzed by the three-parameter model. Table
4-2 shows the direct solution parameters and the estimated
parameters for this problem

Figures 4-4 and 4-5 show graphs of the residuals and
sequential parameters for this problem, respectively. The
sequential parameter estimates give some indication that the
model is not completely appropriate for the data,
particularly in the Biot number curve where the value drops

off with time toward the end. The residual curve, however,

102

 

 

001

0.001%r

 

 

d:
D
..e

Residuals (non-dimensional
temperature)
e: d:
O C:
51‘ 8

 

 

-002

 

0 01 02 03 04 05 06
Non-Dimensional Time

 

 

 

FigureMResiduaisUsingModeltioAnalyzeaSurtaoeRadiationDireotSolution(a=1,Bi=1)

gives virtually no indication that there is any deviation
between the data and the estimate. The largest individual
residual in this case is approximately .02 percent of the
maximum temperature measurement. This model does not seem
to explain the phenomenon observed in the Oak Ridge National

Laboratory measured data.

W

In contrast to modeling a temperature-dependent
variable as an explanation for the behavior exhibited by the
Oak Ridge National Laboratory data, an attempt was made as

part of this research to explain the departure from the

103

 

 

 

 

 

 

+ Diffusivity + Heat Flux + Biot Number
16 ‘
5 14 it \
° 12 .. x
|§‘§ H11 q-p‘i'
'3.E Ill
'35 s 4+
e
3 4 l»
a
a: 2 ..
0 .L s 5 + a .
0 01 02 03 0A 05 05
Non-Dimensional Time

 

 

 

 

 

Fnum44iSemanfiflEMhuMuLkmngdm1unnumMeaSwflneRamNMnEmu:de&m
(a=1. Bi=1)

conventional model in terms of the penetration of the flash
at the moment of heating. This principle assumes that the

flash penetrates the sample and is instantaneously absorbed
by the material internally. This distribution of energy at
the time of the flash can effectively be modeled as an

initial condition.

1l3l1__IEIIIAL_IIII_EIIIIAJMQDIL_1

A simple attempt at duplicating the signature in the
Oak Ridge National Laboratory measured data was pursued by
shifting the time scale of the measured data by an amount,

designated as At, from the given "time zero" of the

104
experiment. In this case, an exact solution was used of the
differential equation described in Chapter 2 and was simply
offset by a pre-determined At. This time shift served to
effectively advance the time of the heating pulse prior to
time zero. In order to estimate an accurate effective time
shift of the heating pulse, the quantity At was treated as a
parameter.

Another way of considering the time-shift model,
designated as Model 4, is to think of it as an initial
condition of penetrated heating. This initial temperature
distribution happens to have a form which is somewhat
similar to an exponential temperature distribution. As in
the cases above, this direct solution was analyzed by the
three-parameter model. Table 4-3 shows the direct solution
parameters and the estimated parameters for this problem. A
considerable error of 14 percent arises in the estimated
value for diffusivity due to the introduction of the time
offset of .01 units of non-dimensional time. The time shift
toward early heating seems to be a phenomenon by which the
more rapid initial heat transfer could be manifest in the
actual laboratory experiment.

The actual laboratory data from Oak Ridge National
Laboratory was also analyzed using Model 4. The standard

deviation of these residuals is 0.009935 which is extremely

105

Table 4—3

Parameter Estimation Using the Model 1 Procedure on a
Model 4 Direct Solution

Diffusivity Heat Flux .Biot Number At

Actual

Values 1 1 1 0.01
Estimated

Values 1.144 0.9102 0.8314 NA

small in comparison with measurements which are typically on
the order of 6. By comparison, the standard deviation of
the residuals generated from using Model 1 as the
mathematical model are approximately 0.04, over 4 times the
magnitude as compared with Model 4.

The most significant aspect of this analysis is that
the estimates of the parameters of interest are
significantly altered between Model 1 and Model 4. The
estimate for diffusivity is 26 percent lower and Biot number
is 65 percent higher using model 4 as opposed to Model 1.

As can be seen from the residuals, the estimates using Model
4 appear to be the most valid of any of the models applied

thus far.

1W
Assuming that the Oak Ridge National Laboratory data

was recorded accurately and that there is no time shift

106
between the measurement time scale and the true time scale,
then a logical physical explanation for the apparent time
shift is radiation penetration. More specifically, even
though there seems to be no evidence of internal radiation
as a heat transfer mechanism subsequent to the flash, it is
still possible that the radiation wavelength of the flash
itself is such that a non-negligible mean free path exists
inside the material for photons of that energy. The
manifestation of this phenomenon would be best modeled as an
initial condition since the flash takes place over such a
short period of time. This model is very similar to Model 4
in terms of a non-uniform initial temperature distribution,
but is based on a much more reasonable physical explanation
as to its origin.

As a first attempt at modeling this phenomenon, an
exponential distribution was chosen since this most closely
corresponds to the principle embodied in Bouger's Law, as
discussed in reference [1]. The mechanism of this
absorption is also modeled in Section 4.4. In order to
calculate the direct solution for this model, a Green's
function solution was chosen for convective conditions at
both boundaries and zero ambient temperature. The Green's
function solution equation for this problem reduces to a

single integral for the initial condition of

107

T(X,t)=fxf'zon33(x,tlx’,0)F(x/)dx/ (4-7)

where F(x') is the initial temperature distribution in the

material, or in this case

 

F(x’) = q° (4-8)

alt-n

ocpa(l-e

where a is a measure of the penetration of the flash. The
denominator of this term is arrived at by normalizing the
exponential temperature distribution to the magnitude of the
heating that brought about the temperature rise. In this

case

L
qo=ocpfF(x)dx (4-9)

x=0

The F(x) defined above satisfies this equation. The Green's
function for X33 is given as
G / _2 2 'Bipit-tlle /
x331X'XIt-T)‘E 1e A,(x)A,,<x) (4-10)
m:
where

x . . x
Bmcos (8,?) +81 Sln (Bu—1:)

An(x)= (4-11)

 

B§+Bi 2+2131

108

and

X, X,
Am(x’) =Bmcos(8m—L-) +Bi sin((3m-f) (4—12)

Integrating Equation (4-7) we have

 

Zqoa m e'BiPt/LZ

 

 

T L,t = c c-{: -
( ) kLa(1-e"")a=1 1( ’- 3) (4 13)
where
_ Bulfimcos (0,.) +81 sin (Bull
1- 2
4-14
(B:+Biz+231{_l_+_fl_“] ( )
a2 L2
62 .B' _L
a J. T; . ..
C2- '3'"; e 31MB“) (4 15)
8' —-€
C3=[-‘?-'-+ B“ l)(e ‘cos(Bm)—1) (4'16)
a L

Figure 4-6 displays the sensitivity coefficients for
the exponential penetration model, Model 5. The sensitivity
coefficient for the penetration parameter, a, appears to be
somewhat correlated with the diffusivity sensitivity
coefficient. Experience has shown, however, that it is
different enough to allow independent estimation of the four

parameters. In spite of the apparent correlation between

109

 

—I--— Dittusiv'ly ——-E3——- Biot —l— Penetrdion —Cl— Hem Flux
Number

 

  

 

 

 

 

f .
.2. g
° '5
(a
v “E
éaor
‘3
2 -02«

0.34

-o.4

o 0.1 02 as 0.4 05 0.6

Non-Dimensional Time

 

 

 

Fbu344lMaflbdSaufifiUdeflfimbflxthflSGflmnghBhtafiafl
diffusivity and a, and between heat flux and Biot Number,
the parameter estimates converged without the benefit of
using derivative regularization.

Next, Models 1, 4 and 5 were tested on CBCF data
measured at 700%: at Oak Ridge National Laboratory. In
contrast to the first three models, the standard deviation
of the residuals for Model 4, which treats the time-shift of
the heating pulse as a parameter, gives significantly
improved results over each of the other models. Table 4-4
shows the parameters arrived at by each method. .A large
reduction in residuals is gained in implementing the time-

shift model, Model4, over the simple three-parameter Model.

110

Table 4-4
Parameter Estimation Comparing Models 1,4 and 5
on actual data taken at 700%:

Diffus- Heat Biot
1 0.3400 19.138 1.2765 NA. 0.03202
4 0.3027 22.203 1.5550 0.0121 0.01212
5 0.3066 19.753 1.5178 0.0682 0.01098

Additional gains are realized when utilizing the exponential
penetration feature of Model 5. The results using Models 4
and 5 are somewhat similar in that both return lower values
for diffusivity than the simpler Model 1. The estimated
Biot number is also higher in each of these cases than that
calculated by Model 1. These factors suggest that both
models are, to some extent, accomplishing the same thing
through different means. That is, they both model a more
rapid heat transport in the early times than would be
predicted by the conventional Model 1 assumptions.

It can be instructive to test the shape of the
residuals generated when using Model 1 to analyze a direct
solution generated from the time shift or penetration
models. Using each of these models as a direct solution and
analyzing them using Model 1, the signatures of the
residuals are very similar to one another and to the
signature generated when analyzing the laboratory data with

Model 1. Figure 4-7 shows a comparison of Models 4 and 5 as

 

111

 

+ 0.01 Second Pre-Heating —“— a-0.1 Penetration

 

002

 

 

D

 

 

Residuals (non-dimensional
temperature)
P
D
8

 

 

s53
can
do:

01 02 03 0A 05 06
Non-Dimensional Time

D

 

 

Figu'e4-7 UsmgModel1toAndyzeDireetSolulionsotModeb4and5(e=1.Bi=1)

analyzed by Model 1. In this example, a value of At=0.01 is
used in the direct solution generated from Model 4 and a
value of a=0.1 is used in Model 5.

In using Model 5 on actual laboratory data, Figure 4-8
shows residuals for the Oak Ridge National Laboratory data.
This figure shows a near-elimination of the signature
resulting from Model 1. The signature, generated from the
Model 1 analysis of the laboratory data, is evidently caused

by some manner of penetration of the flash into the material

at the instant of heating.

112

 

+ Modeli —9— Model 5

01
000*

 

1

F3
8

0041:

0.02 «» ‘
1‘ . , hr 1' J *r m. v.
1 -17, ,.

Residuals (volts)

-002<(
-004
-000

 

 

 

 

0 02 04 08 08 i 12 14 16
Time (seconds)

 

 

 

Figure4-8 CBCFSamplefromORNLaUOO‘CAnalyzedwimModelt andModel5

Al3lJ__IIBIQQ8_IflIIIAL_QQBDIIIQN31_HDDIL8_§:1§

Following the development of Model 5, a series of
models were studied which use various initial conditions
based on the premise that the radiation from the flash
penetrates the surface of the material. The pursuit of the
best approximation for the distribution of this energy
becomes the primary objective in the development of these
subsequent models. The desired outcome is to minimize the
residuals and, in particular, eliminate any characteristic
signature.

A summary of Models 1-4 is as follows

Model 1 Conventional conduction using only the

113
three basic parameters of diffusivity,
heat flux and Biot number.

Model 2 Combined internal radiation and conduction
as described in Chapter 3.

Model 3 Linear temperature variable conductivity
with the fourth parameter being the rate
of change of thermal conductivity as a
function of temperature.

Model 4 Time shift at the time of the heating
pulse with the fourth parameter being the
magnitude of the time shift.

The remainder of the models studied as part of this research

involve various forms of initial temperature distributions

with conventional conduction thereafter. The initial
conditions assumed for Models 5 through 16 are graphically
presented in Figures 4-9 through 4-20. The corresponding
parameters to be estimated are shown. In each case, the
initial condition parameters were estimated simultaneously

with the three basic parameters from Model 1, namely a

(thermal diffusivity), qg (heat pulse magnitude), and the

Biot Number. The additional parameters in each model range

in number from 1 to 3, so that the number of simultaneously

estimated parameters ranges from 4 to 6. None of these

114

TEMPERATURE

INITIAL

 

 

 

x=0 x=L

Figure 4-9 (Model 5) Exponential distribution of initial penetration of the flash in the material with the
fourthparametermeasunngthedegreeofpenetration. lnthisandsubsequentmodeithetermT,
isndneededsinoemistemperatumisindnecuypmsalbedbymemmmxpammeter.

\

 

 

 

TEMPERATURE

INITIAL

 

 

 

 

x=0 x=L

Figure 4-10 (Model 6) Rectangularly distributed initial penetration with the fourth parameter being
the distance of penetration.

\
I

TEMPERATURE

INITIAL

 

 

 

x=0 x=L

Figure 4-11 (Model 7) Exponential squared distributed initial penetration with the fourth parameter
a measurement of the degree of penetration.

115

T=Toe’§s
T

 

 

 

 

LaJ

0:

D

f.-

<£
_lCZ
23:
:5 8.
E:-

I ~ on
x=0 x=L

Figure4-12 (Model 8) Lineartemperamredistributionovertheflrst 10 peroentoftheinoidentside
of the material followed by exponentially dismbuted initial penetration thereafter. The fourth
parameter is the magnitude of the initial temperature at the heated surface and the fifth parameter is

ameasureofthedegreeofexponentialpenetration.

 

 

 

 

/\
g -5
:E 1‘ I-ZZI; E? [a
_icz
:23:
:2 B, 8
EH
‘l Ii 0.1L
, 0.1L
=0

x x=L
Frgure4-13 (Model9) Twelinearandoneexponentielpenetrationzones. Thefourthandlilth
parameters arethetemperaturesatL=Oand L=O.1, respectively. The sixth parameterisa measure
otthe degree ofexponential penetration.

 

 

 

 

 

 

 

/\
LL]
0:
D
I'-
<1
:5 a 8
HQ "" -""_ 5—"
=5
El—
86
I
T
x=0 x=L

Figure4-14 (Model 10) Two constantandoneiinearpenetrationzone astunctionsofpenetration.
Thetourth parameter measures the depth otthe first constant penetration zone. The fifth
parametermeasuresmedepmofmlinearpeneuafionzoneandmesbtthparameteristheinitlal
temperatureoftl'ieseoondoonstantpenetrationzone.

116

 

 

 

 

 

/\
Ed -5
l3 TZToe 85
.10:
T
:2
at: ’3

l ”6

__ l
x=0 I x=L

Flgure4-15 (Modei11) Oneconstantpenetratlonzoneandonecomblnedexponentielplus
constantzone. Thefourflrpammetermeasumsdredepflrofpenetafionhdreflrstconstme.
fleseprammetermeasumsmedepmmmaexpomnualmnadonandmestammeter
measuresthemagnitudeofthe second constantzone.

 

 

 

/\
LrJ
o:
3
|._
_] SE 3.500
<1: Lu
‘3' %
5 g 856<x—L>-\
x=0 x=L

Figure 4-16 (Model 12) Onesurface flashoneach side of the sample. Thefourth parameter
measuresthemagnitudeoftheflashatFL.

 

 

 

 

 

A PARABOLIC
Lu DISTRIBUTION
x
% _ __
:2 T—To e 85
.10:
S.“
:%
EH
8
_. -3. ‘6
x=0 x=L

Figure 4-17 (Model 13) Parabolic distribution at the incident side of the sample with a combined
exponential and constant zone. The fourth parameter measures the depth of the parabolic
penetration. The fifth parameter measures the depth of the exponential penetration and the sixth
parametermeasures the magnitudeofthe second constantzone.

 

117

\
I

 

TEMPERATURE
1b

 

 

 

 

 

J
S
E
E 856(X’L)\
36
l
x=0 1 x=L

Figure 4-18 (Model 14) One linear and one constant zone with surface heating at the x=L face.

Thefmxhpammetermeawmsmedepmofmelimarpeneuaflm.meflmwammetumeasums
themagnitudeoftheconstantzoneandthesixmparametermeasuresthemagnitudeoftheilashat

x=L.

 

 

 

firkx)
32 x
3 T:To e 85
<
_J&
2:
:53
E.»—
86
. 1
l
x=0 x=L

Figure 4-19 (Model 15) Surface heating at x=0 and a penetration zone with combined exponential
and constant distribution. The fourth parameter is the maximum temperature associated with the
exponential componentofthe penetration. Thefrlth parametermeasuresthedepth ofthe
exponential penetration. The sixth parameter measures the magnitude ofthe constant penetration.

 

 

 

[5&1 flth) x
r:— T = To 9 73's
_r or
<1: Lu
E %
5 g 865<x-L>\
x=0 x=L

Figure 420 (Model 16) Surface heating at x=0 and x=L with exponential penetration. The fourth
parameter is the magnitude of the maximum temperature associated with the exponential
distribution. The filth parameter measures the depth of the exponential penetration. The sixth
parameter measures the magnitude of the surface heating at x=L.

118
models was shown to offer significantly improved performance
over Model 5 when analyzing laboratory data and, in most
cases, convergence was not obtained. These models were

essential, however, in the development of Model 17 discussed

in the next section.

 

In the process of refining models 6-16, it became clear
that, in many of the experiments, the sample was heated on
both sides (x=0 and x=L) presumably due to reflection of the
laser flash inside the furnace or test device. This
phenomenon is shown in the graph of the raw data in Figure
4-21 as evidenced by the prompt rise in surface temperature
at time zero followed by the rapid decline in surface
temperature during the first several time steps. The prompt
rise in temperature for the first time step in this figure
is quite significant and amounts to approximately 5% of the
full scale temperature rise.

Additionally, it became evident in many of the
experiments that penetration of the sample surface was not
complete. A large percentage of the energy was absorbed at
the immediate surface and the remainder seemed to be
deposited in an exponential fashion.

As a slight variation of Model 16, Model 1? assumes

119

 

MN

 

50

1

Measured Temperature (volts)
6‘»
C3

 

 

 

0- . s? . .
0 DJ 02 03 0A 05 05 a?
Time (seconds)

 

 

 

 

Figure 4-21 The First 100 Points of File A__R1 Showing initial Temperature Decay

that a there is a deposit of energy on both surfaces of the
sample, that is, at x=0 and at x=L. In this model, at the
time immediately following the flash, the temperature just
inside the sample is substantially less than that at the
surface. In other words, T(x=0)>T(x=0*) and T(x=L)>T(x=L’).
The initial temperature distribution inside the material is
assumed to be an exponential, with maximum temperatures at
x=O+ and x=L‘. Expressed mathematically, the initial

temperature distribution for Model 17 is

T(x)=T1Lo(x) + Tze: + T eT + T4L6(x-L) (4‘17)

120
where T1,'nu I3, and T4.are each different contributions to
the initial temperature distribution. These parameters must
be estimated simultaneously with the three basic parameters
sought in Model 1.

Model 17 assumes that the sample is actually being
heated on both sides, presumably due to a reflection of the
flash inside the furnace. If the mechanism of heating is
the same on both sides of the sample, then the degree of
penetration will also be the same. Using this concept of
symmetry, a redundant parameter in this model can be

eliminated by assuming that

T

T=T -3 (4-18)
‘ 1 T2

This is reasonable since the mechanism of radiation and
penetration and absorption should be the same on both
surfaces even though the radiation magnitudes on each side
are different. The Green's function solution equation is

the same as used in Model 5 above, which is

T(x,t)=fxion33ixyxllt."-I)17'(X/)dxI (4-19)

where F(x') is the initial temperature distribution in the

material, or in this case

121

l x’ L—x'
-— — T
qo[T1L6(x) + Tze “ + T3e ‘ +T1—1-3-L6(L-x)
F(x’): 2 (4-20)

T -3
ocp [L(T1+T1‘I'.l) +(T2+T3)a(l-e ‘)]

 

2

The denominator of this term is found by normalizing the
exponential temperature distribution to the magnitude of the
heating that brought about the temperature rise. In this

case

L
qo=pcpfleidx (4—21)

x=0

The F(x') defined above satisfies this equation. The

Green's function for X33 is given as

 

Gxa3(x,x’, t-I) Jig e-B'2"a(t-U/L2Am(x)An(x’) (4-22)
m=1
where
Bmcos( Bail +Bisin( (Bu-’51
Amlx) = L L (4-23)
Bi+BiZ+ZBi
and
, x’ . x’
Am(x )=Bmcos Bm-f +31 sin (Sm-Z- (4-24)

Integrating this with respect to dx' gives

122

 

 

T(L,t)=

2qcx a (_fi uzh Bacos(8n)+Bisin(Bm)
° e “a C; (4-25)

kLa(l-e"/‘) m=1 Bi+BiZ+ZBi

 

where

IEaL 2 . -£ . -3
Cm=-————— (aBm-LBi)e ‘sin(Bm)-(L(3m+aBmBi) e ‘cos(Bm)-l +
L2+a2(32

L

T3aL _ [ -
(LBm-aBle) cosBm-e ‘

 

-——————— J+(aB:+LBi)sianl+
L2+azBi

T
BmT1L+Tl-Z-gqflmcostBisian) (4-26)
2

Using Equation 4-25 as the direct solution for analysis
of experimental data, the residuals are among the lowest of
any model attempted. Model 17 has the additional advantage
of being consistent with the physical explanation for the
mechanism of the flash penetration on both sides of the
sample. In other words, both sides of the sample are
treated equally in Model 17. Figure 4-22 shows a comparison
of the residuals from analyzing the same data file using
Models 1,4,5 and 17. A great reduction is brought about in

the signature by models 4 and 5, however Model 17

123

 

‘_**—‘hkxkl1 “—”*—lwmkfl4 ‘_**“hkxkfl5 T—P—_'Mbdd17

 

 

 

 

 

15 -;
A10
£
3 5.
O
'3
-3 of vufigfl. .igJflAthlitlud‘_ _.nagg...
.8 I : lbw-”at.
a?

6 4L.

40

0 Q5 1 t5 2 25 3 35

Time (seconds)

 

 

FbueAdutRuflhnbflomflfleAgR1CampuhgflbdflsttSamdfii

essentially eliminates any signature in the residuals. This

is especially so in the early time measurements.

.1a3a5__BHBIIGI_IBAE3H18§I!IIX

A concept which makes Model 17 more manageable in terms
of physical relevance is that of "surface transmissivity".
Developed as part of this research, this concept is employed
as a parameter in Model 17 so as to make the results of the
parameter estimation method more physically tangible. This
parameter is defined as the energy penetrating the surface

per unit energy incident on the surface. This parameter has

 

124
a range in value from O to l and is estimated in Model 17 as
parameter 5 in contrast to considering a maximum temperature
coefficient of the exponential penetration. Parameter 4 is
the same factor "a" in the exponential penetration term used
in Model 5. Finally, parameter 6 is the magnitude of the

incident energy on the "non-heated" surface.

