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This is to certify that the
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SEQUENTIAL E; i IMAI IUI\ . . -MPERA‘IURE
DEPENDENT THERMAL PROPERTIES

presented by

LUKE HOLLISTER

has been accepted towards fulfillment
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SEQUENTIAL ESTIMATION OF
TEMPERATURE-DEPENDENT THERMAL
PROPERTIES

LukeHollister

ATHESIS

Suhmitted to

Michigan State University

in partial fulfillment
of the requirements

for the degree of

MASTER OF SCIENCE

Department of Mechanical Engineering

1990

é¢pm-3.sbq

ABSTRACT

SEQUENTIAL ESTIMATION OF TEMPERATURE DEPENDENT
THERMAL PROPERTIES

BY
Luke Hollister

The determination of thermal properties (such as thermal
conductivity and volumetric heat capacity) as a function of
temperature in transient tests is described. In some
experiments, an individual test covers only part of the
temperature range of interest and subsequent tests cover more
of the temperature range. If temperature-independent thermal
properties are estimated from the results of individual tests.
there is a question of what.temperature to assign the property'
values, since the temperature varies during each test. On the
other'hand, if the thermal properties are assumed to vary with
temperature in the analysis of a given test, there can be
considerable uncertainty in the calculated temperature-
dependent values. A method is presented that solves these
problems by utilizingtwo new concepts. One of these concepts
is the use of regularization to restrict the temperature
variation in the thermal properties of the first test. The
second concept incorporates the information from prior tests
in succeeding ones; it is a sequential concept. '

The method has been implemented in a computer program.
Results of using the method are given for a composite
material. In addition to providing values of the thermal
conductivity and volumetric heat capacity with temperature,
the computer program provides estimates of the confidence

regions of the estimates.

In memory of my brother. John Coburn Hollister

10/04/63 -8/18/85

iii

ACKNOWLEDGEMENTS

I would like to thank Dr. James V. Beck for his moral and
financial support throughout my graduate program.

iv

TABLE OF CONTENTS

List of Tables
List of Figures

Nomenclature

Chapter 1- Introduction
Chapter 2- Mathematical Formulation and Theory-

2.5
2.6
2.7

Linear Case

Introduction/Motivation

Model and Statistical Assumptions
Ordinary Least Squares
Sequential-Over-Experiments Estimation
and Regularization

Sequential Method

Discussion and Comparison of Methods
Conclusions and Remarks

Chapter 3- Mathematical Formulation and Theory-

3.1
3.2

3.3

3.4

Nonlinear Case
Introduction/Motivation

Thermal Property Determination and
Regularization

Nonlinear Sequential-Over-Experiments
Derivation

Program PROPlD Description

Chapter 4- Description of Experiment

4.1
4.2
4.3
4.4

4.5

Introduction/Motivation

Experiment Description

Discussion of Measured Temperatures
Methodology of Sequential-Over-Experiments
Approach

Choice of Regularization Parameters

PAGE
vii
viii

ix

18
18
20

24

25

29
39

41
42
44

44
46

Chapter 5-
5.1
5.2
5.3
5.4
5.5
5.6
5.7
5.8
5.9

Chapter 6-
Appendices

Estimation of Thermal Properties
Introduction/Motivation
Sequential-Over-Experiments Results
Confidence Intervals
Transferral of Prior Information
‘Backwards Run'

Residuals
Sensitivity Coefficients
Sequential-Over-Time Parameter Estimates

'Direct' vs. Sequential-Over-Experiments

Parameter Estimates

Conclusions and Recommendations

PROGRAM LSQ
PROGRAM SEQUEN
PROGRAM SEQUENZ
Figure 13

List of re

ferences

PAGE

53
54
57
58
58
60
62

68

71

74
75
77
79
80

Table

Table

Table

Table

Table

Table

Table

LIST OF TABLES

Sequential-Over-Experiments
Results (linear case)

Sequential Method Results
(linear case)

Simulated Measurements for
Sample Case

Diagonal Components of A Matrix
for First Data Set (27°C)
(varying :11 values)

Diagonal Components of A Matrix
for First Data Set (27°C)
(varying a, values)

Sequential Analysis, Including
Estimates and Confidence Regions
(nonlinear case)

Diagonal Components of A for the
Sequential Analysis

vii

PAGE

22

22

23

51

52

55

59

Figure

Figure

Figure

Figure

Figure

Figure
Figure
Figure
Figure
Figure
Figure

Figure

Figure

10

11

12

13

LIST OF FIGURES

PAGE

Simulated measurements for thermal

conductivity versus temperature 7
Experimental apparatus for measuring

thermal conductivity and density-

specific heat 43
Temperature histories of specimen at

five different interfaces 45

Variation of thermal conductivity

components with regularization parameter

al for 27°C data set 49
Variation of density-specific heat

components with regularization parameter

(:3 for 27°C data set ' so
Sequential parameter estimates and

confidence regions 56
Residuals for 27°C and 96°C data sets 61
Sensitivity coefficients for 27°C data set 63
Sensitivity coefficients for 96°C data set 64
Sequential-over-time parameter estimates

for 27°C data set 65
Sequential-over-time parameter estimates

for 96°C data set 67

'Direct' vs. sequential-over-experiments
parameter estimates 69
Photograph of experimental apparatus

and hardware used to collect data 79

. viii

a :4 x Jfi n-cn-e i? z :r m cm o In 0's: 5

Ho

Nomenclature

Prior information matrix
Regularization parameter

Estimated vector of parameters
Parameter vector

Specific heat

Criterion for parameter convergence
Regularization matrix

Thermal conductivity

Number of measurements

ith value of mathematical model
Density

Magnitude of sum of squares function
Time

ith temperature

Position

Sensitivity matrix

Measured value for ith component

ix

Chapter 1
Introduction

In transient parameter estimation of thermal properties
of new’materials and composites, there is a need for a method
for analyzing experiments over a wide range of temperatures.
In many cases, it is difficult or impractical to cover the
entire range in one experiment. Several experiments commonly
are used to cover the whole range with possible overlaps
between experiments. In one transient method for parameter
estimation a heat flux pulse or step is applied at a surface
and temperature histories are measured in the tested specimen.
Since the temperature varies during the test, it is not clear
how to assign temperatures to the parameter estimates. The
standard procedure is to analyze each experiment separately,
without taking into account prior information from previous
experiments or data acquired at different laboratories.