 

Tables 4-5 through 4-7 summarize the models used in
analyzing the flash problem and the results from three of
the experiments using all of the models. The majority of
the models developed did not prove to be appropriate tools
for the three experiments detailed above. As shown in)
Tables 4—5 through 4—7, the standard deviation of the
residuals for virtually all models is greater than those
resulting from the application of Model 17 to the laboratory
data. Additionally, for experiments which exhibit a back-
side flash, Model 17 is the most physically sound in terms
of providing insight into the mechanism of the flash
penetration. Many of the models which use multiple stages
of penetration have sensitivity coefficients for the
penetration parameters which are highly correlated. This is
true in many cases to the extent that unstable parameter

estimates can be generated yielding nonsensical results.

Model Diff.
Ruhr. ..1dl
1.33180
4 .29928
5 .29937
6 .29958
7 .31241
8 .31069
9 .29848
10 .31194
11 .31228
12 .32738
13 .31231
14 .31189
15 .31733
16 .31545
17 .31538

Model Diff.

Embrl __lol
1 .31136
2 .4021
3 .3878
4 .27615
5 .28193
6 .27274
7 .29248
8 .27861
9 .27024
10 .27486
11 .27493
12 .30982
13 .27668
14 .27657
15 .27824
16 .27525
17 .28360

Table 4-5

125

Purdue University Experiment C_ R1
L= 1. 9685mm At=. 0603 omfl= 0.001164

Heat Biot
_Elux number __£1__
6.3889 .12978 NA
7.0236 21692 .08822
6.9157 21912 .16151
6.8577 21470 .39999
4.3072 20412 .11995
6.7626 18876 .00605
3.5096 22239 10. 580
3.3882 18395 .00252
3.4093 18325 .05792
6.4632 14264 .01857
3.4099 18319 .08033
3.4085 18412 ”33946
3.3912 17089 .28895
3.4057 17573 .89560
3.4061 17589 .10026
Table 4-6

He

at

.Ilux

17

15.
14.

21

18.

18

25.

18

19.
19.
19.

17

19.
19.
19.
17.
19.

.855
20

17

.289
642
.503
190
.576
615
034
287
.979
283
397
094
913
422

Biot

Number

ia+4r4hrhu~14i4+4F4haewaiecoc>ha

.2332
.824

.793

.5609
.5030
.5956
.4650
.5360
.6322
.5730
.5725
.2472
.5548
.5561
.5398
.5692
.4860

__fih__

NA

-.O648
-.02

.02106
.06963
.20383
.10474

.70794

316. 24
.16514
.19440
.01586
.19910

.26725

505.36
1317.5
.0669

__fia.

NA
NA
NA
NA
NA

.67694
.01411
.33749
.00087

NA

.00074

.00102

1. 68- 6

.36966

.3633

__fia_

NA
NA
NA
NA
NA
NA

3.191
.0010
.0899

NA

.0940
.0001
.6109
.0031
.00333

0.008687
JS—JG—
NA
NA NA
NA NA
NA NA
NA NA
NA NA
NA NA
.05905 NA
12.383 .1013
.15531 .0002
.00023 .0008
NA NA
.00037 .0012
1.0E-6 .0001
1.0E-6 .0601
.04530 .0084
1.0 .00723

Resid.

.131.—
.01492
.01032
.00910
.01076
.01073
.01020
.00542
.00298
.00297
.00831
.00294
.00298
.00315
.00294

.00294

Oak Ridge National Laboratory CBCF Sample at 700%:
L=.956mm At=.01408 omfl=

Resid.
.131..
.03698
.10958
.11098
.01250
.01245
.01303
.01371
.01227
.01514
.00971
.01105
.03227
.00967
.00904
.01224
.00902
.01074

126

Table 4-7
A;R1 Palaiseau France
L=3mm At=.0348 owfi= 0.4520

Model Diff . Heat Biot Res id.
Nmbn. _m1 .311): Number _ii4__ _B.s_ J6. _La.i__
1* 1.6017 2920 .08633 NA NA NA 4.8113
4* 1.119 3325 .20423 .06850 NA NA 2.4166
5* 1.418 3236 .1943 .26207 NA NA 1n9452
6* 1.392 3253 .2059 .69647 NA NA 2.6606
7* 1.457 1286 .1925 .14913 NA NA 2.5760
8* 1.480 1046 .1572 .03392 .4825 NA 1.3444
9 1.449 1057 .1738 1.7598 .00823 1.326 142523
10 1.441 1056 .17868 .00001 .7781 .0011 1.1840
11 1.4561 1057 .17049 .00001 .00052 .2123 1.1242
12* 1.5819 2953 .09828 7.4990 NA NA 3.3982
13 1.4598 1054 .16807 .03955 .00032 .2197 1.1235
14 1.4241 1070 .18795 .8721 .00025 .0001 1.3551
15* 1.4573 1056 .16985 4453 2.491 .2214 1.1235
16* 1.1673 1057 .16909 2246 .28605 1J176 1.0311
17*‘ 1.1674 1058 .16893 .2782 .63487 1.4244 1.0417

* Parameter estimates for these models reached convergence

For these reasons, four models were retained for

further investigation and comparison in the other
These are models 1,4,5 and 17.

experiments. In experiments

where no back-side flash is exhibited and the surfaces of
the sample are not coated, Model 5 is the best choice. In

these cases, there is no non—uniform absorption of the flash

at the surface and no direct flash heating at x=L. As such,

the additional two parameters estimated by Model 17 beyond
the four estimated by Model 5 become a handicap and serve
only to make the parameter estimation problem.more unstable.

The three experiments illustrated by Tables 4-5 through 4-7

127
all exhibit back-side heating, making Model 17 a logical
choice in each case. Experiments performed at Oak Ridge
National Laboratory using the Anter test equipment, however,
exhibited no back-side heating, making Model 5 the best
choice for analysis. Surface transmissivity is evident in
the Purdue and French experiments, but not in the
experiments performed at Oak Ridge National Laboratory.
Were it not for the back side flash heating evident in the
experiments performed on Holometrix equipment, there would
have been no reason to use Model 17 in this case.

Model 1 is the most appropriate to use where the sample
is either completely opaque or is well coated such that no
penetration is made by the flash. This is the case with
many of the experiments analyzed from laboratories in other
countries shown in the appendix. Model 4 was retained as a
type of second-check method to investigate for penetration
of the flash. If neither Model 4 nor 5 shows any
improvement over Model 1 for a given experiment, it can be
reasonably assured that no type of flash penetration is
present.

In order to examine the four salient models in more
detail, a temperature correction was applied to each of the
three experiments analyzed above in order to determine the

effect on the estimated parameters and the residuals. The

128
temperature corrections are found by averaging the
temperature readings prior to the time of the flash and
subtracting this value from the temperature readings after
the flash as a correction to a measurement bias. The use of
this technique assumes a bias in the temperature measurement
instrument which, if compensated for, should yield better
results in the analysis of the experiment. The compensation
was applied to the same three experiments analyzed in Tables
4-5 through 4-7 as a basis of comparison. The errors
compensated for were very small, approximately 0.1 percent
of the peak temperature reading in each experiment. No
appreciable improvement was gained in adding these small
correction factors. A.summary of the parameter estimation

results for the three experiments are shown in Table 4-8.

 

In the general case of incident radiation on an
interactive medium, as given by ref [27], the radiation
intensity, as it transfers through the medium, is given by
the equation of transfer

6I(IL.6.¢)
at;

 

= I(IL,6,¢)-S(IL:9:¢) (4'27)

where I represents the local intensity as a function of 6,

129

Table 4-8

Using Temperature Compensation

Model Diff. Heat Biot Resid.
mm... .4120. Jinx Number _ii.__ .115- _Lie_ _(.s_i__
CBCF (error -.00458)

1 .31169 17.829 1.2295 .03791

4 .27573 21.338 1.5641 .02153 .01321

5 .28157 18.656 1.5056 .07042 .01283
17 .283535 19.423 1.4857 1.0000 .00853 .06722 .01057
FRENCH DATA A;R1 (error -.962)

1 1.2841 2916 .08422 5.0793

4 1.1146 3340 .20754 .07168 2.6297

5 1.1306 3248 .19736 .26774 2.1022
17 1.1676 11058 .16862 .60612 1.5783 .2866 1.0356
PURDUE DATA C_R1 (error -.03199)

1 .33907 6.3139 .10953 .02472
4 .28746 7.3500 .25017 .14289 .01790

5 .28789 7.1967 .25453 .20094 .01473
17 .31582 3.4356 .17325» .08669 .00490 .5117 .00296

the polar angle of the radiation incident to the control
volume; o, the incident angle of azimuth and IL, the optical

thickness defined as follows

X
ILefxdx’ (4-28)
0

Also, x, the extinction coefficient, can be expressed as

x = a + o where a and o are the absorption and scattering
coefficients, S is the source

respectively. Finally,

function and is defined as

130

U

0 2n
5(e,¢)=—°-[ fp<6.¢.e’.¢’)I(IL.6’.¢’)sine’de’d¢’ (4-29)
4no o

where p represents the phase function.

In an emitting medium, the source function also depends
upon the local absolute temperature and emissivity. As
discussed in Chapter 3, however, emission within the media
studied in this research has not been observed. For this
reason, scattering and absorption are the only two
mechanisms dealt with in the treatment of the flash
penetration. At this point, the following assumptions are
made

1. The materials are considered grey. .All scattering
and absorption takes place independent of frequency.
In the type of experiment being studied, the
measurements are not spectrally sensitive since
temperature measurements are made using all
frequencies simultaneously.

2. The material will be considered isotropic. The
radiation transfer, therefore, will be one
dimensional. In this type of experiment, there are
no photon detectors which are angularly sensitive and
the azimuth dependence of the radiation, if any,

cannot be measured.

131
3. The duration of the flash will be assumed to be
instantaneous in comparison to the time scale of the
experiment. The temperature distribution inside the
sample will therefore will be treated as an initial

condition.

Using these assumptions, the transfer equation now becomes

1
dI(t.u) Qo / /
.———————=1' , ———- I , d -
p dt (I u) 2‘£ (I u) 11 (4 30)

where no is the albedo and u=cos(6) as discussed in Chapter

3. The solution of this equation is of the form

in
C%e

III Ill):
L l-wu

 

(4-31)

where C5 is an arbitrary constant and w is the root of the

following equation

o - 2v

0
ln(l+1lrj (4-32)

 

1'0

Table 4-9 shows some sample values of w for corresponding
values of no. Both positive and negative values of w
satisfy this condition. Since the solution of Equation 4-31
expands without bound as TL increases for values of positive

w, the negative values of w will be used.

132

ENKBLflill-Q

Values of w

_Qm. __fl;_
0 0 1.0
0.2 0.99991
0.4 0.98562
0.6 0.90733
0 8 0.71041
1 0 0.0
For moderate to low values of the albedo, the value of
w is near negative 1. Only for values of the albedo near
unity, where the dominant radiation interaction is

scattering, is the value of w small. In any case, since the
material is homogeneous, the distribution of the radiation
inside the material will be of a pure exponential form. For
cases of large albedo, and since the local angle of the
intensity is not a concern for the purposes of determining

the distribution of temperature, the solution for intensity

is
I(IL)=C1e'r‘ (4-33)
The extinction coefficient, as described in Chapter 3,
has units of inverse length and corresponds to the

reciprocal of the mean free path of a photon in the medium.

This is expressed mathematically as x=o+a where o is the

133
scattering coefficient and a is the absorption coefficient

Assuming that K is constant throughout the material, the

relationship becomes EFKX.

Using the boundary condition of I(x=0)=Io, where I0 is
the incident intensity, we have
I(x)=15e"‘ (4-34)

Considering now a differential element inside the

material, dx, the total deposition of energy into this

element by the radiation from the flash using the above

model is

t
q=flI(x, t’) -I(x+dx, t’)]dt’ (4—35)
0

where t represents the duration of the flash. This can also

be written as

t I
q=dI(x,t’)-(I(x.t’)+§-I—(;-'{-E—-)-dX) dt’ (4-36)
0

which in turn becomes

/
———aI("' t ’ dxdt’

q=- 6x (4-37)

0%.."

If the flash is taken as having constant intensity during

134
its very short duration, we have q=r<t1(x)dx and since I(x)

has already been defined as

I(x) =Ioe""‘ (4-38)

the energy deposited in the differential element becomes

q(x) =I0Kte“"dx (4-39)

The temperature rise resulting from this deposition of

energy is as follows

 

q
T = _
(x) pcpdx (4 40)

Substituting the equation for q(x) into this equation we

have

 

(4-41)

This equation is a closed form expression of the initial
temperature distribution following the instant of the flash.
Therefore, the parameter "a" being evaluated in Model 17 is
essentially the reciprocal of the extinction coefficient

since the initial temperature distributions were assumed to

be of the form.e**“’. In other words, the parameter a is

the mean free path of a photon in the material. In the case

cm'all of the experiments analyzed as part of this research,

135

this mean free path is expressed in millimeters.

WM

Each of the four principal models 1,4,5 and 17, is
utilized in the appendix tables for analysis of laboratory

data taken in various locations around the world on various

materials. Basic physical information is included on each

table, including the applied null value, which is the

nominal value of the temperature measuring instrument at

ambient temperature with no flash heating.

Models 4 and 5, the time shift model and the
exponential penetration model, respectively, are similar in

performance as shown by the standard deviation of the

residuals.

In utilizing the models on 16 experiments performed in

various laboratories throughout the world, the results seem

to agree with prior analyses performed by the three

experiments studied previously. In analyses performed by

other laboratories, heat loss was not accounted for in many
cases. For laboratories which did not account for heat
loss, the values presented here for diffusivity are lower
than the values reported by those laboratories. This is to

be expected since an analysis which does not account for

heat loss would interpret the temperature reaching an “early

136
peak" as being attributable to a higher diffusivity. The
laboratories which accounted for heat loss produced results
which were more closely aligned with the results of Model 1
presented here.

As a rule, Models 4 and 5, which account for flash
penetration, produced values for diffusivity which were even
lower than those produced by Model 1. This is also to be
expected since, with flash penetration, the temperature rise
on the measured surface comes more quickly than with no
penetration. A.model which does not account for penetration
would interpret this early rise as a higher diffusivity.

Model 17, which accounts for both surface penetration
and reflective heating on the measured side from the flash,
seems to produce values of diffusivity which are between the
values given by Model 1 and Models 4 and 5. This is
partially due to the fact that Model 17 allows for partial
penetration and Models 4 and 5 can deal with full
penetration only. The following experiments exhibited back-
side heating and were most appropriately modeled by Model 17

A;R1 Palaiseau, France
C_R1 West Lafayette, Indiana
L_Rl Poitiers, France

M_R1 Hunan, China

137

Q_R1 Manchester, United Kingdom
Oak Ridge, Tennessee (on Holometrix System)
in“; following experiments exhibited no significant back-side
heating but did involve flash penetration. They are most
appropriately modeled by Model 5

I_R1 Buenos Aires, Argentina

Oak Ridge, Tennessee (on Anter System)
The following experiments exhibited no appreciable back-side
flash or flash penetration and are best suited to be
analyzed using the three-parameter Model 1.

A;R2

Palaiseau, France

E R1

E R2

G_R1

H R1

J_Rl

K R1

N_R1

Vandoeuvre les Nancy, France

Vandoeuvre les Nancy, France
Bombay, India

Trappes, France
Talence, France
Belgrade, Yugoslavia

Stuttgart, Germany

S R1

Ardmore, Pennsylvania

One experiment exhibited such an extremely low value of heat
loss that it is best dealt with using a two-parameter model
involving only heat flux and diffusivity. This experiment
was C_R1 LeBarp, France.

There is a great deal of variation in the degree of

 

138
surface flash penetration from one sample to another.
Experiments performed on materials which were opaque to the
flash resulted in no appreciable surface transmissivity
whereas more transparent materials exhibited surface
transmissivities of 1. Some of this variability,
particularly the cases of partial surface absorption, came
about because some samples were treated with gold foil
coatings prior to running the experiments, for the expressed
purpose of minimizing flash absorption. The amount of gold
foil applied varied considerably between samples.
Deposition thicknesses and application methods can be
somewhat arbitrary.

For cases where little or no penetration occurred,
there was no advantage to utilizing the advanced models
since the parameters measuring penetration were estimated as
being very small in magnitude and the confidence regions for
these parameters were extremely large. Moreover, the values
estimated by the advanced models for diffusivity were
virtually unchanged from those estimated using Model 1, with
no improvement in the residuals. In these cases, the
advanced.models proved to be more of a liability than an
asset.

.Although the values estimated for diffusivity using the

advanced models for cases exhibiting penetration were not

139
radically different from those estimated using Model 1,
there is a higher degree of confidence in the accuracy of
the estimates. More importantly, the differences in
estimated diffusivity are large enough to affect design
considerations for applications of re-entry heat shields.
The basis of the higher degree of confidence is the removal

of a characteristic signature in the residuals, the lower

standard deviation of the residuals, the narrower confidence

regions and the stability of the sequential parameter

estimates.

CHAPTER 5

DERIVATIVE REGULAR]: ZATION

W
As explained in Chapter 3, the four-parameter internal
radiation model for the flash diffusivity problem is a very
ill-posed model when used in parameter estimation. This is
due to the strong correlation between kinetic conductivity
and radiative conductivity, making the two parameters nearly
indistinguishable from one another when trying to estimate
them individually, even when using errorless data. In many
test cases, a converged solution was not obtainable unless
the initial seed value of the parameters used was within one
percent of the known parameters using errorless data. Even
with an unreasonably large heat flux applied, which tends to
accentuate the non-linear nature of the radiant
conductivity, the most successful test did not exhibit any
stability with initial seed parameters differing from the
true values by greater than 10 percent.
Utilizing a method called derivative regularization,
devised as part of this research, parameter estimates were
obtained for the internal radiation cases as readily as for

the three parameter case using ordinary least squares. In

140

141
tests utilizing four parameter exact solution data files,
the method successfully converged on the correct solution
with initial parameter seed values of up to 200 percent
larger than the true parameter values. The same convergence
criteria were used as with ordinary least squares, whereby
each parameter was brought to within 0.1 percent of the
actual value used in the direct solution. The success in
analyzing this type of problem contrasts the un-aided least
squares method which would not bring about convergence
unless initial parameter estimates were within one percent
of the true values using errorless data. This method was
also used in analyzing the data measured at Oak Ridge
National Laboratory using the internal radiation model.

The ill-conditioned nature of this parameter estimation
problem is manifest in the final set of equations being
nearly singular in form. This unstable condition results in
extremely wild estimated parameters when attempting to solve
the equations. Various methods have been used in the past

to bring about stability in this calculation. Reference

[25] provides a template by which regularization may be
applied using prior information about an experiment and the
anticipated parameter values. Reference [27] addresses the
mollification method, by which instability can be mitigated

though a reduction in the measurement errors. Even with no

142

measurement errors in simulated experiments, however, the
estimation of internal radiation parameters has shown to be
highly unstable. By contrast, the type of instability which
derivative regularization attempts to reduce, is the
numerical singularity which comes about in the parameter
estimation equations.

As a means of adding stability to the system of
equations, derivative regularization is applied in the form
of a matrix pre-multiplier. Similar methods, normally

referred to as matrix pre-conditioning, have been used for

other ill-conditioned systems. Reference [62], a general

treatment of matrix computations, describes the application

of pre-conditioning. The use of a pre-conditioning method
is fairly straightforward, however the difficult and most
critical aspect of the this method is the development of the
pre-conditioning matrices for the specific type of problem.
As the name implies, derivative regularization employs the

time derivatives of the measured data and the sensitivity

coefficients in reducing the ill-posed nature of the

parameter estimating equations. This information is used in

developing the pre-conditioning matrices.

Section 5.2 of this chapter discusses the underlying
assumptions for the modifications made to ordinary least

squares in order to improve stability using this method.

———_—:—. 1 .— ... ‘1- 1,4,1.th.

 

143
Section 5.3 discusses a step in the development of
derivative regularization which was later refined and
improved. This method involved using single cubic splines
to approximate whole domain curves. Section 5.4 presents
the refined version of the spline approximation method using
piece-wise parabolic splines. Aspects of the bias and
covariance associated with this method are discussed in
Section 5.5. The results of the method are presented in
Section 5.6, including the magnitude of reduction achieved

in the condition number of the parameter estimation

equations.

 

One principle of the method of ordinary least squares
which is also employed in the derivative regularization
method is the assumption of locally linear sensitivity
coefficients. This means that the sensitivity coefficients
are assumed to vary linearly with respect to the parameters
within the narrow range of interest between the initial
guess for the parameter values and the actual values of the
parameters. For the flash diffusivity problem, studied in
this research, the sensitivity coefficients are non-linear.
For this reason, the parameter estimation process is

conducted in iterations, re-calculating the sensitivity

144
coefficients between each iteration. Within a single
iteration, however, the sensitivity coefficients can be

thought of as being linear.

This concept is employed in the following expression

presented in Chapter 2

. S: 6T (rt-+1) (kl
1:1 abj ‘1 j (5 1)

Expressed in words, the k+1 iteration temperature is the new
calculated temperature which will minimize the sum of
squares of the difference between the calculated and
measured temperatures. In graphical terms, if the
difference between T””” and T“” were plotted as a function
of time, it would be very much like a plot of the residuals
for the applicable iteration. This assumes that the T”””
curve is very close to the measured temperature curve. .As
the equation above states, this curve must be equal to the
sum of the sensitivity coefficient curves, each multiplied
by the respective change in its estimated parameter from the
previous iteration to the present. The approximation sign
is shown because the sensitivity coefficients are not truly
linear.

The method of derivative regularization takes this

concept one step further and requires that the time

145
derivatives of these curves be matched as well as the simple
magnitude of the temperature values themselves. In the case

of the first time derivative, we have

. ’,_ I (9T, (k+1) (k)
T” 1’ ~T‘k) +1231—abijj Tbj (5-2)

where the “prime” symbol designates the first derivative
with respect to time. Likewise we have

(k+1)”.. m” i: 3T” (k+li_ (kl)
T "T +j=1—a-g; j bi (5‘3)

where the “double prime” symbol designates the second
derivative with respect to time. Using derivative
regularization, a set of b,”‘*“ parameters is estimated which
minimizes the difference between not only the measured and
calculated temperatures, but their time derivatives as well,
in least squares fashion. In this way, additional
information regarding the shape of these curves is employed

in bringing about stability.

WW
Several methods were attempted, as part of this
research, in an effort to minimize the effects of the

measurement errors and estimate the derivatives of the

146
measured data and the direct solution. At first, multiple
cubic splines were used to approximate the curves of the
sensitivity coefficients, the calculated temperature and the
measured temperature. These splines, spanning approximately

10 data points each, were defined as being piecewise

continuous with continuity in the first derivative at each

end of each spline. These spline junctions are also known
as “knots”. At first, the splines were calculated
sequentially. At the right hand side of the first spline,
i.e. the first "knot", the position and slope of the spline
end were used as boundary conditions for the position and
slope of the left end of the second spline. This method
proved to be unworkable when used sequentially because the
slopes at the end of each spline tended to become too wild,
causing extremely large deviations of the spline from the
data being approximated.

As a second attempt at approximating first and second
derivatives of the residual curve and the sensitivity
coefficient curves, one cubic spline was used for each curve

over the entire time domain of the experiment. Each of

these curves was expressed as a polynomial with four
constants. The constants were determined via least squares
analysis of the applicable data points. In the particular

case of four unknown parameters, five equations can be

147
written as follows:
6 (t) =aot3+bot2+cot+do
X1 (t) =a1t3+b1t2+c1t+d1
x2 (t) =a2t3+b2t2+c2t+d2 (5-4)
x3 (t) =a3t3+b3t2+c3t+d3
X4 (t) =a4t3+b4t2+c4t+d4
where Xi(t) refers to the four sensitivity coefficient
curves and e(t) designates the measured minus the calculated
temperature. Examples of these approximations are shown in
Figures 5-1 through 5-3. Figure 5-1 is a plot of the
residual curve when comparing the direct solution from the
previous iteration to the measured data, along with its
cubic spline approximation. Figures 5-2 and 5-3 each show
two of the four sensitivity coefficient curves along with
their respective spline approximations.
In order to find the parameters from these curves, the
principle used is that the four sensitivity coefficient

curves should all sum to the residual curve when multiplied

by their respective values of bi‘km-bi‘k’. As a shorthand

notation, we may define Abi=bi“‘*“-bi”". In equation form, we

have

e (t) =Ab1X1 (t) +Ab2X2 (t) +Ab3X3(t)+Ab.,X4 (t) (5-5)
Since this is only one equation with four unknowns,

(Ab1,Ab2,Ab3,Ab4) the equations of the cubic splines are used

 

148

+ Peddds —9— spline

 

 

 

 

 

 

 

[16
E 2 0""
:5 :12
E t
r 5 “‘

e

o I" {12
Z

{14

0 [HS [ll 015 (12
Nan-Dnensiond Tine

 

From 5-1 Spline Approximation of Residual Curve

 

r/r +Dflrsiwty +sdire +i-bdFlut —‘3——3:line

)

 

 

 

 

 

 

 

 

/ a6 1 /..__..._r I '*'—'-' ' ‘I—-
- /".
/ E a ‘14" ’ '
.5. a \i‘“
C E 0.2 \u
a . \ ,.
-- g o-" . -- 4‘“
j/ e O ' k‘ffi" 15"—
. za ’- {12-
/ {14L
0 ELIE 01 [115 (12
hut-Unansiond Tune
1 Figures-2 SMMWWMWCWWJ‘

 

149

 

 

 

 

 

 

l
+ kr/k —9~— Spline —'— Biot —D—- Spline
Number
0.6
05‘
'a 04-»
.5 e 0.3» ___;‘__0_
g g 0.2 0 ‘"“/ fl E\.a\f
E a 0.1» / T“ a
el- '0-1“ ' ' ' - .
Z '02 " i T I . .
-o.3- ' '
-0.4
0 0.05 0.1 0.15 0.2
Non-Dimensional Time

 

Figure 5-3 Spline Approximations of Sensitivity Coefficients

to generate four equations, namely

ao=Ab1a1+Ab2a2+Ab3a3+Ab4a4

bo=Ab1b1+Ab2b2+Ab3b3+Ab4b4

co=Ab1C1+Ab2C2+Ab3C3+Ab4C4 (5-6)

do=Ab1d1+Ab2d2+Ab3d3+Ab4d4
The only unknowns in these equations are ac, be, cc, and do.

In practice, this method was found to be ineffectual due to

the poor approximation of the sensitivity coefficients made

by the splines. Moreover, there was no observed improvement

in the stability of the parameter estimation problem brought

about by the use of the spline approximations. An improved

approximation of the curves was achieved by using one

piecewise spline for each measurement point. The whole

 
   
 

.. -.—_.— —— —--I|-—-——-—— n; -l-

150
domain cubic spline method has been presented here strictly
in order to chronicle the development of the derivative
regularization method, since the final derivative

regularization method is built on the same principles.

W
A more successful approximation of the curves of the
measurements, the sensitivity coefficients and their
derivatives, is a set of parabolic splines, with one spline
defined for each time step. Each spline in this method is
centered at an individual data point in the curve. Normally
seven points are used to define each spline, in which case,
the first three and last three data points from the measured
data are not used, except in obtaining the splines for the
neighboring points. For a typical temperature measurement
Y;- the spline approximation for the measurement, designated
11, is of the form
11“” = Aitz + Bit + C1 (5‘7)
with the first derivative expressed as
Xi'(t) = 2Ait + Bi (5-8)
and the second derivative as
L"(t)= 2A1 (5-9)
For the sake of clean calculations, the center point of each

spline, the point of interest, is assigned the arbitrary

 

151
time value of t=0 and the domain of the spline is then
-32t23. Considering the above equations at t=0, the values

of the functions and their derivatives become

11(0) = C1
11'(O) = Bi (5-10)
X1"(0)= 2A;

In order to find the coefficients A, Bi, and C1, the
method of ordinary least squares is utilized. Using seven
points as described above, the over-defined set of equations

for the spline centered at I=4, for example, is

 

 

 

 

 

cf 1:1 1
Y1
2
t2 t:2 1 Y2
t32 t3 1 A4 Y3
2
t, t:4 1 B. = Y. (5-11)
t3 t5 1 C4 Y5
Y
t2 t6 1 6
2 Y7
t7 1:7 1

This can be expressed in compact form as [1:] [s]=[y] and the
least squares values of [s] can be found by pre-multiplying
both sides by [t]T. We then have

[tITItI [Bi=[t1TIY] (5-12)

From this equation, the values of A4, B4, and C4, are

152
calculated, expressed in terms of the components of [t] and
[y] only. The value of 14, the point of interest, is equal
to C4. Since the time steps are evenly spaced, they can be
expressed as multiples of each other and divided out. This
leaves 1., to be expressed in terms of the seven ifi values

alone. Constants designated with the letter m can be found

such that

which can also be put in matrix form as

 

 

 

 

 

Y1
L 7111 m2 m3m4 m5 mm, 0 0 0Y2
E5 0 m1 m2 m3 mym5 msm7 0 013
1’5 = 0 0 11111222 1123 m4m5m6m, 0Y4 (5-14)
3 : : : : : :Ys
139-ii 0 0 0 m1m2m3m4m5 m6 171,:
Y,“

Equation 5-14 can be written this way since the mi arguments
are not functions of Y1 and are the same for each spline.
Note again that the first 3 and last 3 data points are lost
because they are needed to form the first and last splines;
hence the points are only approximated by splines from
points 4 to n-3. Equation 5-14 can also be written in

compact form as

I¥]=[Mo] [Y] (5-15)

153
Similarly, approximations for the derivative values are
[x' i=[M.i [y]
IX"I=IMzI [y] (5-16)
Likewise, for the curves of the calculated values of
temperature we have
[Tl=[Mo] ['1']
[I']=[“1] ['1'] (5-17)
[1"]=IM2] ['1']
and for the curves of the sensitivity coefficients we have
[X]=[No] [X]
[3']=[M;] [1] (5-18)
[8"]=[Mz] [X]
Exact values for the individual terms of the Mo, MI, and It,
matrices for parabolic splines over seven points are shown
in Table 5-1
As a note of interpretation, the first and last
coefficients of the Mo matrix are negative due to the
parabolic shape of the spline. Since the coefficients shown
affect only the center point, low measurement values at the
end points would tend to make a higher center point of the
spline approximation and vise-versa. This allows the
parabolic spline to address the end points by affecting the

curvature of the spline in order to cause the spline ends to

154

Table 5-1

Parabolic Spline Matrix Terms

Individual For Mo For Ml For M,
Suuma Matrix Matrix Matrix
m1 -2/21 -3/28 5/84

m2 1/7 -1/14 0

m3 2/7 -l/28 -1/28
m4 1/3 0 -1/21
m5 2/7 1/28 -1/28
m6 1/7 1/14 0
m7 -2/21 3/28 5/84

pass closer to the end measurements. In this same vein,
note that the end points provide the largest contribution to
the second derivative of the spline.

We now define new vectors, [T], [1"] and [X‘] where wl
and w§.are weighting coefficients in Equation 5-19. Another
means of defining these vectors is to use a composite
.matrix, which.incorporates all three of the spline matrices
[Mo], [34,] and [34,]. This matrix is designated as [no] and

can be used as a direct conversion from the [Y] and ['1']
vectors to the [r] and [r] and vectors, respectively. The

MC matrix is defined in Equation 5-20.

 

 

 

 

 

 

 

L L is
Z: I: 5:
{a 3:. 55
Y _3 T _3 X -3
w1Y4, "13.4: "1951/
"13.5: "125.: "1355:.
I I I
[1" = "135 III = "132 Ix‘l = "128; (5-19)
[w / x I
119:; Til Tia-3 TQM
// // I/
"239. ”2.1.". "255.
"215/: "2.15:, "25:
“72!: ‘«2.2g: WZEGZ
// I/ //
F252] ["2 Tn-3' lwzxn-ai
n.
u, = win; <5—20)
W2":

 

The conversion using [Mn] is accomplished using

[Tl-’0‘.) [Y]

[r1=m,1 ['r] (5-21)
Similarly, the [1] matrix can be converted to the [r]

matrix by the conversion

156
[r]=[M.] [X] (5-22)
It should be noted that the dimensions of the [Y] and ['1‘]

vectors are dimensionally n x l where n is the number of

measurements. The [T] and [T] vectors are dimensionally

3m x l where m=n—6. The reason for the difference of the 6

points between m and n is that the first and last three
points are lost because they are ends of a 7 point spline.
These vectors contain 3m terms because there is a filtered

point, a first derivative approximation and a second

derivative approximation for each point.

The [1] matrix, the sensitivity coefficient matrix,
undergoes a similar dimensional transformation to that

experienced by the [Y] and [T] vectors in the derivative

regularization process. The [X] matrix is dimensionally

n x p, where p is the number of parameters, and the [3*]

matrix is 3m x p. The [no], [31,], and [31,] matrices are each

of dimension m x n. If the individual terms of the [Mo]

matrix are denoted as mow, the [11,] matrix terms are denoted

as ml,n and the [31,] matrix terms are denoted as m2,“ then

the [33,] matrix in expanded form is written as

mam 0.2 1110.3 [no.4 mo,s mO,6 mom 0 0 0 0
0 0,1 mo,2 m0,3 mo,4 "13.5 m0,6 mom ° ° ‘ O 0 O
0 0 0,1 mo,2 mo,3 “70,4 mo,s "10,5 ' O 0 O
O O 0 ”0,1 ”0,2 mo,3 ”0,4 "70,5 . 0 O 0
0 O O 0 ”0,1 In”2 1110'3 ”0,4 . . O O O
o o o o o mo,1 mm 1110.3 . . . o o o
o o o o o o o o .. 1110', o o
o o o o o o o o . mo 6 mo,7 o
o o o o o o o o mo,5 1110's mm,
wlml,l “'1th w1m1,3 "1mm "1311.5 w1m1,6 wlmlfl O 0 0 0
0 w1115.1 w1"‘1,2 “Emma "1m1,4 w1m1,5 "11111.6 wlm'l," 0 0 0
O 0 w1"‘1,1 w1"‘1,2 w1""1,3 "1m1,4 "135,5 w1m1,6 0 0 0
0 0 0 w1m1,1 w1115.2 "1mm Vim-1.4 w11111.5 0 0 0
0 O O 0 wlml'1 wlmm wluzl'3 witnl'4 O 0 0
O O 0 0 O witni'1 "11111.2 "1%,: O 0 O
o o o o o o o o wlml', o o
0 0 0 0 0 0 0 O 1mm; witnl,7 0

0 O O 0 0 0 0 O 11111,5 "1mm. witnl'7
wzmzu w2"‘z,2 W2m2,3 w2m2,4 w2m2,s w2“’2,6 wzmzn O O 0 0
0 w2‘"2,1 w2m2,2 w2%,:5 w21112.4 "235,5 w2m2,6 w21112.7 0 0 0
O O w2"‘2,1 w2m2,2 W2m2,3 w2m2,4 w2mz,5 w2115.5 O 0 0
0 0 0 wzmz,1 "2115.2 wzmza wzmzu "2115.5 O O 0
0 0 0 0 "2m2,1 w2mm w2%,3 w2m2,4 0 0 O
0 O 0 0 O wzmz'1 wzmz'2 wzzuz'3 0 0 O
o o o o o o o o 21:12,, 0 o
o o o o o o 2:212,6 wzmz', o

O O 0 0 0 0 21112.5 W2m2,5 wzmz7

(5-23)

Now that the [T] and [1"] vectors have been defined,
with respect to the parameters, the modified sum of squares

function

158

3:5 (Yg-Tgfi (5-24)

i=4
can be written in matrix form as

s=[r"- 1"']T [r- 1"] (5-25)

By the same means as shown in Chapter 2, the parameters can

be found under this method as follows

b (k+1) =b (k) + (X" (I)!x(t) ) -lx~(l)!( r“-r' (3)) (2—26)

This equation has the same form as the conventional least
squares method outlined in Chapter 2 except that the

matrices involved have been treated with the pre-

conditioning matrix.

At this point it should be noted that the term it“: can
be rewritten as

x" X“ = ugm’ugm whammy) wingmrugx) (5-27)
which in turn can be rewritten as
x" x“ = x’m’uox + wfx ’16qu + wfx 'xfxtx (5-28)
Factoring out the 1: matrices we have
x" X” = x'mo’tg + wf’ufng + wfufngm (5-29)

For the sake of compact notation, if we define

159
R = no" + wqu’ng + ”22%,“; (5-30)
Then we can use the shorthand notation of

Jr'x“ = x’n 1: (5-31)

This notation can also be used in the definition of the 8

term being minimized as described above

S=[Y'-1’"]T[Y'-T'] = [Y-l'] TINY-1'] (5-32)

Finally, the equation for the estimated parameters can be

rewritten as follows

b(k+1’ =b (3) + (x" (t) It (3’ ) -1x“(hf(r~_r~(l))
(5-33)
:b (k)... (xCHfR x‘”) -1 xCfl!R(r_r(n)

In order to examine the nature of the "R" matrix, it is
useful to generate a sample of the HOMO, M159, and um,
matrices that make up the "RP matrix. These are each n x n
matrices, as is "R", where n is the number of measurements.
In Tables 5-2 through 5-4, three 16 x 16 examples are shown,
one for each of the three matrices noted above. As can be
seen from these examples, each matrix is diagonally
dominant, is repetitive along the diagonal and symmetrical
about the diagonal. The repeating portion along the

diagonal will be the same for any matrix 13 x 13 or larger.

 

160

3989.8 88.0.. 8. no. ...-Sm: in: mum 39m...

 

     

 

 

 

 

 

 

 

 

 

      

 

 

 

 

 

 

 

 

 

 

888.8% 8 88.888- 8.8.... $8...- 8 8 8 8 8 8 8 8 8

$88.. $88. $88.. $88.8 88.88...- $82. $88... 8.8...- 8 8 8 8 8 8 8 8

$8881.98? $8... 8.88.. 88.8.8. 88.8.8- 8...88- $88...- $8...- 8 8 8 8 8 8 8

$888 $88.. 8.8... 8.88. $88.. $8.... 8.88.8. $88... $8...- 8 8 8 8 8 8

$8I8 8- 88.88..- 88.88.. $88.. 8.88.. 8.88.8 $88... 8.8.8. 8.8.8.? $88..- $8E7 8 8 8 8 8

$88..- 88.8$ $8.8- 88.88.. 8.888 8.88 8.88.. $82 $8....- 8...8.8. $88..- $8...- 8 8 8 8

$8...- $88.- 8.88.8- $8..8. $88.8 $88.. $888 8.88.. $82 $8.... 8...8.8- 8.8....- $8...- 8 .I8 8

8 $8...- $88.- 8...8.8. 88.8.8. $82 8.88.. 8.88.8 $88.. $82 8.8....- 8.888. $8.....- $8.. - 8 8

8.. 8 $8...- $88..- 8.88.8. $8...-H 88.82 8.88.. 8.88.8 $88,. $82 88.8.8. 8.88.9 $88..- $8...- 8
8 8 8 $8...- $88..- 8.888. 88.8....- 8.82 8-88. 8.88.8 $88.. $82 88.8.8. 8.8.8.8- $m88..- $8...-
b 8 8 8 $8...- $88..- 8.88.8. $82. 8882 8.88.. 88888 8.888 88.88.. 8.8.? 88-88.8- $88..-
8 8 8 8 8 $8...- $88..- 8.88.? 8.8.? $82 8.88.8 $88.. $88.. $88.. 88.88..- $888.
8 8 8 8 8 8 $8...- $88..- .8...8$lm.-Ium..-W $88W888. $8... $88L 8.88.8 8 -
8 8 8 8 8 8 8 8.8...- $88..- 8.88.8- $8.8. 88.88.. 8.88.. 8.8... 8.88.. 88.888
81. 8 8 8 8 8 8 8 W888..- $8m.- 88.888. 88.88..w-.8..w8]8.-8II$88.. 8.88.. 88.88..
8 8 8 8 8 8 8 8 8 8.8.... $88..- 88-888- 8 8.88.8 8.88.. 8.8....

3989.8. 88.0.. 8. no. xflumz 02...: mam 8.38....

$88.. M888..- $8...- $8.8- 8-8...- $88..- $888 8 8 8 8 8 8 8 8 8

$88..- $88.. $8.. $88.. $88.8 $88.? $8...- 8.88.8 8 8 8 8 8 8 8 8

$mm... ......8 838.8 $88.8 $88.8 8.8.8- 88.... 88.888 8 8 8 8 8 8 8

83.8.8 ......8 .88....8 8888.8 $88.. 8.88.8- $8...- 88.u.8.8 8 8 8 8 8 8

. $88.8 .88....8 88888.8 88188.8 83.8.8 8.88.. $88.8- $8...- 88.888 8 8 8 8 8

8.83 88:8 8888 88.8.88 8883.8 82...; $88.. $88.8- $8...- 88.888 8 8 8 8

$8.. 8- 8.88.. 8:8... 8883.8 8888.8 8883.8 8:88 8.88.. 8.88.8- $8. .- 88.888 8 8 8

8 $88.8 $8. .- $88. 8.88.. 83.88 88888.8 8888.8 8883.8 88...; 8.88.. 8.88.8- 88..- 8.88.8 8 8

.8 8 $88.8 $8...- $888- 8.88. 88...; 88888.8 88888.8 8883.8 838.8 $88. $88 8- $8...- 88.888 8
8 8 8 $88.8 8.8...- $88 8- 8.88.. 83....8 8883.8 8888888 88838 838.8 $88.. $88.8- 8.8...- 8.888
8 8 8 8 8.88.8 $8. .- 8.88.8- 8.88.. 838.8 88888.8 88.8.8.8 8888.8 888:8 8.83 8.88.8. 8.88..-
8 8 8 8 8 $888 $8...- $8...8- $88.. 8:...8 8888.8 8888.8 .88...8 .8888 88.888 8.8...-
8 8 8 8 8 8 8.88.8 8.8...- $8_..8- $88.. 88:8 88:8 ......8 8.8.].w8 8.88.. 8.8.8-
8 8 8 8 8 8 8 8.88.8 $8...- 8.88.8. 8.88... .8888 888.88 ......8 8.8... $8...-
8 8 8 8 8 8 8 8 $88.8 $8...- $88.8- 88-888 $88.. $8... $88.. $88..-
8 8 8 8 8 8 8 8 8 88.888 8.88... 8.8...- $8 ..8- $8...- 8.88..- 88.888

 

 

 

 
 

 

 

 

161

3988.8 8508 8. no... x838. .86.: 78. 3am...

 

 

 

 

 

 

 

 

 

 

 

$88.8 8 8.88..- 88-88..- 88.8...- 8 8.88.8 8 8 8 8 8 8 8 8 8
8 8.88.8 8 88.8...- 88.88..- 88.8...- 8 88.888 8 8 8 8 8 8 8 8
88.8...- 8 88-888 88... 8.88.8. $88..- $8...- 8 88.888 8 8 8 8 8 8 8
$88..- 88.8...- 88.8... 88.88.. 8-88.8 8.88.8- 88-88...-_- 8.8.... 8 8.88.8 8 8 8 8 8 8
$8...- $88..- 8.88.8- $828 $82 $8.88 88.88..- 88-88.8_. 8.8....- 8 88.888 8 8 8 8 8
8 $8...- 88.88..- 8.8.8.8- 888 8.82 88.88 8.88... 8.88.8- 888...- 8 88.8.88 8 8 8 8
88.888 8 $8...- $88.8- $88..- $88 $8... 88.8.88 88.88..- 88.88....- $8..v 8 $88.8 8 8 8
8 88.888 8 8.8....- 88-82. 8.88... 88.888 8.8... $88 88.88..- 88.88.8- 88.8....- 8 88-888 8 8
8 8 88.888 8 88.8.8- 88.88.8- 88.88..- 88.8...8 8.8 ... $8.88 88.88..- 88.88.8- 88.8....- 8 88.888 8
_8 8 8 $888 8 8.8.... 88.88.8- 88.88..- 88.88 8.8... $88 88-88..- 89888- 88.8....- 8 $888
8 8 8 8 88.8.48 8 88.8...- 88.88.... $88..- $82 $88.8 8.88.8 8.82- 88.88... 88.8...- 8
—8 8 8 8 8 8.88.8 8 88.8...- 8.88.8.- 88.88..- $88 $82 88.8.8 88.88.8- 88.88..- 8.8...-
8 8 8 8 8 8 88.888 8 88.8....- 88-82. 8.88.8. $88 $88.. 88.8... 8.8...- $88..-
—8 8 8 8 8 8 8 88.888 8 88.8.? 8.88... 8.88.8. 88.8... 88.88.. 8 $8...-
8 8 8 8 8 8 8 8 8.88.8 8 88.8...- 88.88..- 88.8...- 8 88.888 8
E 8 8 8 8 8 8 8 8 88.888 8 88.8.... 88.88..- 881.88...- 8 88.888

 

 

162
For virtually all experiments in parameter estimation, there
are many more than 13 measurements over the time of the
experiment.