The present work is an extension of parameter estimation
which improves the estimates of the temperature-dependent
thermal conductivity and density-specific heat by allowing
data from subsequent experiments to be combined with the
results of the initial analysis. This process, ”sequential-
over-experiments estimation”. makes it unnecessary to retain
all the details of the initial data base. (There is also a

time-sequential aspect, which is included). Also, the order

1

in which experiments are analyzed is not important. An
additional feature of the method, regularization, is used to
smooth the estimates of parameters at temperatures outside the
range of the initial test results. Regularization improves
the subsequent convergence of parameter values when additional
experiments are added to extend the temperature range.

Similar experimental and analytical work was performed by
(Rooke and Taylor, 1987). (Loh, 1989). (Beck, Hollister and
Osman 1990), and (Scott and Beck, 1989). The concept of
combining experiments is a topic of a statistical concept
called 'group sequential analysis'. This concept has been
explored by (Chang, Therneau, Wieand, Che, 1987), in a study
of the therapeutic efficacy'of anticancer drugs. .Although the
logic is similar to the sequential-over-experiments estimation
method, the group sequential method allows for early stopping
of the analysis when a treatment is considered effective or
inneffective.

The thesis consists of six chapters. Chapter 2
illustrates and proves the validity of the sequential-over-
experiments method for a simple linear example. Chapter 3
illustrates the nonlinear theory'of the method” along with the
regularization technique. Chapter 4 offers a description of
the experimental apparatus, a description of the data, and a
determination of a suitable magnitude of regularization to be
used in the sequential analysis. Chapter 5 shows the results
of this complete analysis. Finally, Chapter 6 gives some

conclusions and recommendations.

Chapter 2

Mathematical Formulation and Theory-Linear Case

2.1 Introduction

The basic concepts of sequential-over-experiments
parameter estimation are illustrated in this chapter. The
method is important because it extends the capabilities of
parameter estimation. In the usual parameter estimation
problem. it is implicitly assumed that there' is enough
information contained in the data of the associated experiment
to estimaterthe parameters of interest. In some cases, it may
not be possible (or may be very difficult) to estimate the
needed parameters for a single experiment or even a group of
experiments that are utilized separately to estimate the
needed parameters. However, by combining experiments in a
sequential fashion. the parameters of interest can be acquired
by transferring information through a sequence of experiments.

Some of the sequential-over-experiments concepts are
illustrated by discussing the theory and then applying it
first to a simple linear example. This example is the
measurement of thermal conductivity as a function of
temperature.

The outline of the remainder of Chapter 2 is now given.
' 3

Section 2.2 gives the mathematical model of the simple linear
example and the statistical assumptions for the parameter
estimation theory. Section 2.3 describes the ordinary least
squares theory applied to the analysis of the data. Section
2.4 illustrates the sequential-over-experiments estimation
theory and the regularization technique. Section 2.5
describes the sequential method. illustrating the accuracy
acquired with the addition of more measurements. Section 2.6
discusses and compares the methods presented in Sections 2.3-

2.5. Lastly. Section 2.7 offers some conclusions and remarks.

2.2 Model and Statistical Assumptions

In this chapter the thermal conductivity is considered as

a linear function of temperature according to the following

model,

“i - 61+82Ti; i-lozoooov" (2.1)

where

I“ - true value of the ith thermal conductivity
81 . y-intercept of k vs. T profile

82 8 slope of k vs. T profile

Ti - ith temperature

N - number of measurements

This model can be written in matrix form as,

1| . XS (2.2)

5
where the sensitivity matrix x (see Beck & Arnold. pg. 17) is

.1 T1.

1 T,

X8 (2.3)

.1 Ts

 

 

The vector of true values of thermal conductivity, n, is

Ni
11- "3 (2.4)

 

The parameter vector. 8, the model given by eq. (2) is

642:]

The underlying theory of parameter estimation is based on
several standard statistical assumptions. These assumptions
basically describe how the errors between the mathematical
model and experimental data are modeled. The first of these
assumptions is that the errors are additive according to the

following model.

Y- = fli + ti . (2.6)

where Yi is the measured value associated with the ith value.
and ti is the error associated with the ith value. The errors
are assumed to have zero mean (sup - 0) and constant
variance (V(ti) I 0°) . Lastly, the errors are modeled as being
uncorrelated (Eui :j) . 0, iii).

In order to illustrate and prove the validity of
sequential-over-experiments estimation. a simulated set of
data is constructed. Simulated conductivity measurements are
found for five values at each of four different temperature
levels, 25°C, 50°C, 75°C. and 100°C. The simulated measurements
are calculated by assuming a random error from the straight
line of k-O.5 N/m-C at 25°C, to k-0.6 film-C at 100°C. Twenty
simulated measurements are used, thus requiring twenty random
numbers whose standard deviation 0-.4472 was derived from an
actual set of experiments. The simulated measurements are
shown in tabular form (See Table 3) and plotted in Figure

1.

2.3 Ordinary Least Squares

The sum of squares function for ordinary least squares is

s; (r-nmr-n) (2.7)

owsusuonEou menses huwswuosvcoo
Hosanna you musoEoLsnsoe oous~=Eum IH museum

Gov 838388.