Several other items of note related to the three
matricies exhibited in these figures include the existence
of both positive and negative values and the sums of the
interior rows. Since the factored component matricies of
each of the three product matricies contain both positive
and negative values, it is natural that the product
matricies also contain numbers of both signs as brought
about in the various cross products. Another similarity of
the three product matricies to the factored component
matricies is that the interior rows of the “0" matrices sum
to 1 and the interior rows of the “1" and “2" matrices sum

to zero .

5i§__lfllflLlflfiLJZBalfifluflfll

In order to determine if bias exists, we find the
expected value of b””“ as calculated using Equation (2-26),
and determine if it is equal to the expected value of the
true parameter vector, B. As the non-linear iteration
process nears convergence, the estimated parameter vector
becomes very close to the true parameter vector. As such,

the estimated parameter vector can be approximated as

163

1:“""’==[3+(X’RX)‘1 x’mr—r) (5-34)

where all non-superscripted matrices are assumed to be of

the k iteration. As such, the expected value of b is

E(b)z B + (x'nxrl x’mmr—m] (5-35)

where I-T is equal to 3. Assuming that the errors are
additive and have zero mean, we can say that the expected
value of e is zero. Substituting this into Equation (5-35),
the expected value of b becomes equal to B. This shows that
there is no bias associated with the method, which is
fortunate, since all commonly used regularization methods
come at the price of bias.

Next it is desired to calculate the covariance matrix

for the estimated parameter vector b. This is defined as

covw) =3 [(19-8) (b-B) ’] (5-36)

where B is the true but unknown vector of parameters. Since
the measurement errors are additive, we have

Y=T+e (5-37)
where e is the error vector. As we approach convergence
using the non-linear least squares method, we expect the
differences between calculated temperature values from one

iteration to the next to be on the order of the measurement

164
errors. If this is the case, we can approximate the error

vector as
Xb“*“=2b""+e (5-38)
We now define
n=(m)'1 n (5-39)
then AX=I. If we pre-multiply both sides of Equation 5-38
by A-we have
b“"’zb“"+he or b‘””-b""=he (5-40)
Substituting this into the definition of covariance given
above we have
cov(b“‘“’)=E[ Me) (“VJ (5'41)
Using the covariance matrix definition of
w=E[e€T] (5-42)
we now have
cov(b"‘“’)= MIN (5-43)
Writing this in expanded form and understanding the b vector
to be of the final iteration we have
cov(b)= (m)‘1mRX(M)’1 (5-44)
'This is a p x p matrix where p is the number of parameters.
'The w matrix is an n x n matrix where n is the number of
measurements. For simplicity, the w matrix can be modeled
as
w = 02m (5-45)

VHnere o2 is the variance of the measurement errors.

165

Using this assumption to compare the covariance of the
regularized parameters with that of the non-regularized
parameters, two models of the sensitivity coefficients were
used as a means of comparison. The cubic model utilized a
simple cubic curve of

“NH = B.t3 + thz + Bat + B. (5-46)

as the assumed temperature measurement curve with BU.[b, B3
and [34 being the four unknown parameters. Using a sample
case of 16 equally spaced measurements from t=0.1 to t=1.6,

the covariance matrix of b for this example is

1.662088 -7.49542 9.065934 -3.20513
-7.49542 40.5623 -53.218 19.70266
9.065934 -53.218 73.44436 -28.1152
-3.20513 19.70266 -28.1152 11.02555

For the regularized case, the covariance matrix is

3.073613 -13.1621 15.62973 -5.47699
-13.l621 64.92957 -83.1152 30.63121
15.62973 -83.1152 111.7891 -42.6842
-5.47699 30.63121 -42.6842 16.73889

.As can be seen from comparing these two covariance matrices,
the regularized case has higher values by 50 to 80 percent
when compared to the non-regularized case. This indicates
that the regularization method will give larger errors in

‘the parameters than will the ordinary least squares method.

166

This is to be expected since the Gauss—Markov Theorem, as
given in reference [25], states that the ordinary least
squares method will provide the minimum covariance matrix
for parameter estimation, assuming the standard statistical
assumptions, as given by that reference, are valid for the
measurement errors.

Another set of covariance matrices is compared, using
100 points from the internal radiation model presented in
Chapter 3. The covariance matrix for the ordinary least
squares case for this model, using the thermal parameters

a=1, qo=1, Bi=1 and 18./180.1, is

7648816 125557.5 -7655539 -8.2E+07
125557.5 2117.377 -125607 -1344766
-7655539 -125607 7662336 81976237
-8.ZE+07 -1344766 81976237 8.77E+08

Comparing this once again to the case where derivative

regularization is used, the covariance matrix is as follows:

13654443 223716.7 -1.37E+07 -1.46E+08
223716.7 3748.345 -223818 -2395863
-1.37E+07 -223818 13678306 1.46E+08
-l.46E+08 -2395863 1.46E+08 1.57E+09

Once again, these values fall somewhere between 50 and 80
percent larger than the covariance for the non-regularized

case, indicating that the errors generated using the

167
derivative regularization method are expected to be larger
than those generated from using ordinary least squares.
Both of these covariance matrices are significantly larger
than in the previous sample problem, indicating that the
errors expected in estimating the parameters from the
internal radiation test case are significantly larger than
those generated from the cubic model.

.As a Monte Carlo test of this tendency, 100 simulated
experiments were conducted by imposing errors of zero mean
and gaussian distribution on a file of an exact direct
solution. The true parameter values of the exact solution
were unity for diffusivity, Biot Number and heat flux. The
standard deviation of the imposed errors was approximately
0.01 and the peak temperature reached was approximately 0.4
non-dimensional temperature units. The exact solution was
generated for a simple flash experiment assuming no
penetration of the flash. Figure 5-4 shows three
distribution curves of the results for the 100 test cases.
This graph illustrates the unbiased nature of ordinary least
squares and the sharp resolution rendered for each of the
parameters. With errors larger than two percent of peak
Ineasured temperature, there were essentially no cases with
errors in the estimated parameters larger than 5 percent.

Figure 5-5 is a representation of the same test

168

 

"—{*“'flw*ll ——**—'Ekxhhnflnw "**——l+uanu:

 

 

4L
T

 

 

Numhat of Occurrences
:3 012515 8313 $35315 5355
\

...J/ \

Paranoia! Value

0.5 0.9 05 1.1 1.15

 

 

 

FbuuSALMawaCahflunuuthgOnfiunlsunSqmnn

TABLE 5-5

Monte Carlo Results Comparing Derivative
Regularization to Ordinary Least Squares

Wad

Mean Diffusivity 0.999775 1.00027
Mean Biot Number 1.00091 0.99853
Mean Heat Flux 1.000885 0.999075
Diffusivity Std Dev 0.005521 0.013181
Biot Number Std Dev 0.01127 0.037045
Heat Flux Std Dev 0.007924 0.025711

[performed using derivative regularization. The same
tanbiased feature of this method is apparent as evidenced by

the symmetrical appearance of the curves about unity. The

169

 

 

    
  

 

 

 

——4*—‘Auwa '——*—-Emahhnber'——*-—liuanx
m 4 8
45*r
a
8 w ... 1..
E 35. .
g a). 1
c>25«-
'g a, .. .-
.E 15 .~
5210*-
5 ..
O
Ofi 1 15
Parameter Value

 

 

 

HmmassIkmMKkMoRmnMsUflmgDuwmweRqammumaw

resolution provided by this method, however, is clearly
inferior to ordinary least squares as evidenced by the wider
distribution of estimated parameters. Table 5-5 further
contrasts the difference between the two methods by
comparing the standard deviations of the estimated
parameters. The larger standard deviations associated with
the derivative regularization method, reaffirm the cost of
reduced accuracy in using the method. The unbiased nature
(bf the method is illustrated by noting the average estimated

LDarameters and their close proximity to the true parameter

\ralues of unity.

170

Interestingly, a comparison of the residuals using both
methods suggests almost identical performance. The average
value of the standard deviation of the residuals is 0.00402
for both methods. This fact might give the impression to
the user of the method that the results from one method are
just as accurate as the other. As can be seen in the
figures and tables associated with this test, however, this

is not so.

W

Although the errors associated with the parameter
estimates using derivative regularization are expected to be
larger than those generated using the method of ordinary
least squares, the advantage of the derivative
regularization method lies in the reduction of the condition
number of the 1’! matrix. In many cases in parameter
estimation problems, the 1"! matrix is so ill-conditioned
that no parameters can be calculated; the numerical results
become nonsense. It is under these conditions that
derivative regularization becomes profitable.

The condition number is a measure of the stability of a
set of linear equations. As stated in reference [26], the
<:ondition number provides a measure of how reliably the

relative residual of an approximate solution reflects the

171

relative error of the approximate solution. The lowest
possible condition number is 1, which is the condition
number for the identity matrix. The means of calculation of
the condition number is somewhat subjective, hinging on the
method used in calculating the matrix norm. The definition
of condition number given in reference [26] is the product
of the norm of the matrix multiplied by the norm of the
inverse of the matrix. The condition number of a general
quare matrix A would therefore be

Condition Number = IIAJI lLAdll (5-47)

where the norm of the matrix A is defined as

__JL_. (5-48)
.1

In this definition, the i subscripts designate any row of
the matrix A. The norm of one vector x as a function of the
individual elements of the vector is
I]!!! = (lxlln +llen +IX3|n +...+I&J“)““ (5-49)

Since n can be any whole number (1,2,3,...,w), the most
commonly utilized norms are the 1,2 and infinity norms. For
the sake of convenience, the actual routine used to
calculate condition number in the subsequent graphs is found
in the commercial mathematical product MATLAB. The results
lasing this method are comparable to the other methods of

calculating condition number, particularly when comparing

172
regularized to non-regularized cases.

The condition numbers for the test cases noted above
were found to be sharply reduced by the use of derivative
regularization over a wide range of weighing factors.
Reductions on the order of approximately 50 are attainable.
This allows calculations to be performed to solve the

equation

1) (1+1) =b (k) ... (x~(k)!R x“) ) -1x~(kuR( 1'”-1'"”) (5-50)

for the parameter vector b when otherwise no solution is

obtainable from the non-regularized equation

b (k+1) 3b (k) + (x~(k)fx(k) ) ~1x~(k)1'( r~-r~(”) (5—51)

Figures 5-6 through 5-9 are plots of the condition
numbers for various sample problems. These graphs show the
effect of the weighting factors, w1 and w2 on the “condition
number ratio” which is defined in these examples as the
ratio of the condition number of the m matrix to the
condition number of the x’x matrix. Figure 5-6 is an
example using 16 simulated data points from a cubic model
from the equation

Y=b1t3+b2t2+b3t+b4 (5-52)
with bffih=bffix=l as the parameters. The time scale used is

from 0 to 1.6 seconds. Figure 5-7 is an example of the same

173

 

“"“”WH=O _‘£F“‘Vflbfl00 "4F"‘VWEQDO ’—{*—"VM¢iDO

 

 

 

 

 

 

 

01 “i /

.o I
3 0.08
E
E 008
E I
S 004 -
:e " ...—v"
E 0.02 1 K —"—"
8

o 4 L i .

0 50 100 150 am)
MhfighfingiauxorHflBOGOOO)

 

 

 

Fnue54!CammhnhMmbufihfloflxCMfichnunmu16Pdmb

model using 100 points over the same time domain as the 16
point example. Larger weighting factors must be used in
order to bring about the same reduction in condition number
for this case, but approximately the same magnitude of
reduction in condition number is obtained. Figure 5-8 plots
the same factors for the internal radiation model described
in Chapter 3. This figure also is taken from a sample
direct solution utilizing 16 time step points. Finally,
Figure 5-9 is the same as Figure 5-8 except that 100 points
are used instead of 16. The basic shape of these curves is
the same as that of Figures 5-8 and 5-9 with the same
approximate reduction in condition number. Once again, the

examples which utilized more time steps required larger

174

 

"—4'—"V~0=0 ‘“—*“——‘VNW-1000 '——4*——‘VV1-2000 '“—{}“‘VN1-5000

 

 

 

 

01 1
.° ,//////
g 000 /,/
g 006 " /
5 if. ////'//
z 41 I
g 004 . ,//'
:e _"___________,__..
3 002 ‘8...—
8
0 t : t l
0 500 1000 1500 2000

Weighting Factor W2 (x1000)

 

 

 

Fme5J(kmdflmhMmUukadflHflnoUflerDPde

weighting factors in order to bring about the same reduction
in condition number.

The results of the derivative regularization method
have also shown to be successful in obtaining convergence in
comparison to the ordinary least squares estimation schemes
used previously. The decrease in the ill-conditioned nature
of the matrices has facilitated the calculation of answers
which were previously unavailable. As a first attempt at
using the method, the results shown in Table 5-6 were
achieved in seven iterations.

Following this test of initial values 20 percent above

the actual values, a similar test was run with initial

[V

 

175

 

+W1-0 +W1-100 +W1-500 +W1-5000

 

 

 

 

 

 

01 V

a i
£0.00!|
-§ 006 fl
5 . q \I //*

] . ' _..-"'“
g 004 1' \fl . {J
'u
c 002~
8

0 ‘- i 1
0 500 1000 1500 2000
Weighting Factor W2 (x1000)

 

 

 

Fumesa(kmdumhMmhuihfiHUWMumflRmfifiauunemwfi169mm:

values 50 percent above parameter actual values. This test
did not converge. As an additional attempt at stabilizing
the spline parameter estimation procedure, a limit was
placed on the maximum allowed change in parameters per
iteration of 10 percent. The test run with initial values
50 percent above actual parameter values was re-executed
with this restriction in place and convergence was obtained
in the same 7 iterations with the same final parameter
estimates as shown in Table 5-6. Using this additional
restriction, parameter estimates were obtainable with errors
for the initial values as high as 200 percent. A marked

improvement over ordinary least squares where convergence

176

 

—‘*'——‘\~0-0 ‘_—*}“—‘V~fl-100 “—**—"V~0-500 "—43—"'VN0-5000

 

 

 

 

 

 

 

Condition Number Ratio

 

 

 

0 500 1000 1500 2000
Weighting Factor W2 (x1000)

 

 

 

Fumese(ammumtumbawmmmwmuRammanmneUflmrwnPdmu

Table 5-6

Parameter Estimation Using Derivative Regularization

Diffusivity Heat Flux Biot Number kro/k

Actual

Values 1 1 1 0.1
Initial

Values 2.0 2.0 2.0 0.20
Estimated

Values 1.00000 1.00000 1.00000 0.10000

was not obtainable beyond a one percent deviation from the

initial parameter seed values to the true parameter values.

CHIBPEIHR 6

OPTIMIZING THE ANALYSIS METHOD

§L18_IHIBQDHCIIQN

The identification of the most appropriate model for a
particular experiment can be a difficult problem since
factors which determine model appropriateness are often
contradictory between competing models. The length of the
measured time scale to be used in the analysis is an example
of a difficult experiment design aspect as well. Clearly,
temperature readings taken at excessively late times in the
experiment contribute virtually no useful information to the
analysis and may serve only to degrade the accuracy of the
estimates for the parameters of interest. Finding the time
window which provides the best information for estimating
the parameters can be difficult to identify. The
elimination of the heat flux parameter is a popular method
used in analysis of flash diffusivity problems, but the
effectiveness of this technique is somewhat debatable. This
chapter deals with miscellaneous issues affecting the
accuracy of various flash diffusivity analysis methods.

Section 6.2 of this chapter deals with the

mollification method as a means of smoothing measurement

177

178
errors and as a means of estimating the standard deviation
of the errors. A method which eliminates the heat flux
parameter is discussed in Section 6.3 in the interest of
eliminating unnecessary parameters to simplify the parameter
estimation process. Section 6.4 deals with the effects of
changing the time duration of the experiment in order to
optimize experiment design. This section also examines new
and existing methods of determining model appropriateness.
Various models are compared using these methods. The
technique of parameter estimation by sequential experiments
is applied in Section 6.5. This method analyzes parameters
utilizing data from multiple experiments simultaneously.
Section 6.6 discusses the application of frequency
distribution routines using fast Fourier transforms on the
residual curves generated from various experiments performed
in Europe, Asia and the United States. Finally, Section 6.7
provides a summary of analyses performed on samples of
various thickness of the same material from Oak Ridge

National Laboratory.

$.2__HBIEEJMQLLIIIQAIIQB
The method of mollification, as described in [26], is a
weighted method of smoothing data in order to minimize the

effect of measurement errors. In utilizing this method, a

179
"blurring radius", 6, is selected based on the nature of the
measurement errors. This blurring radius is normally
expressed in terms of a number of measurements either side
of the measurement being mollified. The mollified value of
each point in the measured data is determined as follows
=5
fungi: p(i)Y(n+i) (6-1)
1=~36
where Y(n) is the value of the measured temperature at
measurement point n and p(i) is the weighting function for
the measurement point n, a distance 1 measurements away from
the point n. The weighting functions are determined as

follows

pun-Le .— (6-2)

Using this method,
=5
J‘25.p(i)==1 (6-3)
1=-36
Selecting the blurring radius can be somewhat of an
imprecise process. One way of doing this is to gradually
increase the blurring radius from 1 to 2 to 3 measurements
and graph the mollified points on the same axes as the non-
mollified points. In some cases, correlated data will

require a larger blurring radius in order to compensate for

180
the correlation in the errors. If the blurring radius is
too small, the mollified data will tend to follow the
correlations in the errors. If the blurring radius is too
large, however, the mollified data curve may not follow the
true data path.

In flash diffusivity experiments the area of the
measurement curve most susceptible to a misrepresentation of
the data by a large blurring radius is at the peak
temperature measurement point. Figures 6-1 through 6-3 show
this area of the curve for three blurring radius selections.
Figure 6-1 depicts a blurring radius of two measurements.
Figure 6-2 depicts a blurring radius of three measurements.
Figure 6-3 depicts a blurring radius of five measurements.
If a an overly large blurring radius is chosen, this area of
the curve will reveal the mollified points continually lower
than the measured data. In this case, a blurring radius of
5 measurements seems to be apprOpriate. The mollified curve
appears smooth but still appears representative of the
overall measurement magnitudes, even at the peak of the
curve.

Figure 6-4 shows a comparison of two residual curves,
comparing a mollified case to a non-mollified case using the
same mathematical model in analyzing the data. The

calculated standard deviation of the residuals using

181

 

 

 

 

 

 

 

 

 

 

53‘ i
‘5 002 .. 0“"':*.VA ..
0 6.8 q» 6! .
a. . I 1'- .
'3 6.781 ,A'. ._
e 676 ~ H
g '1’. "'4 [If
a $74 "
h- i' ‘
6.72 l a: :20
6.7 - .L + i r . J-
0.6 055 0.7 075 08
Time [seconds]
FugueB-1AComparieonotRawDetetoMollifledDate.8hrfingRedm=2
1
l
!
8. g
.- i
A 5.82 ‘r “ ”2'" .. ‘ “I” 1
g i 8.“ i '-
a -' =- , i
g m . .- 5
E 6.76 ~ :1. A é
o 6.74 J 1 - ‘ i
'- ] V i .- I
6.72 m ..T’
' a 3
5.7 r i % 4, ‘W 1'
0.6 0.65 0.7 0.75 0 8 ‘

Time (seconds)

Figures-2 ACompafisonofRawDetatoMomfledDeta, Blurring Radius-=3

 

182

 

 

$0)
.m,,
36:88

 

Tompemmre (Volts)
a) 9’ 9’ 9 OD
. a! 3' a!

 

0%

.° -
a:

0.7
Time (seconds)

0.75

 

Figures-3 ACompaneonofRawDetatoMolifledDam,BuMRad.-m=5

 

 

 

0.025 it
0.02 ~
0.015 4' ‘ .-
.- l'i ; "- I I“ g a
j r" . I F fl' “NRA ”‘ 1' J a L" "I 'l 1 -» 1

Temperature (Volts)
b o
b ' p
nggfi

b
“.b
58

1.
-0.015 ‘1 v ,.

 

O

0.6 0.8 1
Time (seconds)

 

 

Figure 6-4 A Comparison or Residuais from mum and Non-Mollified Data A

 

183

Tfliflfll 6-1.

Comparing Mollified to Non-Mollified Experiment Results

Model Diff. Heat Biot Resid.
Nmbr. (a) Flux Number 84 B5 86 (s)

ORNL 700°C MOLLIFIED DATA

1 .31205 17.760 1.22475 .04068
4 .27217 21.727 1.60236 .02408 .00996
5 .2789? 18.772 1J53200 .07366 .00857
17 .28077 19.545 1J51312 .07088 140000 .00773 .00568

ORNL 700°C NON-MOLLIFIED DATA

1 .31136 17.855 1.23326 .03697
4 .27615 21.290 1.56101 .02107 .01250
5 .28193 18.643 1.50312 .06964 .01244
17 .28364 19.422 1448574 .06681 1.0000 .00751 .01074

FRENCH.DATA A_R1.MDLLIFTED DATA

1 1.2816 2917 .08583 4.9883
4 1.1121 3345 .21013 .07187 2.4465
5 1.1288 3250 .19919 .26716 1.8968
17 1.1645 1060 .17134 .29105 .59666 1.4077 0.8719
FRENCH DATA A_R1 NONHMOLLIFTED DATA

1 1.2813 2920 .08633 4.8113
4 1.1190 3325 .20423 .06850 2.4166
5 1.1344 3236 .19432 .26207 1.9452
17 1.1673 1057 .16894 .27820 .63479 1.4243 1.0417

mollification is shown in Table 6-1. Although the
parameters calculated do not differ significantly from the
non-mollified cases, the residuals are substantially lower
than in the non-mollified cases. Moreover, the differences
in the standard deviation of the residuals between models is

much more significant in the mollified cases.