 

 

 

 

 

(O-UJ/M) wagpnpuog [owJaql

which can be written in summation form as

5=£(Y1-n1)2 (2.8)

Taking the matrix derivative of eq. (2.7) with respect to B,

V,S - -2x’(r - In) (2.9)

Similarly, differentiating eq. (2.8) with respect to the

parameters 81 and 82 yields the following expressions,

as

T91: ='-12£(Y1“T| fl1)%:- (2.108)
as =‘_ _ a":
'36—: 21§(Y1 m) E (2.10b)

which are equivalent to the matrix normal equation given by
eq. (2.9). Incorporating eq. (2.1) into these expressions,
replacing 81 and 82 with the estimates b1 and b2, and setting

equal to zero yields,

A091 + 517‘sz a £171 (2.11a)
£13191 + £12313, . gym (2.11b)

The solution for this set of equations is,

. 21521? - Entry;

b (2.12 )

1 mm - (2:7,): .
NZYT -ET£Y

b: a —Li___1_._1 (2.12b)

NET} " (2T1):

or, in matrix form, solving eq. (2.9), replacing B with b, the

estimated vector of parameters, and setting equal to zero,

b= (1’1?me (2.13)

2.4 Sequential-Over-Experiments Estimation and Regularization

The basic methodology of sequential-over-experiments
estimation is to utilize information from previous experiments
in subsequent ones. It is the intention to show that.using

sequential-over-experiments estimation (canbining experiments)

10

will result in virtually identical results as analyzing all
the data simultaneously using a least squares approach.

The first set of data will be analyzed using a technique
called regularization as a part of the least squares which
will result in feasible parameter estimates. Since only the
first set of data will be analyzed at this stage, we do not
expect the estimates to be accurate.

The acquired parameter estimates from this analysis, and
corresponding prior information (see Beck & Arnold) will be
used in the analysis of the next set of data. Similarly,
these results will be implemented in the third set of data,
and the process repeated whereupon the final parameter
estimates*will be obtained in the analysis of the final set of
data, which includes data from all experiments. The sum of

squares function for the first experiment is
sl-(vu) - nun’mn - 13(1)) + (amine (2.14)

The second term in eq. (2.13) is the regularization term, and
is used in the analysis of the first experiment of a
sequential-over-experiments analysis. Essentially, this term
regulates or controls parameter estimates for which very
little information is known. The regularization term is
flexible in the sense that the regularization matrix B can be
arranged to minimize an appropriate sum, corresponding to a
particular problem. Since it is impossible to estimate 31

and 8: simultaneously if measurements are at a single

11

temperature, regularization is required for the analysis of
the first experiment. Regularization on 82 can produce
reasonable results for the analysis of the first experiment.
Consequently, the following E matrix is used

3 - [g ‘35] (2.15)

where a is called a regularization constant.
An analogous expression to eq. (2.14) in summation form

is

51 . Emu) - n,(1))= + up: (2.15)

Taking the matrix derivative of eq. (2.14) with respect

to B, and setting equal to zero yields,

v.51L - -2r"(1) (r(1) - r(1)b,) + 2mm)1 (2.17)

Analogously, taking the derivative of eq. (2.16) with respect

to 81 and 8, yields,

as1 _ _ _ emu)
3!: 23mg» mun—5r: (2.12»

351 _ _ 3mm
3: 22.;(y1u) n1(1))——$2—+2sp, (2.1m)

12

Incorporating eq. (2.1) into eqs. (2.18a,b) , replacing BI

and 82 with the estimates b1 and b2, and setting these

expressions equal to zero yields,

Nbi + £13m», - Erin) (2.19s)
2 g
gnaw: + (gnu) + a», 1§Y1(1)T3_(1) (2.193)

The solution for this set of equations is

9171(1) (fryer + a) - 3323(1) iy,(1)r,(1)
b1 _ 1-1 1-1 1-1 12

 

(2.”)
"(firm)” + a) — 03-13(1))2
b2 8 1'1 1'1 1'1 (2.”)

 

mgnuvm) - (2313(1))2

Notice that if (:80 and Ti(1) equals a constant value,
such as 25°C, the denominator of both eq. (20a) and (20b)
equals zero. In such a case it is not possible to estimate

either 51 or 82. But by using a non-zero value of a, values

13

of 81 and 82 can be found.

The solution can be expressed in matrix form as

 

b(l) - 1'1(1)(x’(1)v(1)) (2.21)
1(1) = x‘(1)x(1) + E’s (2.22)
where
r( ) N 271(1) (2 23)
x 1 :1 = .
) ( 213(1) 213(1):
0 s

In eq. (2.23), N denotes the number of data points for the
first experiment. By adding terms to the diagonals of the XI}!
matrix, we are in effect improving the condition of the
problem. The effect of the regularization is to permit
estimates of both 81 and 92.

The sum of squares function associated with the second

experiment is.

52 - (2(2) - x(2)e)’(v(2) — x(2)e) + (13(1) - a)"A(1)(b(1) — e)
(2.25)

and the analogous expression in summation form is

14

a: - 3 .. -
5, guns mm) +§1£(b.(1) B.)A.(1)(bn(1) B.)

(2.26)
Taking the matrix derivative of eq. (2.26) with respect to the

parameters 81 and 8; yields,

as2 an,(2)
3: 2132mm -n,(2)) 331 - 2Au(1)(b1(1) - (31)

 

-2An(b,(1)-B,) (2.27a)

as2 ‘ _ _ 6mm
33—: 21§(Y,(2) n,(2)) as: -2A,,(1)(b,(1) - B.)