184

The mollification of the data allows the contribution
of the measurement errors to be effectively separated from
the contribution from model non-compatibility. This allows
the differences in accuracy between models to be contrasted
more effectively.

For some forms of regularization in the field of
inverse problems, it is important to know an "expected"
magnitude of the errors in the experiment. Additionally,
this information is useful in evaluating the adequacy of a
model for a particular set of measured data. If the
standard deviation of the residuals is on the same order as
the anticipated standard deviation of the errors, a measure
of assurance is gained in the validity of the model.

One means of estimating the standard deviation of the
errors is to use the mollified curve as the "true values" in
the experiment. In this way, the standard deviation is
found by computing the sum of the squares of the differences

between the mollified data and the raw data or

Calf (mm-fun)2 (6-4,
0:1

 

 

N-l

Table 6-2 below shows the results of this method used
on a file of simulated data typical of that measured from a

CBCF sample at 700°C measured at Oak Ridge National Lab.

185

TEEHHB GFQE

Estimated Measurement Errors Using Mollification

Known 0 Percent of Estimated (3)
E E II II ! H . H JJ'E' l'
0.05123 0.753 0.04778
0.02049 0.301 0.01953
0.01028 0.151 0.01057
0.00512 0.075 0.00652
0.00204 0.030 0.00484
0.00020 0.003 0.00445
0.00002 0.0003 0.00445

The diffusivity used was .31, heat flux 19, and Biot Number
1.4, calculated as a direct problem with 400 points. The
peak measurement in such a case is approximately 6.8 volts.
Errors with a Gaussian distribution and a known standard
deviation were superimposed on this exact data. The
mollification method was then used as stated above in an
effort to approximate the standard deviation of the errors.
When the errors are large, the mollification method
performs quite well in estimating the standard deviation of
the errors. When the standard deviation of the errors drops
below 0.1 percent of the maximum measurement, however, the

mollification method is unable to accurately estimate the

standard deviation of the errors.

 

A distinct handicap of the more advanced models is that

186
estimating a larger number of parameters simultaneously is
an inherently less stable process because a larger number of
simultaneous equations must be solved. Whenever possible,
parameters which are of no interest should be eliminated
from the model, provided model accuracy is not degraded in
doing so.

Researchers from the "Institut National Polytechnique
de Lorraine et Universite de Nancy" in France have set forth
a method by which the heat flux magnitude need not be
calculated simultaneously with the other parameters. Since
this work was shown to be successful for a three parameter
model, reducing the effective number of parameters
simultaneously estimated to two, an attempt was made as part
of this research to expand this conCept to the higher order
models.

Using Model 1 as an example, the direct solution as

given in Chapter 2 is

“L, t): 2% 2": e-efipc/Lz Bmtfimcosmm) +Bis1nu3mn
1+_£;1__2.]+Bi (6'5)

 

 

CPD-11 (B:+BiZ) 2
Bm+Bi

 

If the maximum point of this curve is found, and referred to
as Tm“, and the corresponding time is referred to as tn”,
then a scaled solution can be obtained by dividing the above

solution by Tm“, specifically

187

T(L,t1
9 L,t =—————— -
( ) T (6 6)

MAX

This ratio then becomes

2 e -Biflt/Lz BJBDCOS (Bu) +BiSin(B“H

 

 

 

 

 

"1 (3:33:12) 1+———_Bl +31
Bfi+312
e(L,t)= . . (6-7)
2": e'sztfl/‘z Bmtfimcos (8,) +Blsm (8,)1
”4 (B:+Biz)lfl' Bl +Bi
Bi+Bi2

 

This function has exactly the same shape as the original
temperature solution, is dimensionless, is no longer a
function of heat flux and has a maximum value of 1.

The sensitivity coefficients are shown for this model
in Figure 6-5 and contrasted to those of Model 1. As shown
in this figure, both of the sensitivity coefficient curves
for the two parameter model cross the zero temperature line
at the same point. This is because the maximum temperature
measurement is reached at this point and the derivatives
with respect to all parameters are zero there. This tends
to make the parameters correlated, but is not a severe
problem in the two parameter case. When moving to higher
order models, however, this correlation becomes more of a
handicap and the performance of the parameter estimation
procedure becomes very poor.

In Table 6—3, a test case is considered using a

188

 

 

 

 

 

 

+ Alpha (2) —+— Biot —-+— Alpha (3) —0—— Sci
Number (2) Number (3)
ea:
.. soc *-
% 40° 4
25 augl
2 an ~
‘3 100 J
a 0 ‘ """ ”VFW
5 4m " "“"“""“"«ume
'— m ..
4pc
0 05 1 15 2 25 3 35
Time (seconds)

 

 

 

Fumee4iSunfiWVdeflfimmxmnpamn34Mnnnflflb2¥hmnnflranxn

calculated direct solution with imposed errors of standard
deviation o=0.00854 and a maximum temperature “measurement”
of 0.7577. .A comparison of the two versus three parameter
method is shown. .As indicated in the table, the 3-parameter
model estimated the parameters more accurately than the two
parameter model. In regard to diffusivity, the primary
parameter of interest, a 2.1 percent error from the true
value was reported using the three parameter model. By
comparison, the 2-parameter method reported a diffusivity
value which was in error by 5.4 percent.

As an additional measure of investigation, a sample

189

THUILElii-3

Contrived Test Case Comparing 3-Parameter to

Sample Thickness:
Maximum Measured Reading:
Anticipated Residual Std Dev:

Model
Nmbr.

Actual
2-Parameter
3-Parameter

Diff.
(a)

0.3000
0.2838
0.2937

Biot
Number

2.000
2.458
2.170

2-Parameter Method

L=1.0mm
0.7577
o=0.008854

Resid.
(8)

0.00000
0.01340
0.00827

problem was studied where heat losses from the sample

surface were known to be zero.

This problem provided an

opportunity to compare a two-parameter model, which

estimated diffusivity and heat flux, with a one-parameter

model estimating diffusivity only. In a contrived trial

problem with diffusivity equal to 0.3 and heat flux equal to

1.0, the sensitivity coefficient curves are virtually

identical when comparing the one and two parameter models.

The performance in the estimating routines is shown in Table

6-4. In this case, the performance of the two methods is

comparable, with the two parameter model having a slightly

more accurate estimate of diffusivity than the one parameter

model.

In a test comparing the two methods against one another

using actual laboratory data, The Oak Ridge National

190

EHUILELli-4

Contrived Test Case Comparing 2-Parameter to

l-Parameter Method

Sample Thickness: L=1.0mm
Maximum Measured Reading: 1.0
.Anticipated Residual Std Dev: o=0.008854
Model Diff. Resid.
Nmbrl __1dl ..bil
Actual 0.3000 0.00000
1-Parameter 0.2954 0.00994
2-Parameter 0.2989 0.00848

uuuanzzci-s

ORNL Data at 7OOWC‘Using 2-Parameter Method

Model Diff. Biot
Nmbr. (or) Number 83 [34 85 (3)
RESULTS EROM'DIVTDING OUT THE HEAT FLUX PARAMETER
1 .30995 1.24841
4 .28226 1.49013 .01789
17 .29243 1.39531 .05582 1.00000 .00049
95% Confidence Intervals
1 .025643
4 .038407
17 .063664
RESULTS FROM COMPUTING THE HEAT FLUX PARAMETER
1 .31136 1.2332
4 .27615 1.56101 .0210?
17 .28364 1.48574 .06681 1.0000 .00751

95% Confidence Intervals

1 .004431
4 .005980
17 .004025

Resid.

.038129
.015763
.015342

.036976
.012506
.010741

191
Laboratory data was used for a CBCF sample at 70033. The
means by which the two methods can be compared are by the
standard deviation of the residuals and by the width of the
confidence intervals for the parameter of interest. As
shown in Table 6-5, the standard deviation of the residuals
and the width of the confidence regions for all three models
tested are smaller when the heat flux is computed as an
independent parameter rather than being divided out in the
reduced parameter method.

A concern related to sensitivity coefficients is the
way in which they are calculated. The method used in
calculating sensitivity coefficients, for all of the
experiments studied as part of this research, has been a
numerical approximation method. Using this method, the

derivatives are approximated by the following expression

3T ~ T(l.00181,L, t) -T(BI,L, t)
‘55:" .001

 

(6-8)

where 81 is the parameter corresponding to the applicable
sensitivity coefficient. In order to test the validity of
this approximation, several other values were chosen for the
magnitude of the perturbation used. The value used in the

above equation of 0.1 percent perturbation was tested in

192

 

 

 

 

 

 

F
+ Alpha —‘}— HeatFlux + BiotNunbot
0.2
2 2:-
i5 005* 7' 5 uiié‘Mhi‘w IA 1 1 . ‘1
e 0 ‘ ‘ 1,1 I I“. i ‘11 ", “X“. LE1 L1 '1: I! an"
- .. l.“ mm
E 41125“ ’1! "4 “if? 51‘s,! ”f" f; ’H
a 411+ L!
a {115
0 02 114 0.5 118 1 1.2 1.4 1.8
Thnelsecondsl

 

 

Fbm3641PaganDMUumnoflfimmmeOmmhbmxBdwuuflknummhmumxnflunison

comparison to other values. The percent difference in
sensitivity coefficients between using 0.1 percent and 0.01
percent perturbation of the parameters is shown in Figure 6-
6. This percent difference seems to be fairly uniform over
the length of the experiment. The standard deviation of
these differences for each of the parameters is as shown in
Table 6-6. The differences at the smaller perturbation
values are clearly negligible. At a 10 percent perturbation
value however, two of the sensitivity coefficients are
different enough from those calculated at the finer
perturbations, that convergence time could be affected due
to inaccurate estimates at the intermittent iterations.

Examining the use of these sensitivity coefficients in

 

193
UHUNUB (ifis

Various Sensitivity Coefficient Calculations

Percent Standard Deviations of Pct Difference
Perturbation Alpha Heat Flux Biot Number
0.001 0.06099 0.041907 0.046979
0.01 0.06099 0.038379 0.049313
1.00 0.42086 0.003837 0.237397
10.00 4.40292 0.004222 2.486596

TIEHH! 6-7’

Results of Calculation Method on ORNL Data

Percent Number of Heat Biot
Perturb. Iterations Diff. Flux Number
0.001 .33628 18.9632 .85425

4
0.01 4 .33628 18.9628 .85422
0.1 4 .33628 18.9629 .85423
1.0 4 .33634 18.9563 .85374
10.0 5 .33687 18.8957 .84923
estimating parameters from actual laboratory data, the
results from the CBCF samples measured at 600°C at Oak Ridge
National Laboratory are shown in Table 6-7.

As with the sensitivity coefficients, the difference in
converged parameter values is negligible until large values
of perturbation are used, such as 10 percent. This gives

confidence that the perturbation value of 0.1 percent used

in the preceding calculations produces satisfactory results.

 

In the interest of avoiding the use of excessive

 

 

 

H
(D
U)

194

measurements, analyses were performed over varying time
scales on the same experiment and the results were compared.
The objective of this comparison was to determine which
window of time is most appropriate. The data file studied
was the Oak Ridge National Laboratory CBCF sample measured
at 70083, as studied extensively in Chapter 4. The file
contains 463 measurements, and was studied in five windows:
the first 100, 200, 300, 400 then all 463 points. The
results were compared for appropriateness using three
primary measurement methods:

1. The standard deviation of the residuals.

2. The width of the confidence region for the

parameter of interest, diffusivity.

3. The stability of the sequential parameter

estimates for diffusivity.
The numerical results of these analyses are shown in Tables
6-8 through 6-12. Finding the most appropriate model among
these results is not immediately obvious, since some
measurement criteria are superior using one model and
another criterion is superior using another model. For
example, comparing Model 1 between Table 6-8 and Table 6-9,
the 200 point sample produced a lower standard deviation in
the residuals but the 300 point sample resulted in estimates

with a narrower confidence interval.

 

Max mutt
Ant cioa

ppiied
Model “1
\1mbr {
q 'x
4 01:1:
5 .2
17 2
95% Copf
1 a
o b

4 L
5 :
17 i
0 K.

~531p1e '3.
MaximUm .
AntLCi 1
Pa

MOdel

w E
.‘I‘lbr. I
l a
4 ':
5 'f
17 -<
2

to
m
op

F)

\JLnAr—A

   

 

p

(3(DCDC)

 

195
EQUELEIGi-B

ORNL Data at 700°C Using First 100 points

Ambient Temperature: 700%:
Sample Thickness: L=.956mm

Maximum measured Reading: 6.8258

Anticipated Residual Std Dev: 0.0091074
Applied Null Value: 0

Model Diff. Heat Biot

Nmbr. (d) Flux Number 8, 85 86

1 .36543 8.98728 .01922

4 .23113 42.0567 3.42080 .03659

5 .27622 20.6111 1.81798 .07007

17 .34643 11.3035 .31135 .0506 .2713 .0092

95% Confidence Interval

1 .011550

4 .127430

5 .041888

17 .026662

TEEHHB (i-Q
ORNL Data at 700°C Using First 200 points

Sample Thickness: L=.956mm
Maximum measured Reading: 6.8258

.Anticipated Residual Std Dev: 0.0091074

Model Diff. Heat Biot
Nmbr. hi) Flux Number' 8, 85
1 .32889 14.8203 .91080
4 .27394 21.7314 1.60360 .02219
5 .28632 18.0951 1.43088 .06739
17 .29242 18.3297 1.34449 .0611 1.000 .0090

95% Confidence Interval

1 .005501
4 .018057
5 .010263
17 .007461

Resid.
(S)

.02574
.02268
.02178
.01130

Resid.
(3)
.02857
.01660
.01589
.01035

 

Mode; [3

Nmbr.
1 .3
4 .2
1: i
. 4

KO
()7
w

O
O
F3
7*)

0 I
/"\ n (1 r“

 

196

IABLE 6-10

ORNL Data at 7OOWC'Using First 300 points

Sample Thickness: L=.956mm
Maximum measured Reading: 6.8258
Anticipated Residual Std Dev: 0.0091074
Model Diff. Heat Biot
Nmbr. (d) Flux Number 84 85
1 .31837 16.7081 1.1200
4 .27727 21.0698 1.5409 .02102
5 .28432 18.3207 1.46340 .06901
17 .28761 18.9482 1.42614 .0645 1.000

95% Confidence Interval

.0097

1 .004716

4 .009646

5 .006141

17 .004790

IABLE 6-11
ORNL Data at 700°C Using First 400 points

Ambient Temperature: 700%:
Sample Thickness: L=.956mm
Maximum measured Reading: 6.8258
Anticipated Residual Std Dev: 0.0091074
Applied Null Value: 0
Model Diff. Heat Biot
Nmbr. (a) Flux Number 8, [35 86

1 .31136 17.8554 1.2333

4 .27615 21.2900 1.56101 .02107

5 .28193 18.6432 1.50312 .06964
17 .28364 19.4220 1.48574 .0668 1.000 .0075

95% Confidence Interval

1 .004431
4 .005980
5 .004338
17 .004025

Resid.
(3)
.03412
.01538
.01437
.01023

Resid.
(S)
.03698
.01251
.01245
.01074

197
TABLE 6-12
ORNL Data at 700°C Using All 463 points
Sample Thickness: L=.956mm

Maximum measured Reading: 6.8258
Anticipated Residual Std Dev: 0.0091074

Model Diff. Heat Biot Resid.
Nmbr. Rx) Flux Number 84 85 86 (s)
1 .30897 18.2850 1.2736 .04132
4 .27337 21.6633 1.59526 .02290 .01349
5 .27936 18.8599 1.53652 .07224 .01398
17 .28097 19.6629 1.52054 .0694 1.000 .0081 .01192

95% Confidence Interval
.002205
.002732

1
4
5 .002088
7

1 .001878

One thing that is clear from Tables 6-8 through 6-12 is
that using only the first 100 or 200 points of the data file
results in a much wider confidence interval than the tests
using more points. Although the standard deviation of the
residuals is lower in the 100 and 200 point cases when
comparing the Model 1 tests, there is no advantage in the
residuals when comparing the Model 17 tests. The wider
confidence region therefore makes the 100 and 200 point
cases undesirable, as depicted in Tables 6-8 and 6-9,
respectively. This trend seems to continue through the rest
of the tests involving 300, 400 and 463 points in Tables 6-

10 through 6-12, respectively. These three higher

198

 

“""IWN$N1 “**_‘hkxk+4 '——*““Ikai5 ‘—O——1MD¢fl17

 

 

 

 

 

0 : : L t : i +
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
Tim (seconds)

 

 

 

Figum 6-7 Sequential Estimates of Diffusivity (CBCF 700°C)

measurement data files, however, are more difficult to
discern between. It is helpful to bring information from
the sequential estimates into consideration at this point.
As a means of quantifying the degree to which the
sequential estimates vary in the later times of the
experiment, the last 50 sequential estimates from each model
are averaged by model and the standard deviation of the last
50 points is compared from model to model. For the
sequential estimates shown in Figure 6-7, the standard
deviations from the last 50 average of the sequential
estimates is 0.003642, 0.001493, 0.000909 and 0.001672 for

models 1,4,5 and 17, respectively. In this particular

199
situation, the most stable sequential estimates were
associated with Model 5, however Models 4 and 17 also showed
significant improvement over Model 1. The most likely
reason that Model 17 exhibits slightly less stable
sequential estimates than does Model 5 is in the inherent
loss of stability brought about by a greater number of
simultaneous parameters to compute. The standard deviation
of the residuals from Model 17 indicates a superior fit for
the laboratory data used to generate these sequential
estimates. Once again, it is difficult to consider
information from all three of the evaluation criteria
simultaneously in order to evaluate the appropriateness of
one analysis over another.

Since many factors come into play when comparing
results from competing models in parameter estimation, the
identification of a model which is “most appropriate” can be
elusive. One highly respected method of determining model
appropriateness is the Akaike Information Criterion as
described in Reference [34]. This method assigns a
numerical value to the performance of a model in terms of
its fit to the measured data using a reasonable number of
parameters. Specifically, the criterion is

AIC = -2(maximum log likelihood) + 2k (6-9)

where k is the number of independent parameters. This

200

expression can also be written as

AIC = n-ln(2n) + n-ln(v) + n + 2k (6-10)
where v represents the variance of the residuals and n is
the number of measurements in the experiment. The model
considered to be most appropriate is the one which generates
the lowest AIC value. Since the variance of the residuals
is typically a small number, the AIC is usually negative.
Using this method, the variance must decrease by a factor of
7.39/n in order to offset the penalty incurred by adding
another parameter to a model.

One of the primary comparisons which make a criterion
method desirable is the ability to compare the effects of
varying the number of measurements on the parameter
estimation adequacy. The AIC discussed above is highly
dependent on the number of measurements and, as such, is not
well suited to comparing models used on varying numbers of
measurements for the same experiment. It is used here
primarily to test the adequacy of one model against another,
specifically, Models 1,4,5 and 17.

Because of this and the fact that the AIC involves only
one measured performance criterion, the variance of the
residuals, a new method was considered which makes use of
three criteria simultaneously. These are the standard

deviation of the residuals, the width of the confidence

201
region, and the standard deviation of the sequential
estimates over the last half of the experiment.

Since the lowest magnitudes of each of these three
factors is the most desirable, the lowest number of each of
the three categories is used as a scaling factor for its
respective category. Each of the three categories is then
scaled to this number, so that the lowest number in each
category is 1. The scaled factors are then multiplied
together to give the total "grade". The model with the
lowest grade is then considered the most appropriate model,
the lowest possible score being 1. Expressed in equation

form,

 

 

 

Si cri 391
Grade: (6-11)
fiun Guam. sfiun

where smfl, crmfi, and semin represent the lowest standard
deviation of the residuals, confidence region and sequential
estimate standard deviation of any of the models considered,
respectively. The terms 31, cri, and se,-L represent the same
factors for the model being graded. The results of this

analysis are shown in comparison to AIC in Table 6-13.

5l5__AflLLIZIHBCBIQDINIIBL_IIEIBIMIEI§

As experiments are performed over a range of

 

Li:
J) U) N H
.17.

A t) 0
O C)
ox C)

CC).
0C)
010

463

Model
100
200
00
400
463

mDer
no Una 1
Can be
the Va

alt

 

Number Std

of Dev
Msrmnt Residuals
Model 1
100 0.025739
200 0.028565
300 0.034117
400 0.036976
463 0.041321
.Mbdel 4
100 0.022682
200 0.016596
300 0.015377
400 0.012506
463 0.013493
.Mbdel 5
100 0.021779
200 0.015892
300 0.014371
400 0.012448
463 0.013986
.Model 17
100 0.011297
200 0.010352
300 0.010233
400 0.010741
463 0.011924
temperatures,

normally reported at each temperature.

Comparison of Model Selection Schemes

202

IABLE 6-13

Conf
Region

.01155

.005501
.004716
.004431
.002205

00000

.12743
.018057
.009646
.00598
.002732

00000

.041888
.010263
.006141
.004338
.002088

00000

.026662
.007461
.00479

0.004025
0.001878

000

Std Dev
of Seq.
Param.

0.003715
0.00584

0.004769
0.003642
0.004069

0.011064
0.005039
0.003666
0.001493
0.000954

0.006062
0.000974
0.000838
0.000909
0.001603

0.002264
0.006274
0.001541
0.001672
0.002513

McMasters'

Grade

68.5788
56.9831
47.6463
37.0526
23.0210

1985.741
93.76717
33.76516
6.93325
2.18371

343.4005
9.864368
4.592271
3.047968
2.906799

23.02667
16.36306
2.550586
2.440876
1.900242

Akaike
Infrmtn
Criteria

-442.16

-848.649
-1169.41
-1496.83
-1630.64

-465.447
-1063.86
~1645.56
-2362.08
-2665.02

-473.572
-1081.2
-1686.16
-2365.8
-2631.79

-600.854
-1248.65
-1885.91
-2479.79
-2775.49

individual values of the parameters are

Polynomial curves

can be approximated using these points, in order to express

the value of these parameters as functions of temperature.

An alternative to using the individual experimental results

203

to construct the polynomial relationships is to analyze
individual experiments sequentially in a collaborative
fashion to obtain the temperature dependence as discussed in
[35]. One advantage of this method is that individual
experiments are automatically weighted in comparison with
one another based on the number of data points collected in
each experiment. Moreover, the experiments can each be
assigned weighting factors in proportion to the degree of
reliability of each experiment.

In the flash diffusivity experiments, a non-dimensional

temperature can be defined as

 

T‘= 1 (6-12)

In the particular case of a set of experiments measured at
Oak Ridge National Laboratory with the Holometrix equipment,
the temperatures at which measurements were made were 100,
400, 500, 600, and 700%:. For this example, it is desirable
to set T1=100, T2=400 and T3=700°C. The non-dimensional
temperature then becomes

= r-100°c
600°C

T‘ (6-13)

so that T*==0 at T=100°C and T'=1 at T=700°C which correspond

to the temperature limits of the experiments performed. Due

204
to the nature of the temperature dependence of diffusivity
and Biot number for these experiments, it is desirable to
fit the parameters each to respective parabolic curves as
functions of temperature. In order to simplify the
parabolic equations, we can define parameter coefficients as
d1= a(100), d2= (1(400) and d3= d(700)

Biy= Bi(100), Bi§= Bi(400) and Bi§= Bi(700) (6-14)
Expressions used for the parameters as a function of
temperature for Model 1 then become

oz(T“)=dl+(-oz3+4ozz-3o:1)T‘+2(013-20:2+011)T‘2
(6-15)

Bi(T‘)=B.i1+(-Bi3+4Bi2-33.i1)T‘+2(B.7'.:,-ZB.1".,_+B.1'1)T’2

There are a total of 7 parameters to be estimated in
this case: the six noted above and the magnitude of the
heat flux, qb. The heat pulse term has no dependence on
temperature, however, and the information regarding this
parameter must be discarded from one experiment to the other
so as not to "contaminate" successive calculations from
other experiments.

In the analysis of the first experiment, only three
parameters can be estimated, namely a1, Bi1 and Heat Flux.
This is because only one temperature is used per experiment.
Estimation of the first three parameters, as in previous

cases, is accomplished by approximately minimizing the

205
following expression
51=tt (Yi-Ti); (6-16)
1:1 1:1
where the index j refers to the individual experiment and
the index i refers to each individual measurement within the
applicable experiment. In vector form, this becomes
- _ T _ -
S-g (Y, 1") (Y: 1") (6 17)
where Yj and '13 denote the vectors of measurements and
calculated values, respectively, for the experiment denoted
by the index j.
Taking the derivative of the above equation and setting

it equal to zero we have

Von5=§ 2 Bi(rj-rjflhrj-rj) (6-18)

since the equations are non-linear, the problem must be
solved in an iterative fashion where the iteration is
denoted by the superscript (k). The calculated solution
T(B) can be approximated by the first 2 terms of the Taylor

series as follows

(3’1, _ (k) (3) (3’1) (3)

where x, is the sensitivity matrix and t:j is the calculated

parameter vector as opposed to B which refers to the true

 

206
but unknown parameter vector. Substituting this expression

into the above equation gives

(3)! (I) (3*1) (3) _ (3)! (3)
L; x1 x1 lb! “b. ) ‘ 12x1 (ti-T1 ) (6‘20)

where the vector bfl‘m’ is the unknown.

The approximation of the minimum for S is found using
this procedure, rather than the true minimum, because the
non-linear sensitivity coefficients for each previously
analyzed experiment are not updated to reflect the value of
the parameters for subsequent estimates. The parameters
cannot be calculated simultaneously because of the random
nature of the g0 parameter from one experiment to the other.
The D vector, the parameter vector, is calculated for each
experiment until convergence.

Since, three experiments performed at different
temperatures are required in order to estimate three
temperature sensitive parameters for diffusivity and Biot
number, the values for 01,, d3, Bi2 and Bi3 cannot be
calculated at this stage. Some regularization will be
required in order to facilitate the calculation. This could

be in the form of

3:; (Y—T)2+w1(dl-da)2+m2(Bil-Bi3)2 (6-21)

207

The additional constraint of setting (13 and Bi3 equal to 012
and Biz, respectively makes the problem stable enough to
calculate the first set of parameters. Finally, in the
calculation using the third experiment, the regularization
terms can be dropped and all parameters will be calculated.
Thereafter, the entire set of 7 parameters will be re-
calculated with information from the previous experiments
being added as prior information to the XTX and the XTY
matrices. Column and row 7 in each case will be set to zero
prior to being added to the next experiment.

Table 6-14 compares the estimates arrived at for
diffusivity and Biot Number using three methods. These are

1. Individual experiment results averaged for each
temperature at which more than one experiment was performed.

2. Fitting a least-squares parabola through the
individual experimental results.

3. Using all of the experiments in a sequential
estimation Scheme as described above.

As shown in the table, the results are very close
between all of the methods. In the case of these particular
experiments from Oak Ridge National Laboratory, the number
of measurements for each experiment was approximately the

same. Had this not been the case, or if one experiment had

208

TABLE 6-14

Sequential vs. Individual Estimation

Ambient Average Least Sqrs Sequential
Temp. Individual. Curve Fit Method

Di ffusi vi ty
100 .32461 .32447 .32748
400 .31351 .31154 .31288
500 .31184 .31517 .31785
600 .32691 .32277 .32775
700 .33277 .33434 .34257
Biot NUmber

100 .15379 .15874 .14343
400 .53483 .52030 .50472
500 .74180 .70679 .69697
600 .86154 .92627 .92513
700 1.20691 1.17873 1.18921

been weighted more heavily than another, the results may not
have been this close.

Figure 6-8 shows the results from this table in
graphical form. Superimposed on the sequential results are
the results from analyzing experiments individually. The
results from the sequential experiment method basically
conform to the parameter values obtained by analyzing the
experiments individually. One area of exception is in the
region of the three highest temperatures at which the
experiments were conducted. Particularly, the results for
Biot Number taken at 600°C are not in conformance with the

assumed parabolic temperature dependency used in the

209

 

 

 

 

 

' Dibflfihr "Soquuia| A Bknhhntor "Soquufifl
lamnukn Eflhmmn
1A
a 12“
s 1 V
'3
E as «» 7”
0 08+
E 0.4 J? h
02 .1-
0 1 : I. a : :
O 100 an: an: 400 500 8d) 703
Degrees (C)

 

 

 

Figures-8 Parabolic Fitfor Parameters Using Simultaneous ExpenmentAnalysls

sequential experiment method.

In addition to fitting a parabolic curve through the
parameters as a function of temperature, these experiments
were also analyzed sequentially assuming a cubic dependence
of Biot number on temperature. This was investigated due to
the radiative nature of the heat transfer from the surfaces
at high temperature in a vacuum and the natural tendency
toward a cubic relationship in this type of physical
phenomenon. The conformance of the data to the cubic shape,
however, was less acceptable than that shown in Figure 6-8
which assumed a quadratic dependence. The nature of the

heat loss from the surfaces may therefore have a convective

210

component from partial pressures of gasses present.

Diffusivity, the parameter of interest, is remarkably
constant throughout the temperature range in which these
experiments were conducted. This has proven true in both
the sequential and individual experiment analysis cases.
Should sequential estimation be pursued further, a simple
and more reasonable model for diffusivity would be as a

constant with respect to temperature.

W

Many of the residual curves exhibit characteristic
signatures which indicate a disparity between the measured
data and the mathematical model. Figure 6-9 shows residuals
from three experiments. Each of the three experiments shown
in this graph measured different materials. In order to
compare the three experiments directly on the same set of
axes, the residuals for each experiment were normalized by
scaling to a non-dimensional time. Additionally, the
vertical axis was scaled to the percent of maximum
temperature rise. The three experiments shown in this
figure are

A;Rl Palaiseau, France
C_R1 West Lafayette, Indiana

M_R1 Hunan, China

 

211

 

 

 

 

 

 

 

+ a_r1 + c_r1 + m_r1

.8 3 1'
'3 254’
a? 24»
5 1.5-
g 14,
3 0.5»
2 o

fi-usi
:13 -1

0 DJ 02 03 DA 05
Non-Dimensional Time

 

 

 

 

FbunfidiRafihflbthoflbdd1thmfiaBTMDBUmDhMOEnlmmDMI

Although the experiments represent different materials at
different thicknesses, the characteristic signatures, when
compared with one another, are strikingly similar. These
curves all exhibit the early temperature rise not predicted
by the non-penetrating model, as evidenced by the large rise
at approximately 0.05 units of non-dimensional time.
Another prominent common feature among these curves is the
back-side flash, or flash heating at time zero and x=L.
This is evidenced by the non-zero start temperature at the
first time step.

Figure 6-10 likewise shows residual curves from three

different experiments measured in different laboratories and

212

 

 

 

 

 

 

—'— a_r2 —9— e_12 + o_t1

:? 35

i 23"

a 2..

5 1.5,

.E 1

5 0.5 .

3 o

'5 .

s ”-1

-1.5
3 -2
0 0.05 0.1 0.15 0.2
Non-Dimensional Time

 

 

 

 

Flame-10 RoddualsUsthodsHbAndyonhrosUmflatsdEmomm

analyzed using Model 1. The experiments featured in this
figure are

A;R2 Palaiseau, France

E_R2 Vandoeuvre les Nancy, France

C_R1 LeBarp, France
In contrast to the previous figure, these curves tend to
exhibit no distinguishable signature. This suggests that
there is no appreciable penetration of the flash nor back-
side heating at time zero. For these experiments, Model 1
gives the best overall performance for estimating the
parameters. In fact, C_R1 is best suited two a model which

assumes Bi=0. This is evidenced by the lack of discernable

213

 

+ a_r1 + c_r1 + m_r1

(.9

 

1"
mm

Pm: Speclml Density
5‘.

 

 

 

 

0 5 10 15 20
Non-Dimensional Frequency

 

 

 

Figlfl'e6-11 quuencyDistibufionofResiduelGraphShownhFlgureB—Q

improvement in the residuals and wider confidence regions
when using the higher order models.

In order to gain more insight into the meaning of the
signature, a frequency analysis was performed on these
residual curves utilizing the fast Fourier transform
provided in the MATLAB software. Figure 6-11 is a plot of
the frequency distribution of the residuals shown in Figure
6-9. The predominant frequency is approximately 4 non-
dimensional frequency units. This validates the dominant
frequency evident in Figure 6-9 which exhibits a period of
approximately 0.25 non-dimensional time units in the primary

path of the residual graph. The residuals shown in Figure

214

 

+ e_t2 + e_t1 + o_r1

 

1.4

..s
L
T

E 0.81-
g 0.6«
03

5 Q4;

1 1 '
g (12 31 _ 1| 1' ..
U ‘ LE" "1 r"

50

Density

T

 

 

 

   

I L 'r -. I LI
103 150 200 250
Non-Dimensional Ftequency

o .‘

M

0

 

 

 

Horne-12 FrequencyDistrbuflonofReflduelGraphShownhFigweO-11

6-10 on the other hand, do not have an easily detectable
pattern or signature. Therefore, the frequency distribution
for this graph, featured in Figure 6-12, does not exhibit
the same consensus in terms of dominant frequencies among
the experiments. Aside from some very low frequencies
evident in experiment e_r2, the only dominant frequencies in
this set of experiments seem to be in experiment o_r1. In
this case, the frequencies are so high that the most likely
source of these signals is measurement noise rather than a
mechanism related to heat transfer.

A prominent effect of flash penetration on the

estimated parameters, using Model 1 as the analysis model,

215

 

+ Diffusivity —*—- BiotNunbet —°— HeatFlux

12 : 1 1 t
1.15 ‘1’ u H '1"
1.1 .. . 1*
1.6 .11- ‘ 73 4.: 1i. '5. #-

 

 

Estimeted Parametets
g:
3 ..

 

 

 

0.9 ~~

0051*

0.3 1

0.75 1-
07 P 1 1 4 r
0 0.02 0.04 0.13 0.00 0.1

Flash Penettation [mm]

 

 

 

Figures-13 EsfimatedPeremeters.UsingModel1,eseFmonofFleshPermafion

is that the diffusivity estimates are higher than the true
values and Biot number values are lower than the true
values. With small exceptions in cases of shallow
penetration, the estimated heat flux magnitude is also lower
than the true value. In order to determine the extent to
which these trends exist, 20 sample experiments were
simulated by generating direct solutions which included
various depths of penetration using Model 5. The sample
thickness (mm), diffusivity (nmfi/sec), Biot number

(unitless) and heat flux (joules/mm?) were all set to unity.

The depth of exponential penetration was varied from 0 to

216

 

+ Bi-O —*— Bi=1

 

0.04 ~
003 1
0.02 « p
0.01 4»

 

Residuals

-0.01 1

.flflz :RWflU‘ A 1 1
0 0.2 0.4 0.5 0.0 1

Non-Dimensional Time

 

 

 

 

 

 

 

Fuu3644lhnflmmnmmmammuwedSbmmmswmnemWMMNMRHuanu

0.1 mm between experiments. Figure 6-13 shows a plot of the
three estimated parameters for each case, using Model 1 to
analyze experiments with known penetration. With an
exponential penetration depth of 0.1 mm, the error in
estimated diffusivity, the parameter of interest, is 17
percent. The error in the Biot number parameter is even
larger at 27 percent. The error in estimated heat flux is
smaller at only 8 percent.

In order to gain more insight into the physical causes
of the unique signature in the residuals, it is helpful to

break the problem into component parts so that isolated

217
effects can be studied individually. To accomplish this, a
simulated experiment was generated with a penetration of 0.1
and a Biot number of zero. The results from this analysis
were compared to those of an identical experiment with a
Biot Number of 1. Figure 6-14 shows a plot of the residuals
from analyzing both of these experiments using Model 1.

As observed from the plot of these residuals, the case
with no heat loss oscillates only once above and once below
the neutral axis. The reason for this pattern is that the
estimated diffusivity is higher than that of the actual
sample because of the tendency of the method of ordinary
least squares to achieve balance in the residuals. In the
process of minimizing the sum of the squares of the errors,
there must typically be a comparable magnitude of residual
points above and below the neutral axis. The only way for
the least squares method to accomplish this using Model 1 to
analyze a penetration case, is for the diffusivity to be
estimated higher than the actual diffusivity so as to mimic
the early rise in temperature brought about by the
penetration in the experiment. The early rise in
temperature cannot be adequately modeled by Model 1, of
course, and the residuals show the premature rise in
temperature of the experiment beyond that predicted by Model

1. Subsequent to this rise, the measured temperature is

 

218
lower than that predicted by Model 1 because the actual
diffusivity is not as large as that estimated using Model 1.
Consequently, the temperature rise does not continue as
would have been predicted by the model. At the end of the
experiment, the model temperature and the measured data are
once again the same since, with no heat loss, the final non-
dimensional temperature is 1.0 in both the penetration and
non—penetration cases.

Making the same comparison, assuming surface heat loss
this time, reveals just one more set of oscillations. In
this case, the same initial behavior is exhibited as in the
zero heat loss case for the same reasons. The exception is
that this pattern is followed by a period in which the
measured temperature is higher than that predicted by the
model. The reason for this rise is onCe again the result of
the method of least squares bringing balance to the
residuals across the neutral axis in order to offset the
latter portion of the experiment. In the last portion of
the experiment, the dominant phenomenon affecting the
residual curve is brought about by the estimated Biot number
being less than the actual Biot number. This causes the
measured temperature to be lower than that predicted by the
model due to the higher heat loss than predicted. .As with

the zero heat 1038 case, the residual curve will eventually

 

 

219

 

—*~"— Alpha-1.17 —+‘— Alpha-1.15 —°— Alpha-1.17
Biull72 Biz-1172 Bi-Cl74

 

0.02
0.01 5 ~

 

Residuals

 

 

{HHS # 1 1 : .
0 0.1 0.2 0.3 0.4 0.5

Non-Dimensional Time

 

 

 

FumeadslhuflmflflkmmmNthfluandfliEWuXSdmbmnbedeuscnunmm
dmwFtEWfltmdPammeumBQ1

go to zero in very late times. This is because both models
predict a final temperature of zero as heat is transferred
to the surroundings.

In order to investigate the nature of the oscillations
further, residual curves were generated comparing direct
solutions against each other. The base solution of
comparison is the same simulated experiment described above;
that is, an exponential penetration of 0.1 mm and a value of
unity for diffusivity (nmfi/sec), heat flux (joules/am?) and
Biot Number (unitless). To this baseline experiment, other

direct solutions were compared as generated by Model 1 using

 

 

220
various parameters. Figure 6-15 shows a plot of the
residuals of three such comparisons. Each curve compares
the baseline experiment with a Model 1 solution using the
parameters indicated in the legend. These parameters each
represent a perturbation of nominally 2 percent on the
baseline case. Since these Model 1 solutions were not
generated by a minimization of errors, the balance across
the neutral axis of the residuals is no longer evident. The
second case shown in this figure exhibits only one segment
of residuals above the axis and one below, even though it is
a case including heat loss. This further confirms that the
nature of the oscillations is not brought about by an
oscillating heat transfer phenomenon of some kind. Rather,
what appears to be a damped sinusoidal oscillation is
actually an attempt by the method of least squares to
minimize errors when applying a model which does not account
for the mechanisms of heat transfer at work in the

experiment.

.E.1_QIQILAHILIEIE_BDMHIBI

Experiments were run on Carbon Bonded Carbon Fiber
(CBCF) samples at Oak Ridge National Laboratory on samples
of four different thicknesses. Two samples were tested for

each thickness for a total of 8 samples. Three experiments

 

221

Table 6—15

CBCF Samples Measured at ORNL 800°C Using Model 1

 

Diffu- Biot Residual Confidence
mmmmmm
1.0mm(a) 1 0.29372 2.16129 0.01743 0.006764
1.0mm(a) 2 0.29521 2.16418 0.01773 0.006605
1.0mm(a) 3 0.29474 2.18856 0.01827 0.006652
1.0mm(b) 1 0.26229 2.44981 0.01483 0.00668
1.0mm(b) 2 0.26362 2.43437 0.01448 0.006375
1.0mm(b) 3 0.26386 2.44566 0.01520 0.006513

(0.276 ave)
1.2mm(a) 1 0.35628 2.10463 0.01726 0.004067
1.2mm(a) 2 0.35028 2.39603 0.01865 0.003912
1.2mm(a) 3 0.35182 2.3996 0.01989 0.003951
1.2mm(b) 1 0.31311 2.7485 0.01251 0.003438
1.2mm(b) 2 0.31985 2.64612 0.01323 0.003086
1.2mm(b) 3 0.31887 2.72847 0.01379 0.00311
(0.332 ave)
l.4mm(a) 1 0.31651 2.80514 0.01102 0.004696
1.4mm(a) 2 0.32126 2.6718 0.01327 0.005097
l.4mm(a) 3 0.32338 2.6703 0.01422 0.005259
1.4mm(b) 1 0.31935 3.05136 0.01287 0.00528
1.4mm(b) 2 0.31728 3.24799 0.01292 0.004922
l.4mm(b) 3 0.31038 3.65064 0.01314 0.005073
(0.320 ave)
l.7mm(a) 1 0.35357 2.2676 0.01328 0.010564
1.7mm(a) 2 0.35779 2.19244 0.01504 0.011163
1.7mm(a) 3 0.36534 2.0309 0.01830 0.013042
1.7mm(b) 1 0.35385 2.92408 0.01316 0.009741
1.7mm(b) 2 0.35418 3.03193 0.01371 0.009742
1.7mm(b) 3 0.36021 2.85595 0.01681 0.011694
(0.357 ave)

were run on each sample at each of three temperatures:

800°C, 1000°C and 1200°C. The experimental measurements were

analyzed using Models 1 and 5. Model 17 was not used since

the Anter flash diffusivity system used in these experiments

222

Table 6-16

CBCF Samples Measured at ORNL at 800°C Using Model 5

Samulefihszt
1.0mm(a) 1
1.0mm(a) 2
1.0mm(a) 3
1.0mm(b) 1
1.0mm(b) 2
1.0mm(b) 3
(0.
1.2mm(a) l
1.2mm(a) 2
1.2mm(a) 3
1.2mm(b) 1
1.2mm(b) 2
1.2mm(b) 3
(0.
1.4mm(a) l
l.4mm(a) 2
1.4mm(a) 3
1.4mm(b) l
1.4mm(b) 2
1.4mm(b) 3
(0.
1.7mm(a) 1
l.7mm(a) 2
1.7mm(a) 3
1.7mm(b) 1
1.7mm(b) 2
l.7mm(b) 3
(0.

allows no back-side heating of the samples.

Diffu- Biot
mm
0.25193 2.99722
0.25389 2.98195
0.25335 3.0202
0.22765 3.3298
0.22959 3.28179
0.22935 3.31184
240 ave)

0.31202 2.82292
0.31327 3.02372
0.31443 3.03208
0.284 3.39654
0.29128 3.23404
0.28988 3.35047
300 ave)

0.27583 3.94772
0.27723 3.84954
0.27628 3.94693
0.27685 4.36154
0.27527 4.64669
0.27184 5.12257
276 ave)

0.2704 5.05119
0.26935 5.12367
0.26371 5.42516
0.28006 5.80302
0.28151 5.93131
0.27342 6.4647
270 ave)

Pene-

Residual Confidence

WWW

0.08086
0.08028
0.0803
0.0773
0.07658
0.07701

0.08832
0.08372
0.08401
0.07632
0.07545
0.07584

0.09577
0.0992

0.10211
0.09668
0.09587
0.09234

0.14102
0.14462
0.15304
0.13282
0.13201
0.14184

0.00955
0.009624
0.009908
0.013172
0.01307
0.013565

0.006764
0.006259
0.006966
0.009332
0.008824
0.009762

0.006668
0.007426
0.00698

0.008534
0.008069
0.010204

0.007662
0.008426
0.009548
0.008521
0.00869

0.009716

0.010766
0.010459
0.010531
0.017718
0.017288
0.01743

0.006343
0.004061
0.004264
0.008463
0.006777
0.007247

0.009531
0.009327
0.008325
0.011719
0.010354
0.01355

0.017773
0.01796

0.019044
0.018495
0.018063
0.018494

Additionally,

the surface transmissivity feature offered by Model 17 was

unnecessary in analyzing the CBCF samples since the samples

were uncoated and tests of this method showed a surface

transmissivity of 100 percent.