 

-2il,3 (191(1) 431) (2 .27b)

Replacing 81 and B; with the estimates b1(2) and b2(2), and

setting these expressions equal to zero gives,

A11(2)b1(2) +An(2)b3(2) ID1(2) (2.28s)

Au(2)b,(2) + A,,(2)b,(2) - D;(2) (2.231;)

15

where

4111(2) -N+Au(1) (2.29a)
413(2) -ET,(2) +A,,(1) (2.29b)
1131(2) =£T1(2) +A,,(1) (2.29c)
222(2) -2r}(2) “1,,(1) (2.295)
131(2) -EY,(2) +Au(1)b1(1) +An(1)b,(1) (2.29e)

192(2) -£Y,(2)T,(2) +A,,(1)b,(1) +An(1)b1(1) (2 . 291')

In eq. (2.29s), the symbol N denotes the number of data

points for the second experiment. The solution for this

set of equations is,

b . 1,,(2)n,(2) ~A,3(2)D,(2)

1 lineman) -A},(12) (2'30"
2 - 2 n 2
b, . WM (2.3%,

4111(2)Azz(2) - 1133(2)

16

The matrix expression analogous to eq. (2.30a) and eq.
(2.30b) can be derived by taking the matrix derivative of eq.
(2.25) with respect to 8 and setting this equal to the zero

VBCtOZ‘ ,

ms, - -2x’(2) mz) - 1(2)b(2)] - 21(1) (6(1) - b(2)) .- o

(2.31)

The solution for eq. (2.31) is.
b(2) - 1(2)'1(x’(2)v(2) + A(l)b(1)] (2.32s)
A(2) - x’(2)X(2) + 1(1) (2.32b)

This same procedure can :now proceed to three and more
experiments. A more general expression for sequential-over-
'experiments estimation is,
b(j+l) - 2'1(j+1)(x‘(j+1)v(j+1) + A(j)b(j)]
(2.33a)

A(j+1) - x’(j+1)x(j+1) + 1(3) (2.33b)

where j denotes the value at a particular experiment sequence.

The analogous expressions in summation form are,

An(j+1)b1(j+1) + A,,(.‘i+1)b,(j+1) - D1(j+1) (2.34s)
AnU+1)b,(:I+1) + A,,(j+1)b,(j+1) - D,(j+1) (2.34b)
where

Au(j+1) - N + And) (2.35a)

17

Auuu) - ET1U+1l + A1, (3') (2.35b)
Az,(j+1) - Dawn) + Anti) (2.35C)
Anon) - 22‘} (1+1) + 1),,(1) ' (2.352)
D1(j+1) - Zyiuu) + Alum!)1 (j) + Au(j)b,(j) (2.35s)

D2(J'+1) - EY1(.1+1)T1U+1) + .in(j)b,(j) + Au(j)b1(.1)

(2.35f)
In eq. (2.35a), N denotes the number of data points for the
(j+l)st experiment. The solution to this set of equations is

similar to eqs. (2.30a) and (2.30b):

A3, (1+1)D1(j+1) - Au(j+1)D,(j+1) (2.36“

 

 

( 1) -
b1 “1+ Auunmnuu) -A§.(j+1)
b,U+1) _ Anuumuu) -6,(j+1)a,,(j+1) (2.36b)

Au(j+1)A,,(j+1) - A}; (1+1)

18

2.5 Sequential Method

The concept. of arriving at. more accurate parameter
estimates as more data is analyzed is an integral part of the
theory of parameter estimation (See Beck & Arnold, pg.475).
To illustrate this idea, the data was analyzed, starting at
the first temperature level and first data point.
Regularization was required for data analysis at this point,
since one can not estimate two parameters using ordinary least
squares using only one measurement or even many measurements
at the same Ti ‘value. The resulting information was
transferred to the next data point. This process was repeated
in sequence until all five simulated measurements were
analyzed. at the first temperature levela The combined
information from the first temperature level was then carried
over to the first point at the second temperature level, and
the process was continued in an algorithmic fashion as in
equations 2.33a and 2.33b. The data is analyzed in the

following section.

2.6 Discussion and Comparison of Methods

In order to acquire a basis for comparison, all the data
(See Table 3) was analyzed simultaneously using an ordinary
least squares. approach, as illustrated in Section 2.3.

Program.LSQ‘was written and utilized (See Appendix A.l). The

19

following results were obtained:
b1-. 49078 w/m-°c
6,2. 00098522 win-“c2

Program SEQUEN (See Appendix A.2, A.3) was constructed
, using the theory presented in Section 2.4. Various magnitudes
of regularization were used on the 82 parameter. The
estimates diverged when the a value was set to zero (see eq.
2.15) . However, there was quite a large range of
regularization which resulted in practically the same results.
This range of . regularization parameter values was from
approximately a-l to a-30. The sequential-over-experiments
estimation results were obtained as given in Table 1, using a
value of a-3 m-°C°/N . Referring to Table l, the results in
the first row (25°C) represent the parameter estimates
associated with the first experiment, as described in eqs.
(2.20s) and (2.20b). The results in the second row (50°C)
represent the parameter estimates associated with the second
experiment, and are described by eqs. (2.30a) and (2.30b).
Note that the regularization used in the analysis of the first
experiment was transferred through, and incorporated in the
results of the second experiment. The third experiment was
analyzed, using the combined information from the first two
experiments (See eqs. (2.34s) and (2.34b). The corresponding
results are located in the third row (75°C). The process was
repeated with the analysis of the fourth experiment, and the

final results are located in the fourth row of Table 1

20

(100°C). These results are in very good agreement with the
ordinary least squares approach. This is an indication that
the method is sound.

Program SEQUENZ was constructed (See Appendix A.4, A.5)
according to the method presented in Section 2.5. The results
are contained in Table 2. It is interesting to note that the
~intermediate parameter values corresponding to i=5,10,15, and
20 are identical to the sequential-over-experiments values
(Table l) at each temperature level. In other words, the same
results were obtained, whether all five measurements were
analyzed simultaneously at each temperature level in the
sequential-over-experiments approach, or if measurements were
added one-by-one to illustrate the accuracy acquired with the
addition of more measurements. Effectively, the methods are
utilizing the same information. .As expected, the results are

the same.

2.7 Conclusions and Remarks

The main advantage of sequential-over-experiments
estimation is that it enables the combination of a series of
experiments in.a situation where the simultaneous analysis of
all data is impractical, or even impossible.

In the analysis of the simulated data contained in this

chapter, the method has been proven viable for a realistic

C888 .