Tables 6-15 through 6-20

 

223

Table 6-17

CBCF Samples Measured at ORNL at 1000°C Using Model 1

Diffu- Biot Residual Confidence
53102133152151.1111: WWW
1.0mm(a) 1 0.34674 2 99464 0.04664 0.005283
1.0mm(a) 2 0.34678 3.07153 0.05063 0.005617
1.0mm(a) 3 0.34721 3.10568 0.04935 0.005448
1.0mm(b) 1 0.30663 3.61463 0.03847 0.005077
1.0mm(b) 2 0.31011 3.53287 0.04210 0.005378
1.0mm(b) 3 0.30763 3.68227 0.04319 0.005513

(0.325 ave)
1.2mm(a) 1 0.38973 3.1199 0.03797 0.003838
1.2mm(a) 2 0.39027 3.14304 0.03709 0.00364
1.2mm(a) 3 0.37718 3.65986 0.04396 0.004467
1.2mm(b) 1 0.3455 3.74152 0.02563 0.003068
1.2mm(b) 2 DNC
1.2mm(b) 3 0.3314 4.61103 0.04394 0.005401
(0.360 ave)
1.4mm(a) 1 0.36514 3.06754 0.03511 0.005999
1.4mm(a) 2 0.36172 3.24589 0.03487 0.006011
1.4mm(a) 3 0.36026 3.28718 0.03216 0.005629
1.4mm(b) 1 DNC
1.4mm(b) 2 0.32796 5.82739 0.06219 0.006308
1.4mm(b) 3 0.32056 6.43341 0.06498 0.006898
(0.345 ave)
1.7mm(a) 1 0.41213 2.30298 0.04380 0.013738
1.7mm(a) 2 0.3812 3.33984 0.05220 0.00903
1.7mm(a) 3 0.39151 2.83499 0.03511 0.011728
1.7mm(b) 1 0.36969 5.2497 0.07193 0.013071
1.7mm(b) 2 0.37837 4.50694 0.07327 0.012368
1.7mm(b) 3 0.36135 5.44462 0.05209 0.009433

flu

 

 

(0.370 ave)
present the results of the experiments using Models 1 and 5
for analysis.
As can be seen in the tables, there is considerable
variation between samples of the same thickness in terms of

the estimated diffusivity. For example, the two samples of

224

Table 6-18

CBCF Samples Measured at ORNL at 1000°C Using Model 5

Diffu-
WWW
1. 0mm(a) 1 0.30535
1.0mm(a) 2 0.30327
1.0mm(a) 3 0.30472
1.0mm(b) 1 DNC
1.0mm(b) 2 0.27514
1.0mm(b) 3 DNC

(0.29 ave)
1.2mm(a) 1 0.35102
1.2mm(a) 2 0.35431
1.2mm(a) 3 0.33179
1.2mm(b) 1 0.3144
1.2mm(b) 2 DNC
1.2mm(b) 3 0.29551

(0.33 ave)
l.4mm(a) 1 0.31467
1.4mm(a) 2 0.30934
1.4mm(a) 3 0.31007
1.4mm(b) 1 DNC
1.4mm(b) 2 DNC
1.4mm(b) 3 DNC

(0.31 ave)
1.7mm(a) 1 0.29236
1.7mm(a) 2 0.30666
1.7mm(a) 3 0.29432
1.7mm(b) 1 0.30673
1.7mm(b) 2 0.3006
1.7mm(b) 3 0.31596

(0.30 ave)
1

Biot
.Alumbsr
3.91577
4.0861
4.1017

4.59171

.89329
.85483
.89176
.62912

Ahww

6.10489

.38358
.75498
.74136

bhb

.16938
.2305

.46643
.08284
9.03639
7.96359

‘OmONm

Pene-

Residual Confidence

WWW

0.07348
0.07493
0.07397

0.07076

0.07982
0.07695
0.0844

0.07322

0.07776

0.10115

0.10237
0.10032

0.15974
0.13004
0.14616
0.12361
0.13212
0.10823

0.024285
0.025658
0.025209

0.040361

0.021702
0.022854
0.016186
0.019423

0.037764

0.026491
0.025232
0.02267

0.024736
0.038158
0.021654
0.057738
0.051719
0.047508

0.
0
0

O 0000

000

000000

008341

.008584
.008446

.016116

.006962
.007212
.005235
.007921

.015732

.014605
.014036
.012956

.02256
.020664
.021235
.02321
.021929
.023421

.2 mm thickness are sometimes more than 10 percent apart.

0n the other hand, differences between experiments, or

shots, on the same sample are very small, typically on the

order of 1 percent.

between sample thickness and estimated diffusivity.

There seems to be no correlation

As a

CBCF Samples Measured at ORNL at 1200°C Using Model 1

Diffu- Biot Residual Confidence
Samalsfihatamm WWW
1.0mm(a) 1 0.34443 6.75526 0.06703 0.016502
1.0mm(a) 2 0.35614 6.03611 0.05748 0.012614
1.0mm(a) 3 0.37376 4.88173 0.03820 0.00782
1.0mm(b) 1 0.35334 4.88497 0.01326 0.006047
1.0mm(b) 2 0.34626 5.40619 0.01596 0.007485
1.0mm(b) 3 0.35146 5.1119 0.01459 0.006797
1.2mm(a) 2 0.42094 4.7531 0.00786 0.004319
1.2mm(a) 3 0.46574 3.19036 0.01617 0.007882
1.4mm(a) 2 0.41315 3.68544 0.00665 0.005605
1.4mm(a) 3 0.47133 2.12266 0.01514 0.011436
1.7mm(a) 2 0.51714 1.57495 0.01311 0.017269
1.7mm(a) 3 0.56239 1.02209 0.01463 0.018571

225

Table 6-19

 

general rule, the estimated diffusivity tends to increase
with increasing temperature. The only exception to this is
in Sample (a) of the 1.0mm thickness. From 1000%2tx>
1200K; the average diffusivity of this sample dropped
slightly.

The wide variability of diffusivity of the material,
due to non-uniformities in the synthesis process, make
validation of the effectiveness of Model 5 over Model 1
somewhat more difficult. If all samples were of uniform
consistency, samples of different thicknesses could be
compared directly in terms of the effect of thickness on
estimated diffusivity. The expected condition in this case
would be for estimated diffusivity, using Model 1 as a

method of analysis, to be lower for the thicker samples.

226

Table 6-20

CBCF Samples Measured at ORNL 1200°C Using Model 5

Sample Shot

.0mm(a)
.0mm(a)
.0mm(a)
.0mm(b)
.0mm(b)
.0mm(b)
.2mm(a)
.2mm(a)
.4mm(a)
.4mm(a)
.7mm(a)
.7mm(a)

HHHHHt—IHHHHt—IH

l
2
3
1
2
3
2
3
2
3
2
3

Diffu-
. .
.28165
.28562
.30796
.30035
.28436
.29242
.37543
DNC
DNC
DNC
0.36277
0.39764

OOOOOOO

Biot
_Numbsr
13.0422
12.0443
8.19113
7.48131
9.52066
8.46244
6.30423

3.90979
2.50712

Pene- Residual Confidence
tratign Stdl_Daxl._lntsrral
0.0745 0.052076 0.03243

0.07807 0.035693 0.021968
0.07826 0.011215 0.007647
0.07259 0.003057 0.004722
0.07596 0.004114 0.00647

0.07527 0.002463 0.003905
0.07643 0.004664 0.008828
0.18041 0.009311 0.03368

0.18992 0.010711 0.035501

The estimated values would be lower than those estimated for

the thin samples but not lower than the values obtained by

using Model 5 as the method of analysis.

Model 5 produces

lower estimated diffusivities because the phenomenon of

flash penetration causes the temperature at the x=L surface

to rise more quickly than in the non-penetration cases.

The

Model 1 method interprets this early rise as a higher

diffusivity, thereby resulting in an unrealistically high

value being reported.

Since Model 5 takes flash penetration

into account, the reported diffusivity is lower and closer

to the true value.

The reason that the thicker samples should

theoretically yield lower estimated diffusivity values is

that the flash penetration should not penetrate more deeply

227
into the surface of a thick sample than that of a thin one.
As such, the flash penetration in the thick sample
represents a smaller fraction of the sample thickness than
does the same penetration in the thin sample. The thick
sample is therefore less heavily influenced by flash
penetration. This should cause a lower diffusivity to be
estimated, which is closer to the true value than that
rendered by the thin sample experiment. Unfortunately, the
high variability of diffusivity from sample to sample
precludes the use of this type of validation for the
effectiveness of the penetration model.

Without the availability of a direct comparison test
between models of varying thicknesses, the primary
validation for the effectiveness of Model 5 in these
experiments is the reduction in residuals gained by the
application of this model. In many of the individual
experiments, the standard deviation of the residuals was
reduced by a factor of three. In other cases, there was a
more modest reduction, but in no case was there a negligible
reduction. The 95 percent confidence regions are wider for
the Model 5 analysis, but that is due in large part to the
fact that there are four parameters to be simultaneously
estimated using Model 5 and only three to be estimated using

Model 1. The higher number of simultaneous variables makes

228

Model 5 inherently less stable to some degree.
Nevertheless, in terms of modeling, it is a more accurate
portrayal of the physical phenomena taking place in the
experiment. Another key indication of the success of Model
5 over Model 1 is the consensus of the uniformly lower
estimated values of diffusivity for Model 5 versus Model 1.

The parameter estimate for penetration depth is fairly
uniform among the thinner samples but increases for the
thicker samples, particularly the l.7mm samples. An
important point to make about the penetration depth
parameter is that the sensitivity coefficients for this
parameter become smaller as the sample becomes thicker.
This is due to the increased time required, during the
diffusive heat conduction process, for information related
to the penetration to reach the temperature measurement
surface. As this length of time increases, the accuracy of
the flash penetration information transmitted through the
temperature rise profile with respect to time, becomes
diminished. Due to the I? dependency of this transmission
time, the thicker samples are at a significant disadvantage
in accurately estimating this parameter. The 1.7mm thick
specimen, for example, requires a transmission time of 2.89
times that of the 1.0mm sample. This results in a

significant amount of accuracy degradation.

229

 

+1.0mr(e) 1.0m) +1.2Me) 1.2M)
1.4Me) 1.4m:]:.—1'7m°) —D——1.hm(b)

uni 10a: 12a)
Temperature (degrees C)

 

0°
m8}

'3
\‘

 

 

 

DWNmNflycmmflfinna
O
81

 

 

 

 

 

 

 

Flames-16 MnTemmforCBCFSWatORNL UsingModeH

Another factor which can result in degradation of
accuracy in the thicker samples is the added surface area
for heat transfer at the edges. As heat loss at the edges
becomes more significant, the one-dimensional assumption on
which the models are built loses some validity which makes
the results less reliable. This is why samples are normally
used which are as thin as possible when analyzing the
material for parameter estimation. The thicker samples
shown here were prepared and tested specifically to
investigate the effect of thickness on the flash penetration
models.

Figures 6-16 and 6-17 provide a graphical portrayal of

 

230

 

——I——1nm«q <:o——13m«m ——i——12muq ——I——12mmm

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1.4mm!) —‘i-— 1.4m) that“) -—D—— 1.7an)
0.55 \“

1? i
3 L
:r' d
5 4+
5 .,
E II
b .

0.2 . .L .

800 1WD 1200

Temperature (degrees C)

 

 

 

 

Figures-17 WwTenvperamforCBCFSamplesatORNLUsmModels

the estimated diffusivities for the four sample thicknesses
using both of the models of analysis. Figure 6-16 provides
results from using Model 1 as a method of analysis and
Figure 6-17 presents Model 5 results. As shown in the
tables, many of the samples did not produce measurements at
1200 degrees for which convergence was obtained when
analyzed. Since both figures have the same scale on the
vertical axis, the figures provide a visual comparison of
the range of the estimated diffusivities using both models.
As discussed above, the Model 5 diffusivity values are
uniformly lower than those estimated using Model 1. Once

again, the general trend of higher diffusivities at higher

 

231
temperatures can be clearly seen in the graphs. Another
conclusion which is clearly evident in the graphs is the
closer agreement between diffusivity values among all
samples using Model 5. Accounting for the flash penetration
in this model seems to reduce the variance of estimated
values between samples, independent of the sample
thicknesses. The closer agreement between samples can be
taken as another validating factor for Model 5 and the

concept of accounting for flash penetration.

 

CHEUIEERLTV

StmflflNRY'lflmD REKXNMMiwfllkTICMHS

7.1 SUMMARY

As outlined in Chapter 1, the goals of this research
were

1. To determine thermal properties of the materials,

specifically thermal diffusivity, from transient

temperature measurements.

2. To investigate the possibility of internal

radiation as an ancillary means of heat transfer to

Fourier conduction and to investigate penetration of

the laser flash beyond the surface of the specimen.

3. To investigate non—radiative effects which may have

been responsible for systematic disparities between the

measured data and the mathematical model.
The common underlying motivation behind each of the above
objectives was to develop and utilize a heat transfer model
for flash diffusivity experiments which would more
accurately conform to the physical phenomenon observed,
thereby giving greater confidence in the parameter values
reported. The primary means of evaluating the degree of
success achieved in this endeavor, was the extent to which
reductions were achieved in the residual signatures, evident
in previous analyses. These signatures indicated an

inadequacy in the models used previously which, in turn,

232

 

“"1

 

233
implied some unreliability in the estimated parameters.
Other measurement methods of model adequacy used in the
evaluation of the models developed in this research,
included a reduCtion of the standard deviation of the

residuals, a reduction of the confidence region, and the

 

reduction in the variation of the sequential estimates.

In the process of achieving these desired goals, this
research investigated several areas which have not been
examined previously: simultaneous estimation of internal
radiation parameters in flash diffusivity measurement
experiments, development of a numerical means for
stabilizing equations used in extracting radiation heat

transfer parameters from experimental data, and modeling of

 

laser flash penetration into the samples. From these new
models, the magnitude of the extinction coefficient for the
laser flash has been estimated in concert with the other
thermal parameters. Even more important is that the
analysis of actual laboratory data has given validation to
these models and has provided explanations for the
inadequacies in previously used models.

The success of the higher order models developed as
part of this research has been validated in several ways.
Model S, as presented in Chapter 4, achieved a higher level
of agreement between samples when comparing the estimated
parameters measured at one temperature. By contrast, the

results obtained using the simpler model described in

234
Chapter 2 produced a wider range of values under the same
conditions. The observed reduction in the magnitude of the
residuals, the reduction of the residual signature, the
reduction of the confidence region, and the reduction in the
straying of the sequential estimates, are all validations of
the higher order models.

Not all types of materials were found to have
benefitted from the use of the additional parameters of the
higher order models. Materials which are totally opaque or
are coated with a gold film are not susceptible to laser
flash penetration. Additionally, flash diffusivity
laboratory systems which do not allow laser leakage within
the sample enclosure do not benefit from the back-side flash
parameter model described in Chapter 4. The samples which
benefit the most from the higher order models are those that
have permeable or semi-permeable surfaces with respect to
the laser flash.

The value of the main parameter of interest, thermal
diffusivity, can be sensitive to the type of model used,
affecting the magnitude of the estimated value by as much as
20 percent. The results generated by the refined models can
be reported with greater confidence, as demonstrated by the
improvements noted above, and are therefore more reliable
for use in subsequent work with these materials. Since the
differences in estimated diffusivity, the parameter of

interest, can be significant when accounting for flash

 

 

235
penetration, the results from this research can have an
impact on other aspects of engineering. From a design
aspect, this ultimately leads to safer and higher performing
equipment and products designed using the published

estimated parameters obtained using these improved methods.

7.2 RECOMEEEQATION§

For future work in this area, the following
recommendations are offered
1. The internal radiation model should be utilized in
analyzing experiments performed on samples which
actually exhibit combined conduction and internal
radiation.
2. Testing should be continued on materials which are
subject to surface penetration of flash heating.
Testing identical materials at varying thicknesses will
provide additional confirmation of the penetration
models, since estimated diffusivity for thick samples
using Model 1 should approach those of Model S for thin
samples.
3. The models presented in this research, particularly
Model 5, should be incorporated in the software
packages provided with flash diffusivity testing

equipment.

 

 

APPENDIX

 

 

DATA FILE (A;R1)

Palaiseau, France
Ambient Temperature: 20%:
Sample Thickness: L=3.0mm
Maximum measured Reading: 856.9746

Anticipated Residual Std Dev: a=0.7955155

Applied Null Value: 0
Model Diff. Heat Biot Resid.
Nmbr. (a) Flux Number 8, BS 86
1 1.2813 2920 .08633
4 1.1190 3325 .20423 .06850
5 1.1344 3236 .19432 .26207
17 1.1673 1057 .16894 .2782 .63479 1.4243
95% Confidence Correlation
Interval Coefficient
1 .005776 .940228
4 .010879 .806608
5 .006361 .709900
17 .006128 .515124
IHKHA FUHHE AL}UZ
Palaiseau, France
Ambient Temperature: 980%:
Sample Thickness: L=3.05mm
Maximum measured Reading: 462.1537
Anticipated Residual Std Dev: 0:.615957
Applied Null Value: 0
Model Diff. Heat Biot
Nmbr . (a) Flux Number 3, BS 35
1 1.23366 1587 .08165
4 1.22658 1594 .08490 .00344
5 1.22665 1591 .08487 .06462
17 1.23182 1521 .08268 .0112 .33750 .19175
95% Confidence Correlation
Interval Coefficient
1 .001490 .42142
4 .004653 .458935
5 .004571 .458409
17 .004606 .491917

236

(8)
4.811
2.416
1.945
1.0417

Resid.
(s)
.74589
.73553
.73579
.70865

 

Ambient Temperature:

Sample Thickness:

Maximum measured Reading:
Anticipated Residual Std Dev:

Applied Null Value:

Model Diff.
Nmbr. (a)
1 .33181
4 .29947
5 .29932
17 .31533
95% Confidence
Interval
1 .005376
4 .013411
5 .008977
17 .003420

Vandoeuvre les Nancy, France

Ambient Temperature: 20%:

Sample Thickness: L=10.34mm
Maximum measured Reading: -0.0356
Anticipated Residual Std Dev: a=.00018395
Applied Null Value: -0.1631
Model Diff. Heat Biot
Nmbr. (a) Flux Number 8, 85

1 2.13194 1.3764 .02821

4 2.12487 1.3772 .02849 .02114

5 2.13169 1.3758 .0280? .07265

17 2.13074 1.1331 .02819 .03750 .02194
95% Confidence Correlation

Interval Coefficient

1 .002399 .585307

4 .002540 .603788

5 .002399 .589470

17 .014275 .911547

237

DATA FILE (C_R1)

West Lafayette, Indiana
400°C
L=1.9685mm
2.718663
=.0047949
0
Heat Biot
Flux Number [3, BS
6.388 .12974
7.019 .21629 .08770
6.916 .21927 .16159
3.406 .17602 .10176 .3603
Correlation
Coefficient
.947993
.885454
.855693
.558632

IH¥EA F1133 (ELJII)

Be

.0033

35

.00004

Resid.

(s)

.01492
.01032
.00910
.00293

Resid.

(s)

.00040
.00040
.00040
.00040

238

DATA FILE (E_R2)
Vandoeuvre les Nancy, France

Ambient Temperature: 20%:
Sample Thickness: L=4.6mm
Maximum measured Reading: 0.3346
Anticipated Residual Std Dev: 0=.0039249
Applied Null Value: -0.3453
Model Diff. Heat Biot
Nmbr . (or) Flux Number 8, BS
1 2.08497 3.431 .06287
4 2.08470 3.431 .06292 .00003
5 2.08489 3.431 .06290 .00098
17 Unstable
95% Confidence Correlation
Interval Coefficient
1 .006296 .962768
4 .006296 .962727
5 .006280 .953760

17 Unstable

llAflfliiFIIJE (GLJII)
Bombay India

Ambient Temperature: 1403%:
Sample Thickness: L=1.0872mm
Maximum measured Reading: 496
Anticipated Residual Std Dev: 0:.3709797
Applied Null Value: 222
Model Diff. Heat Biot
Nmbr . (or) Flux Number [3, BS

1 11.260 350.0 .10934

4 11.260 350.0 .10934 9.76E-10

5 11.284 348.6 .10626 .00076

17 11.259 321.9 .10934 .00005 .01250
95% Confidence Correlation

Interval Coefficient

1 .008716 .956708

4 .008732 .956708

5 .015172 .593499

17 .013529 .956886

Be

Be

.00025

Resid.