21

Regularization is useful in a sequential-over-experiments

analysis. The main function of the regularization is to

obtain reasonable though not necessarily accurate parameter

values for the analysis of the first set of data. Generally,

a large range of regularization is feasible for the analysis.

 

22

 

 

 

Table 1
Sequential-0ver-Experiments Results
Exper. b1 (W/m-°C) 62 (w/m-°c’)
—: _ —)
25 .51770 0.00000
50 .49718 8.22992E-4
75 .49596 8.59615E-4

 

 

 

 

100 .49076 _ 9.84654E-4

Table 2

Sequential Method Results

m 00-4 c\(n e.(» a))4 H?

Rina F'ld h-ou ks)» F‘IJ hi
c>(o e)-4 c)(n 6.10 N)+a 0

b1 (W/m-°C)
.50927
.50523
.51340
.51515
.51770
.48452
.48169
.49838
.50274
.49718
.49794
.50294
.50399
.50140
.49595
.49997
.49526
.49741
.49266
.49075

b2 (W/m-° C2)
2.77555E-17
5.551118-17

0.00000

0.00000

0.00000

1.327048-3
1.444613-3
7.749553-4
6.00084E-4
8.229928-4
8.004298-4
6.50171E-4
6.187828-4
6.962613-4
8.59615E-4
7.63333E-4
8.52586E-4
8.249398-4
9.390428-4
9.84654E-4

 

23

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Table 3

Simulated Measurements for Sample Case
1 '1', (°C) Tl; (W/m-C) e, (W/m-C) I1 (W/m-C)
l 25 .50000 .00928 .50928
2 25 .50000 .00120 .50120
. 3 25 .50000 .02972 .52972
ill 4 25 .50000 .02044 .52044
H 5 25 .50000 .02788 .52788
6 50 .53333 .01809 .55142
I 7 50 .53333 .02355 .55688
H 8 50 .53333 -.03005 .50328
ll 9 50 .53333 -.01380 .51950
ll 10 50 .53333 .02744 .56074
11 '75 .56667 -.00971 .55696
12 75 .5666? -.02759 .53908
13 75 .56667 -.02027 .54640
14 75 .56667 -.00017 .56650
15 75 .56667 .02779 .59446
16 100 .60000 -.03574 .56426
17 100 .60000 -.00021 .59790
18 100 .60000 -.02678 .57322
19 100 .60000 .02082 .62082
20 ‘100 .60000 .00558 .60558

 

 

 

 

 

 

Chapter 3

Mathematical Formulation and Theory-Nonlinear Case

3.1 Introduction/Motivation

The purpose of this chapter is to give the mathematical
theory and formulation for the nonlinear sequential-over-
experiments analysis. Nonlinear estimation is important
because most of the problems for estimating parameters from
transient.measurements are in fact nonlinear. There are many
similarities in the nonlinear procedure and solution with the
linear problem, but one major difference is that the problem
mustmbe solved in an iterative manner; The analysis also uses
the concept of regularization, which was introduced in Chapter
2. The regularization. is even more important for the
nonlinear case than the linear case because the sensitivity
matrix is a function of the parameters in the nonlinear case.

The underlying theory of the method is given both in
matrix and algebraic form. A Taylor series approximation is
incorporated into the theory of the nonlinear problem, so that
an iterative algorithm is used for obtaining parameters. This
is the main theoretical difference between the linear and

nonlinear case.
24

25

Program PROP1D (Beck) was developed for the specific
nonlinear problem of the determination of thermal properties
of materials. In particular, thermal conductivity and the
volumetric specific heat are found as functions of temperature
by analyzing data acquired in transient experiments. The
program is capable of performing sequential-over-experiments
analysis. Both regularization and 'prior information' (See
Beck and Arnold, 1977) are incorporated in the program.

The outline of the remainder of Chapter 3 is now given.
Section 3.2 gives some of the basic concepts of thermal
property determination and regularization. Section 3.3
illustrates the nonlinear sequential-over-experiments
derivation, including the iterative technique. Section 3.4
offers a description of the computer program PROP1D, used to

analyze the data.
3.2 Thermal Property Determination and Regularization

The underlying purpose of the inverse algorithm in
program PROP1D is to estimate suitable parameter values that
will parallel experimental temperature data with a calculated
set of temperature data using a modified weighted least
squares procedure. Calculated values of temperatures are

obtained by solving the one-dimensional transient heat

conduction equation,

M'_ a 81'
so}? 30:7") (3.1)

26

with appropriate boundary conditions (a constant heat flux at
x80, and a temperature boundary condition at x-L, where L is
the position of the farthest temperature sensor). The
describing variables are temperature T, position x, time t.
.thermal conductivity k, density p and specific heat c. A
known heat flux is utilized at x-O as a boundary conditon for
eq. (3.1).

The following derivation illustrates the basic
mathematical theory of regularization and introduces some of
the regularization matrix configurations. It also describes
the mathematical formulation for the nonlinear sequential-
over-experiments analysis. This mathematical formulation is
an extension of some of the more basic concepts of least
squares estimation.

The first step in the sequential procedure is to minimize
for the data of the first experiment the sum of squares

function.

5(1) -(r(1) -n (1) )Tsu) (r(1) -1) (1) ) «mum (3.2)

The various terms are:
Y I measured vector of temperatures and heat fluxes

(N x l)

q - calculated vector of temperatures and heat fluxes

(N x l)

27

H - weighting matrix (N x N)

B - parameter vector (p x 1)

H = regularization matrix (p x p)

The second term in eq. (3.2) is the regularization term. This
.term constrains the parameter estimates, and is useful for
acquiring reasonable values of properties at temperatures
outside the experimental range, as mentioned in Section 2.4.