(s)

.00550
.00550
.00550

 

239

IHVLA.F1]HE (ELJII)
Trappes, France

Ambient Temperature: 300%:

Sample Thickness: L=3mm
Maximum measured Reading: 3.118764
Anticipated Residual Std Dev: 0:.00786077
Applied Null Value: 0

Model Diff. Heat Biot

Nmbr . (or) Flux Number 8, BS

1 14.8448 11.09 .14804

4 11.2068 13.92 .36753 .01475
5 12.3922 12.40 .27802 .33994
17 Unstable

95% Confidence Correlation
Interval Coefficient
1 .084077 .001804
4 .293294 .002279
5 .198615 .003849

17 Unstable

DATA FILE (I_R1)
Buenos Aires, Argentina

Ambient Temperature: 20%:

Sample Thickness: L=3.895mm
Maximum measured Reading: 1.85
Anticipated Residual Std Dev: a=.0178227
Applied Null Value: -2.8656
Model Diff. Heat Biot

Nmbr. (oz) Flux Number [3, BS

1 3.65094 18.11 .00001

4 3.57298 18.14 .00001 .01195
5 3.57561 18.14 .00001 .20357
7

1 Unstable

95% Confidence Correlation
Interval Coefficient
1 .002649 .343374
4 .005880 .169516
5 .005509 .166428

17 Unstable

Ba

Ba

Resid.
(s)
.30019
.29742
.29801

Resid.
(s)
.02405
.02141
.02136

Ambient Temperature:
Sample Thickness:

Maximum measured Reading:
Anticipated Residual Std Dev:

Applied Null Value:

Model Diff. Heat
Nmbr. (a) Flux
1 .59017 5801
4 .59017 5801
5 .59057 5787

17 Unstable

95% Confidence

Interval

1 .011383
4 .011484
5 .013807

17 Unstable

Ambient Temperature:
Sample Thickness:

Maximum measured Reading:
Anticipated Residual Std Dev:

Applied Null Value:

Model Diff. Heat
Nmbr . (a) Flux
1 6.46155 75.89
4 6.46109 75.89
5 6.48789 75.75
17 Unstable

95% Confidence

Interval

1 .029238
4 .060280
5 .029554

17 Unstable

240

DATA FILE (J_R1)
Talence, France

20°C
L=18.05mm
167

a=1.043377
0

Biot
Number B. 35 Be
.49238
.49237
.48915

.00001
.00563

Correlation
Coefficient
.896098
.896099
.630977

llAflfllIFIIJE (KLJCL)
Belgrade, Yugoslavia

20°C
L=2.48mm
27.74
0:.4293872
-1.319

Biot
Number B. 35 B:
.02915
.02915
.02838

.00001
.00035

Correlation
Coefficient
.078318
.078370
.080687

Resid.
(8)
1.0452
1.0452
1.0464

Resid.
(5)
1.0666
1.0667
0.9894

241

[HTDALIFIIJB (LLICL)

Poitiers, France
Ambient Temperature: 20%:
Sample Thickness: L=2mm
Maximum measured Reading: 1.32574

Anticipated Residual Std Dev: a=.004954192

Applied Null Value: 0
Model Diff. Heat Biot
Nmbr . (or) Flux Number [3, BS
1 3.81851 2.826 .04433
4 3.49869 2.885 .05648 .01103
5 3.58016 2.856 .05308 .17115
17 3.96091 1.358 .02624 .00154 1.0000
95% Confidence Correlation
Interval Coefficient
1 .074467 .006850
4 .187615 .005592
5 .162719 .006211
17 .006300 .335489
IDAHHXZFIIJE (FLJRl)
Hunan, China
Ambient Temperature: 23%:
Sample Thickness: L=2.55mm
Maximum measured Reading: 3048
Anticipated Residual Std Dev: 0:5.4985
Applied Null Value: 0
Model Diff. Heat Biot
Nmbr . (a) Flux Number 8, 55
1 20.307 8596 .06568
4 19.654 8788 .08269 .00087
5 19.487 8800 .08762 .14639
17 19.927 3423 .07764 .00420 1.000
95% Confidence Correlation
Interval Coefficient
1 .005280 .847107
4 .015978 .831695
5 .014074 .825329
17 .003583 .899414

3:.

.00050

8:.

2210

Resid.
(s)
.12607
.12588
.12591
.00689

Resid.
(8)
17.732
17.094
16.859
7.255

 

Ambient Temperature:
Sample Thickness:

Maximum measured Reading:
Anticipated Residual Std Dev:

Applied Null Value:

Model Diff. Heat
Nmbr. (a) Flux
1 79.7225 17.58
4 79.7214 17.58
5 79.7793 17.54
17 Unstable

95% Confidence

Interval

1 .008504
4 .009221
5 .013585

17 Unstable

Ambient Temperature:
Sample Thickness:

Maximum measured Reading:
Anticipated Residual Std Dev:

Applied Null Value:

Model Diff. Heat
Nmbr. (a) Flux
1 71.8983 6.182
4 71.8947 6.183
5 71.9000 6.183
17 71.8942 6.276
95% Confidence
Interval
1 .003925
4 .012128
5 .010506

17 .008678

242

DATA FILE (N__R1)
Stuttgart, Germany

366°C
L=1.86mm
8.046875

1.1152338

Biot
Number B,
.28631
.28633
.28443

.00001
.00500

Correlation
Coefficient
.939483
.939475
.644702

DATA FILE (C_R1)

O=.0079494

35

.00006

Le Barp, France
20°C
L=9.85mm
0.63672

=.00514629

0

Biot
Number B, B5
.00001
.00001 .00001
.00001 .02500
.00001 .01055
Correlation
Coefficient
.314520
.316054
.028702
.323732

.00475

Resid.
B6 (3)
.04862
.04862
.04887

Resid.
36 (S)
.00476
.00476
.00476
.00476

243

DATA FILE (Q_R1)
Manchester, United Kingdom

Ambient Temperature: 327%:
Sample Thickness: L=3.24mm
Maximum measured Reading: 2156.688
Anticipated Residual Std Dev: o=15.11283
Applied Null Value: 557.8
Model Diff. Heat Biot
Nmbr. (a) Flux Number B4 B5 B6
1 12.248 5102 .00767
4 10.571 5466 .05176 '.01313
5 10.639 5430 .05113 .36818
17 11.093 1647 .03781 .27186 1.0000 12.253
95% Confidence Correlation
Interval Coefficient
1 .012323 .492996
4 .029308 .247158
5 .021646 .182597
17 .022393 .078951
DATA FILE (S_R1)
Ardmore, Pennsylvania
Ambient Temperature: Unknown
Sample Thickness: L=2.9mm
Maximum measured Reading: 6.39961
Anticipated Residual Std Dev: a=0.1028782
Applied Null Value: 0
Model Diff. Heat Biot
Nmbr. (a) Flux Number B4 B5 B6
1 3.50009 45.05 1.0234
4 2.43694 79.74 2.1573 .05063
5 2.24625 79.63 2.8976 .36752
17 2.23109 27.72 2.9401 .37014 1.00000 .00000
95% Confidence Correlation
Interval Coefficient
1 .196966 .008036
4 1.04830 .016209
5 .490873 .017972
17 .500564 .017879

Resid.
(8)
22.909
18.798
17.939
16.155

Resid.
(s)
.90882
.89569
.89200
.89200

244

CBCF DATA (Holometrix)
Oak Ridge, Tennessee

Ambient Temperature: 100%:
Sample Thickness: L=1.0mm
Maximum measured Reading: 3.4817
Anticipated Residual Std Dev: 0.002629
Applied Null Value: 0
Model Diff. Heat Biot
Nmbr . (or) Flux Number B4 B5 B6
1 .32426 4.231 .14654
4 .29172 4.627 .22840 .02521
5 .29514 4.508 .22029 .08099
17 .29772 4.488 .21352 .08200 .85912 .00271
95% Confidence Correlation
Interval Coefficient
1 .00457 .959097
4 .00556 .794955
5 .003523 .688726
17 .005083 .690548
CBCF DATA (Holometrix)
Oak Ridge, Tennessee
Ambient Temperature: 400%:
Sample Thickness: L=0.96mm
Maximum measured Reading: 7.762737
Anticipated Residual Std Dev: 0.005604
Applied Null Value: 0
Model Diff. Heat Biot
Nmbr. (a) Flux Number B4 B5 B6
1 .31351 13.45 .53483
4 .28644 14.77 .65473 .02034
5 .28858 14.01 .64592 .07311
17 .29210 13.92 .62951 .06521 1.0000 .01561
95% Confidence Correlation
Interval Coefficient
1 .003946 .934237
4 .006498 .751121
5 .004486 .672286
17 .001371 .289023

Resid.

(S)

.01726
.00609
.00495
.00414

Resid.

(s)

.03657
.01694
.01485
.00419

24S

CBCF DATA (Holometrix)

Oak Ridge, Tennessee

 

Ambient Temperature: 500%:
Sample Thickness: L=1.0mm
Maximum measured Reading: 9.217506
Anticipated Residual Std Dev: 0.007871
Applied Null Value: 0
Model Diff. Heat Biot Resid.
Nmbr. (a) Flux Number B, B5 B6 (8)
1 .31217 18.52 .73590 .04472
4 .28552 20.47 .88173 .01950 .02274
5 .28757 19.14 .87119 .07156 .02040
17 .29142 19.01 .84896 .06304 1.000 .02182 .00629
95% Confidence Correlation
Interval Coefficient
1 .003949 .910976
4 .007236 .703975
5 .005090 .627923
17 .001694 .059215
CBCF DATA (Holometrix)
Oak Ridge, Tennessee
Ambient Temperature: 600%:
Sample Thickness: L=0.96mm
Maximum measured Reading: 8.549782
Anticipated Residual Std Dev: 0.007031
Applied Null Value: 0
Model Diff. Heat Biot Resid.
Nmbr. (or) Flux Number B, B5 B6 (8)
1 .30868 18.02 .86752 .04093
4 .27930 20.50 1.07002 .01872 .01946
5 .28255 18.83 1.04763 .06800 .01790
17 .28620 19.46 1.02083 .06103 1.000 .01746 .00723
95% Confidence Correlation
Interval Coefficient
1 .004041 .904892
4 .007416 .642220
5 .005177 .572979
17 .002255 .092708

 

246

CBCF DATA (Holometrix)
Oak Ridge, Tennessee

Ambient Temperature: 700%:
Sample Thickness: L=.956mm
Maximum measured Reading: 6.8258
Anticipated Residual Std Dev: 0.0091074
Applied Null Value: 0
Model Diff. Heat Biot Resid.
Nmbr. (or) Flux Number B, B5 B6 (3)
1 .31136 17.85 1.23326 .03697
4 .27615 21.29 1.56101 .02107 .01250
5 .28193 18.64 1.50312 .06964 .01244
17 .28364 19.42 1.48574 .06681 1.00000 .00751 .01074
95% Confidence Correlation
Interval Coefficient
1 .004431 .908390
4 .005980 .364464
5 .004338 .367844

17 .004025 .329394

REFERENCES

REFERENCES

[l] R. Siegel and J. Howell, Thermal Radiation Heat

Transfer, Hemisphere Publishing Corp., New York (1981).

[2] A. Hazzah, Transient Heat Transfer by Simultaneous

Conduction and Radiation in Absorbing, Emitting and
Scattering Medium, Ph.D. Dissertation, Michigan State

University (1968).

[3] W. Minkowycz, E. Sparrow, G. Schneider and R. Fletcher,

 

Handbook of Numerical Heat Transfer, Wiley, New York,
(1988).

[4] A. Hazzah and J. Beck, Unsteagy Cembined Conduction-
Radiation Energy Transfer Using e Rigoreue Differential

Method, International Journal of Heat and Mass Transfer,
Pergamon Press, (1970).
[5] A. Tuntomo and C.Tien, Trensient Heat Trenefer in a

Conducting Pertigle With Internal Radiant Absotption,
Journal of Heat Transfer, ASME, (1992).

[6] A. Helte, Radiative and Conductive Heat Transfer in

Pourous Media: Estimation of the Effective Thermal
Cenductivity, Journal of Applied Physics, American Institute

of Physics, (1993).
[7] P. Jones and Y. Bayazitoglu, Ragietien, Conduction and

Convection From a Sphere in an AbeorbingI Emitting, Gray
Medium, Journal of Heat Transfer, ASME, (1992).

[8] M. Brewster, Thermal Radiative Transfer and Properties,

247

 

248
Wiley, New York, (1992).
[9] E. Sparrow and R. Cess, Radiation Heat Transfer,
Hemisphere, New York, (1978).
[10] M. Modest, Radiative Heat Transfer, McGraw-Hill, New
York, (1993).

[11] D. Schwander, G. Flamant and G. Olalde, Effects pf

Boundepy Properties on Transient Tempetatupe Dietributions
in Condensed Semi-Traneparent Media, International Journal
of Heat And Mass Transfer, (1990).

[12] M. Naraghi and C. Saltiel, Qompined Mpde Heat Transfer
in Radiatively Partieipating Media: gompdtetionel
Cpneideretions, ASME Heat Transfer Division, (1990).

[13] H. Chu and T. Chen, Transient Conduptipn and Radiation
Heat Tranefer in Ultra-Fine Powder Insuiatipn, Hemisphere,
New York, (1991).

[14] M. Goldstein and J. Howell, Boundaty gonditions for the

Diffusion Solution of goupled Conductipn-Radiatiop Problems,
NASA TN D-4618, (1968).

[15] J. Beck, B. Blackwell, C. St Clair, Inverse Heat
Conduction - Ill Posed Problems, Wiley, New York, (1985).
[16] J. Novotny and K. Yang, The Interaction of Thermal
Radiation in Optically Thick Boundaty Layers, ASME 67—HT-9,
(1967).

[17] R. Edwards and R. Bobco, Radiant Heat Transfer From

Isothermal Diepersione With Isptropic Scattering, ASME 67-
HT-8, (1967).

249
[18] W. Dalzell and A. Sarofim, Optical Constants of Soot
and Their Application to Heat Flux Calculations, ASME 68-HT-
13, (1968).
[19] J. Schornhorst and R. Viscanta, An Experimental
Examination of the Validity of the Commonly Used Methods of
Heat Transfer Analysis, ASME 68-HT-42, (1968).
[20] A Sarofim, I. Vasalos and A. Jeje, Experimental and
Theoretigal Stddy of Absorption in an Anisotropically
Scattering Medium, ASME 71-HT-20, (1972).
[21] R. Bobco, D'r i na Emi sivi ie From a Tw
Dimensional epsorping Sgattering Medium: The Semi-Infinite
eiep, ASME 67-HT-12, (1967).

[22] L. Matthews, R. Viscanta, and F. Incropera, Develgpment

of Inverse Methode fgr Determining Thermoppysigal and
Radiative Propertiee of High-Temperatdte Fibroue Materials,

International Journal of Heat and Mass Transfer, (1984).

[23] A. Neto and M. Ozisik, An Inverse Analysis of

Simultan u E 'm ' Phas unc ' n lb d an O tical
Thicknees, Developments in Radiative Heat Transfer, ASME,

(1992).
[24] G. Forsyth and W. Wasow, Finite Difference Methode for

Pattial Diffeteptiel Egpetions, Wiley, New York, (1960).
[25] J. Beck and K. Arnold, Parameter Estimation, Wiley, New

York, (1977).

[26] S. Conte and C. de Boor, Elemeptepy Numerical Analysis,
McGraw-Hill, New York, (1980).

250
[27] D. Murio, The Mollification Method and the Numerical
Solution of Ill posed Problems, Wiley-Interscience, New
York, (1993).
[28] S. Chandrasekhar, Radiative Transfer, Dover
Publications, New York, (1960).
[29] N. Ruperti, M. Raynaud and J.Sacadura, The Method for
th Solution of th Cou le I v rse He C u ti n-
Radiation Problem, ASME Journal of Heat Transfer, Vol 118,

(1996).

[30] U. Koylu and G. Faeth, Spegtpei Extipgtigp Qpefficients

of Spot Aggregates Ftpm Turbulent Diffusign Flamee, ASME
Journal of Heat Transfer, Vol 118, (1996).

[31] T. Hendricks and J. Howell, Ab 0 ion S terin
Coefficients and Seattering Phage Edpgtipne ip Retigpleted
Poroue Ceramiee, ASME Journal of Heat Transfer, Vol 118,
(1996).

[32] T. Ito and S. Fujimura, Simuitaneous Measurement of

Temperature and Absotption of Distributed Medium by Ueing
Ipfrared Emmieeign CT, IEEE Transactions on Instrumentation

and Measurement, Vol 44 No 3 (1995).

[33] J. Longtin, T Qiu, and C. Tien, Pdlsed Lager Heating of
Highly Absorbing Perticlee, ASME Journal of Heat Transfer,
Vol 117, (1995).

[34] Y. Sakamoto, M. Ishiguro and G. Ktagawa, Akaike
Informatiop Criterion Statistics, Reidel Publishing, Tokyo,

(1986).

251

[35] J. Beck and A. Osman, Sequential Estimation of

Temperature-Dependent Thermal Properties, High Temperatures-
High Pressures, Vol 23, (1991).

[36] O. Hahn, F. Raether, M. Arduini-Schuster, and J. Fricke

Trans'en Co 1 d nd c 'v R dia iv Heat Transfer i

Abeorping, emitting ehd Seetteting Media: Application tg
|
Laeer Flaeh Methgde eh Cepemig Metetiale, International I

Journal of Heat and Mass Transfer, Vol 40 No 3, (1996).

[37] J. Beck and R. Dinwiddie, P r m t E ima i n M d

for Flash Thermal Diftueivity with Twp Different Heat
Transfet Coeffieiente, Proceedings from 23rd Thermal

Conductivity Conference, Technomic Publishing, Lancaster,
Pa. (1996).
[38] Y. Gu, D. Zhu, L. Zhu, J.Ye, Thermal Diffusivity

Measurement pf Thin Fiims by the Peripdic Heet Flgw Method
with Laser Heeting, High Temperatures-High Pressures, Vol

25, (1993).

[39] P. Boulet, G.Jeandel, P. De Dianous and F. Pincemin,
Combined Radietioh and Conductien in Fiherpus Insulation,
Proceedings from 23rd Thermal Conductivity Conference,
Technomic Publishing, Lancaster, Pa. (1996).

[40] American Society for Testing and Materials, Standard
T t Method r Th rmal Diffusivit of So ids b the Flash
Method,

E 1461-92 ASTM Committe on Standards, Philadelphia, Pa.

(1992).

252
[41] H. Wang, R. Dinwiddie and P. Gaal, Multiple Station
Thermal Diffusivity Instrument, Proceedings from 23rd
Thermal Conductivity Conference pp 120-125, Technomic

Publishing, Lancaster, Pa. (1996).

[42] J. Koski, Improved Data RedUgtion Methods for Laser

Pulse Diffusivity Determination with the uee of
Mihicomputeps, National Technical Information Service,

Springfield Va.(1981).
[43] R. Cowan, Pdlse Method of Meeepring Thetmei Qiffdsivity
at High Tempereture, Journal of Applied Physics, Vol. 34,

No. 4, pp. 926-927, (1963).

[44] L. Clark, R. Taylor. EEdiati2n_L2§§_in_tbs_filash_fleth2d
‘ fer Thepmei Difhsivity, Journal of Applied Physics, Vol. 26,

No .2, pp. 714-719, (1975).

[45] C. Strodtman, D ' n 'm' t' ' A 'ed
to a Sgpeeze Film Gae Journal Beering, Ph.D. Dissertation,
Michigan State University, (1968).

[46] W. Parker, Jenkins, C. Butler and G. Abbott, Flash
Method of Detetmining Thermal Diffdeivity, Heat Capacity and
Thermal Conductivity, Journal of Applied Physics, vol.
32(9), 1679-1684, (1961).

[47] R. Taylor, Heat Phlse Thermal Diffusivity Meeeurements,
High Temperatures - High Pressures, vol. 11, pp. 43-58,
(1979).

[48] M. Raynaud, J. Beck, R. Shoemaker and R. Taylor,

Seghentiei Eetimetion of Thermal Diffueivity for Flash

 

 

 

 

 

253
Teete, Proceedings from 20th Thermal Conductivity Conference
pp 305-321, Plenum Publishing, (1989).
[49] C. Lin, C. Hwang, and Y. Chang, The Unsteady Solutione
of a Unified Heet Conductipn Eghation, International Journal
of Heat and Mass Transfer, Vol. 40, No. 7, pp. 1716-1719,
(1997).
[50] L. Vozar and T. Sramkova, Two Data Reduction Methgde
fer Evalpation of Thermal Diffdsivity frgm Step-Heetihg

Measuremente, International Journal of Heat and Mass

 

Transfer, Vol. 40, No. 7, pp. 1647-1655, (1997).

[51] J. Beck, K. Cole, A. Haji-Sheikh, and B. Litkouhi,_geet
Cohduction geing Sreen’e Functipne, Hemisphere Publishing,
Washington D.C., (1992).

[52] S. Andre and A. Degiovanni, Experimental Meesurements
of the Phonig Diffueivity of Semi-Transparent Materiels up
to 800K, Glastechnische Berichte, Vol. 66, No. 11, pp. 291-
298, (1993).

[53] H. Ohta, M. Masuda, K. Watanabe, K. Nakajima, H.
Shibata, and Y. Waseda, Determination pf Thermal
Diffusivitiee pf Continuous Casting Powdere for Steel by
Precicely Exeluding the Contributign pf the Radiative

Compenent et High Temperature, Journal of the Iron and Steel
Institute of Japan, Vol 80, No. 6, pp. 33-38, (1994).

[54] C. Moreau, P. Fargier-Richard, R. St. Jacques, G.

Robert, and P.Cielo, Thermal Diffusivity of Plasma Sprayed
Tungsten Coetihge, Surface and Coatings Technology, Vol. 61,

254
No. 1-3, Part 1, pp. 67-71, Dec (1993).
[55] K. Yamamoto, T. Hirosawa, K. Yoshikawa, K. Morozumi,
ans S. Nomura, Melting Temperature and Thermal Conductivity
of Irradiated Mixed Oxide Fuel, Journal of Nuclear
Materials, Vol. 204, pp. 85-92, (1993).

[56] R. Bailey, F. Cruickshank, P. Kerkoc, D. Pugh and J.

Sherwood, Thermal Conductivity gf the Mpleculer Ctyetal

(-)2-(alpha-methylbenzylaming)-5-Nitrppyridihe, dphrnal of
Applied Phyeics, Vol. 74, Part 5, pp. 3047-3051, (1993).

[57] L. Kehoe, P. Kelly, G. O'Connor, M. O’Reilly and G.
Crean, Me h 010 f r La - s rm 1 Diffusivit

M as r nmen of Advan e Mult'-Chi M ul mi
Matetials, IEEE Transactions on Components, Packaging and
Manufacturing Technology, Part A, Vol. 18, No. 4, December,
(1995).

[58] H. Maleki and L. Holland, Thetmai Diffpsivity
Measurement of N ar-Pseudobina H Te Solid and Melt Te-
Rich HngIe and HanTe melte ahd Pure Te Splid and Melt by
the Laser Fleeh Technigde, Journal of Applied Physics, Vol.
76, Part 7, October (1994).

[59] R. Heckman, Error Analyeie of the Flesh Thermai
Diffueivity Teehnigde, Proceedings of the 14th Thermal
Conductivity Conference, pp. 491-498, Plenum Publishing, New
York, (1974).

[60] R. Taylor, T. Lee and A. Donaldson, Determination of

Thermal Physieel Properties of Layered Composites by the

 

255
Flash Method, Proceedings of the 15th Thermal Conductivity
Conference, pp. 135-148, Plenum Publishing, New York,
(1978).
[61] M. Yovanovich, Simple Explicit Expressigns to;
Calculation pf the Heieier-Grober Charts, Proceedings of the

1996 Heat Transfer Conference, American Institute of

Aeronautics and Astronautics 96-3968, Reston Va., (1996).
[62] G. Golub, Matti; ggmputetione, Johns Hopkins University

Press, Baltimore Md, (1989).
[63] J. Beck, S n 1 'vi C f i i n Utili e 'n Non-

Linear Eetimetioh with Smeli Patametets ih e fleet Trenefer

Problem, Jornal of Basic Engineering, (1970).

 

"‘11111111111“