For example, consider a situation where experimental data
was collected at 25, 50, 75, and 100°C from four different
experiments, each starting at one of these initial
temperatures. Each experiment may only cover a limited range-
perhaps 10°C. Suppose we wish to estimate four thermal
conductivity values: k1 at T1825°C, k, at T2850°C, k, at T3275°C
and k‘ at T‘8100°C. Since the 25° C experiment might only cover
the range of 25°C to 35°C, no information is contained in this
experimental data regarding k, and kw. The regularization
technique provides an algorithm that seeks to minimize the
deviation between specified parameters, so that reasonable
preliminary k3 and k‘ values may be obtained. Better values
are obtained as more measurements are used for the later
experiments. In general, the regularization reduces the
variability of the estimates of the first few experiments, but
does not affect the final parameter estimates. For (this

example, the sum to regularize k(T) is

28

2
23¢r(k: -k-) (3.3)
1.1 1 i+l 1

which is analogous to the H matrix of

-1 1 0 0
- 0

3...,1/ : 01 :11 1 (3.4)
o 0 0 0

where a is called a regularizing parameter. Another case is
for estimating two k components and simultaneously two

components of the pc product. The B matrix in this case

becomes
«:2 5 a2" 0 0
5. ° ° ° O (3.5)
0 43" a: 5
(J 0 (I 0

 

 

and the corresponding sum to regularize k(T) and pc(T) is
al(k2 " k1); 4’ C3‘PC2 " 9C1): (3.6)

This configuration for the E matrix represents the main focus

29

of the data analysis present in this thesis.
3.3 Nonlinear Sequential-Over-Experiments Derivation

Consider using the estimation of two parameters, with the

following E matrix.

_ 0.5 0.5
s-[ ‘0 “0] (3.7)

The analogous expression for eq. (3.2) in summation form is

51 - £(Y.(1)-n,(1))3 + «((02-01)2 (3.8)

for N(l)-I. Taking the derivative of eq. (3.8) with respect

to 61 and 82 gives,

as1 amu)

BF- : -2£(Y1(1)‘fl1(1))T-23(33‘31) (3.98)

1 1

gas; - -22(y,(1)-n.(1))giganuprpp (3.9b)
3 3

In order to minimize 51' the Taylor series approximation is

introduced,

"(wuqvn 13m wonky") (3.10)

The sensitivity matrix x can be written as

30

 

 

(an. an.‘

m I 32'
1.41 ’92 . W. 'aT,’ (3.11)

#5139: iNh, 3N1

R? 7351

 

 

It follows that the ith value for n can be written as

"£701, . “g” + x‘1(b1(W1) _ bf”) + x13(b2(”1) _ 132‘”)

(3.12)

Substituting eq. (3.12) into eqs. (3.9a) and (3.9b) . replacing

81 and 82 with the estimates b1 and b3, arranging terms, and

setting equal to zero yields.

Au(1)(b,‘””(1)-b1"’(1)) + 32(1) an?" (1)-6.")(1n - a‘"

(3.13s)

A,,(1)(b1‘”"“’-b1"’(1)) + 3,,(1) (b."'” (1)-6;” (1)) - a")

(3.13b)
where

31

4111(1) - 2x1!”(1) + a (3.14a)
Ann) - 2x}? (1)1:1‘3" (1) - a (3.14b)
321(1) - 2x19" (1)221,” (1) - a (3.14.)
4123(1) - 22x},"’(1) + a (3.14:1)

Ff" - 1:”(1)(Y1(1) -n£"(1))-a(b,"’ (1)-b,M (1)) (3.14e)

Pam - XXI,” (1) (Y,(1) - n5” (1))+¢ (6,” (1)-b1” (1)) (3.14:)

Using Cramer's Rule. the solution to eqs. (3.13a,b) is

14%;,” ( 1) - Frag," (1)

 

(W11 __ (V3 .

6, (1):). (1) "’(1) ‘;’(1)- ""(1) (3.15.)
(V, (V, _ (V, (V)

bzifll) (1) _b‘(fl (1) - ’3 A11 (1) Fl A12 (1) (3.15b)

 

41‘1”(1)A.‘;" (1)-A1‘Fu)

These equations are solved in an iterative manner until the
changes in the (v+l)st estimates of 81 and 63. denoted bllflll

and 6,"*“, are little changed from 61‘" and 62‘".

32

The criterion for convergence is as follows:

 

 

(mu)._ (»

b1 blmbl (<5 (3.16a)
(wd)__ (fl

13’ 62““ «6 (3.16b)

where 6 is some small number such as .0001.
The analogous matrix solution can be derived by taking
the matrix derivative of eq. (3.2) with respect to the

parameter vector 6 (assuming W(l)=I).

v.5, - -2(v,q{) (r(1)-n(1)) + mm (3.17)

where

v"= 5.2.... a] (3.18)
' 59—. an. '56;

The estimates of the parameter vector 8 are where 51 is a
minimum or where eq. (3.17) is equal to the zero vector,

provided there is a unique minimum,

1‘” (1) (b‘m’ (1) - b‘” (1)) - 1"”(1) (ru) - n‘”(1))-I'D"' (1)

(3 .1.9ll)

where

33

A‘”(1) xr‘”'(1)x"’(1) 1 51-3 (3.19b)

The next step in the sequential-over-experiments analysis

is to minimize for the second experiment for the new S,
S; = (1(2) - n(2))’"(z) (1(2) - (1(2)) +

(b(1) - B(2))’A(l) (b(1) - 9(2)) (3.20)

with respect to the parameter vector 6(2) , which includes the
data from experiments 1 and 2. The A(1) matrix in eq. (3.20)
is given by eq. (3.14) and b(1) is the converged vector given
by eq. (3.19a).

The analogous expression for $2 in summation form is

(assuming N(2)-I).
s, . 2mm - mum . §n§(6_(1)-p.(2))a_(1)(6,(1)-)).(2))

(3.21)

Taking the matrix derivative of eq. (3.21) with respect to the

parameters 81 and 82,

34

 

7235—: - -22(r.(2) - 111(2)) 3‘53) - 2311(1)(61(1) - 01(2))
4313(1) (b,(1) - 52(2)) (3.228,

' 3a? " -22[r,(2) - ““znflaifl - 2A,,(1) (b,(1) - 62(2))

-2A,,(1) (13(1) - 01(2)) (3.22b)

Again, introducing the Taylor series expansion;

111”” (2) - nl”(2)+x,,tb.‘”” (2)-b.”(2)) + raw)“ (2)-6;”(2n

(3.23)
Substituting eq. (3.23) into eqs. (3.22a) and (3.22b).

replacing 81 and 83 with the estimates b1 and b2. and

rearranging terms,

313(3) (6.‘”*’(z) - b.”(2)) + 333(2) (b.‘””(2) - b3“ (2)) - n‘"

(3.24s)

33m. (bf’”(2) - b.”(z)) + 333(2) (b3""" (2) - b.” (2)) - Fa‘”

(3:246)

where

35

AJF‘NZ) -2x£{”(2) +1113" (1) (3.25a)
1113”" (2) - EX}? (2)1:1‘3" (2) +411," (1) (3.256)
Az‘f’“(2) - EXII’(2)X1‘;’(2) +AS"(1) (3.25c)
A3‘2‘""(2) - 13"’(2) +1132" (1) (3.25.1)

Ff" - 2212" (2) (m2) - 01" (2)) + 31‘." (1) (61(1) - 6,"’(2))

+ All.” (1) (15(1) - b.” (2)) (3.23.)

33‘" - 1‘,”(2)(y,(2) - n1”(2)) mg," (1) (61(1) -6."’(2))

AI." (1) (63(1) - b."’(2)) (3.25:)

Using Cramer's Rule, the solution is

F{”A3‘3" (2) - E9115," (2)

3.26
AI.” (2)41” (2) - 2;,"’(2) ‘ 3)

b1” (2) -6."’(2) -

36

r3711." (2) - WASH»

(3.26b)
51‘1” (2)432" (2) - sigma)

 

b."’(2) -b;"’(2) -

The criterion for convergence is stated in eqs. (3.16a) and
(3.16b) . I

The parallel matrix formulation is now given. Taking the
matrix derivative of eq. (3.20) with respect to 8(2) and

setting it equal to zero yields

1‘” (2) 0"” (2)-bm (2))81""'(2) (2(2) 41‘” (2))

+ a") (1) (6(1) - b‘” (2)) (3.27.)

a") (2) -x""'(2)x"’ (2) a") (1) (3.27b)

This matrix expression is analogous to eqs. (3.24a) and
(3.24b). The matrix solution for the parameter estimates_is
symbolically given by eqs. (3.28a,b).

The extension to the sequential analysis of three, four,
or more experiments is direct. Suppose that 3' experiments
have been performed and the estimates from these 3 experiments
is to be updated with the data from the (j+1)st experiment.

Then eqs. (3.27.) and (3.276) would be replaced by

37

A‘” (j+1) 0“” (1+1) - b‘” (1+1) - 1"”’(:l+1) (It/+1) - n‘” (j+1))

+ 1‘” (2‘) 0(1) - b‘” (1+1))

0

(3.28a)

4‘” (3+1) . x‘”’(j+1)x‘”(:i+1) + 1‘” (:I) (3.28b)

The information coming from the previous :5 experiments is

contained in AM”) and b(j) .

In a similar fashion as in eqs. (3.24) through (3.26).
the algebraic equivalence of eqs. (3.28a.b) can be

illustrated. The analogous expressions are

3;.” (3+1) (6?" (3+1) -6."’ (3+1) ) +3.5.” (3+1) (6.” (3+1) -6."’ (3+1))

3 F1", (3.29.)

3.3" (3+1) (6?” (3+1) 4):" (3+1) H.132" (3+1) (6.“) (3+1) -6."’ (3+1))

- 93‘" (3.29b)

where

3.1"” (3+1) + 21:3.”'(3+1) + 3:,” (3) (3.30.)

38

1.2"“ (3+1) - 2:69" (3+1)x1."(3+1) + 1);," (3) (3.30b)
A4?" (3+1) - 22x3? (3+1)x;;’ (3+1) + AS’ (3) (3.30c)
113‘?" (3+1) - 2x1.”’(3+1) + 33‘," (3) (3.30d)

F.” - 210‘? (3+1) (y,(3+1) - nl" (3+1)) + 1);,” (3) (b1(.1)-b1"’(j+1)
+ A? (3) (162 (3) - b.” (3+1)) (3.30.)
F.” - 1‘.” (3+1) (Y.(3+1) - nl"(j+1)) + 3;," (3) (61(3) -61"’ (3+1)

+ 1);." (3) (6,(3) - 132‘" (3+1)) (3.30:)

Using Cramer's Rule, the solution is

(V) - (v)
61‘” (3+1)-6,") (1.1). ’13:: (1+1) F3433 (3+1)

 

(3.31 )

31‘1” (j+1)Az‘2fl (j+1)- ("2‘32”) 8
(v) (v) _ (v) (v)

F, Au (1+1) 1'1 ‘12 ”‘1’ (3.316)

 

(y) 1 _ (V) 1):
b3 (j‘i' ) b2 (j+ 3:1”(jT1)A2‘Zfl(-1+1).A1‘;’2(j*1)

39

The criterion for convergence is stated in eqs. (3.16a.b).

3.4 Program PROP1D Description

Program PROP1D (Beck) utilizes measured temperature and
heat flux histories acquired in transient experiments to
estimate the properties thermal conductivity and volumetric
specific heat. The program (can estimate up to four
properties at once (any combination of the two properties).
The transient data is analyzed in a .sequential-over-time
fashion. utilizing a modified weighted least squares algorithm
similar to eqs. (3.28a.b) . This sequential-over-time analysis
is similar to the approach illustrated in Section 2.5. where
it is shown. that more accurate parameter estimates are
acquired with the addition of more measurements. This
accuracy is illustrated in the sequential-over-time estimates
for the non-linear problem of Chapter 5. The main focus of
the data analysis in this thesis was the estimation of two
thermal conductivities and two density-specific heats. as
discussed in Section 4.4.

The program uses a Crank-Nicolson finite difference
method. to» calculate a temperature history for parameter
values. This calculated temperature history is compared to
the experimental data set. and matched with suitable parameter
estimates that will minimize the error. utilizing the theory

presented in Sections 3.2-3.3.

40

It is possible to vary the calculational time step. the
weighting for each thermocouple. and the maximum number of
iterations. It is also possible to include the features of
prior information and regularization, so that sequential-over-

experiments estimation with regularization can be performed.

Chapter 4
Description of Experiment
4.1 Introduction/Hotivation

The purpose of this chapter is to describe the
experimental apparatus used for the determination of thermal
properties. The data acquired from a series of experiments is
described. Also described is the basic methodology of
sequential-over-experiments estimation that will be used to
combine the data. Furthermore, an analysis to determine an
appropriate range of regularization parameters is performed so
that the effects of relatively 'large' and 'small' amounts of
regularization can be determined for a complete sequential
analysis. Similar experimental procedures and acquisition of
data have been performed by (Scott. 1989) and (Loh. 1989).

The outline of the remainder of Chapter 4 is now given.
Section 4.2 gives the description of the experiment. Section
4.3 discusses the measured temperature histories. Section 4.4
illustrates the methodology of the sequential-over-experiments
approach. Finally, Section 4.5 justifies the choice of two
sets of regularization parameters representing relatively
large and small amounts of regularization within a suitable

range.

41

42

4.2 Experiment Description

An example of the use of the procedure discussed in
Chapter 3 is now given. A Quantum Composites material is
analyzed using data generated under controlled heating
conditions. The thermal conductivity and volumetric specific
heat are both found as linear functions of temperatures.

The chopped fiberglass composite material provided by
Quantum Composites was tested at four different temperature
levels, starting at 27°C. 55°C, 72°C and 96°C, respectively.
The test involved four flat plates, each about .004m thick.
arranged in a stack with a thin square heater between the
middle two elements. See Fig. 2. Thermocouples were placed
on both sides of the heater and at the interfaces between the
plates. The thermocouples were made of fine wire (40 gage,
.00315 m in diameter) and were further thinned by .001575 m.
Silicone grease was added at the interfaces to improve thermal
contact. The plates were held together by an aluminum
housing. At the outside surfaces. aluminum blocks. about .014
m thick were used to provide a nearly isothermal boundary
condition.« Heating was accomplished by applying a constant
voltage for a specified period of time. followed.by a drop to
zero. Hence the heat flux. initially zero. takes a step
increase to a constant value which is followed by reduction to
zero again. Temperatures are measured just before the heating

starts. during the heating period. and after heating.

43

Abmhmnebdc

3 .56 Heater.

 

140nn1 Bbdc

.X"nunmxnuMBUxaflxs

Figure 2- Experimental apparatus for measuring
thermal conductivity and density-
specific heat

44

4.3 Discussion of Measured Temperatures

A typical set of transient temperature measurements is
shown in Figure 3, which shows the temperature range is
approximately 27 to 32° C, for the first experiment. The
thermal conductivity and density-specific heat product are
desired from 27 to 96°C: a linear temperature variation for
both properties is assumed for this temperature range. Three
other experiments were performed for initial temperatures of
55. 72. and 96°C. The magnitudes of the temperature rises for
these experiments is approximately the same as the 27°C
experiment. as illustrated in Figure 3. The transient
temperature measurements exhibit a large degree of waviness.
This waviness is believed to be due to a fault in the heating
element. since the 'humps' in the waves are consistant for
each thermocouple. Also. this waviness ceases after the

heater is turned off.
4.4 Methodology of Sequential-Over-8xperiments Approach

The thermal properties are estimated in a sequential
fashion: four parameters are found. k1 at 27°C. pcl at 27°C. k,
at 96°C. and no: at 96°C. Regularization is used on the 27°C
data. utilizing an a matrix as in eq. (3.5) , and the resulting
prior information (see eq. 2.15) is to be used as input for
the 55°C data.. The information acquired from this analysis

(see eq. 2.19) is used as input for the 72°C data. Similarly.

ture [°C]

Tempero

45

 

‘02 . ' I ' l ' I ' I ' l '
101-
1005
99$
98-}
97-1
964
95 .

 

 

 

 

 

 

 

 

 

 

32-}

 

 

 

 

 

o‘ 260 460 660 860 1600 1200

Time [seconds]

Figure 3- Temperature histories of specimen
at five different interfaces

46

the information acquired from the 72°C analysis is used as
input for the 96°C data. The final parameter estimates are

acquired from the 96°C analysis.

4.5 Choice of Regularization Parameters

A brief study is given of the effect of varying a1 and (:3
(see eq. (3.6)) for the 27°C experiment. Fig. 4 shows
estimated k1 and k2 values as a function of 0.1 for two values
of ('13. The variations of k1 and k, with a1 are even larger for
(11 less than 4,000,000 mmz-Czlwz. with the greatest effect being
upon k2. The thermal conductivity estimates are nearly
independent of 0.3. These observations are consistent with
expectations. That is. the regularization is mainly needed
for the high temperature k since the measurements are
concentrated at the lowest temperatures: also the (:1 parameter
is for k and a3 is for pc (see eq. 3.6). so it is not
surprising that k is little affected by changes in (:3. The
complementary estimates for pc versus a, for two values of (:1
are shown in Fig. 5: these results are consistent also with
the greatest regularization effect being that of a3'and again
the highest temperature pc being affected much more than the
lowest temperature pc.

Further insight into the regularization process can be
obtained by examining components of the A (. x’wx + 8’3)

matrix. The diagonal components of the A matrices for the

47

first data set (see eq. 2.15) and Tables 4 and 5). include
prior information for the next data set. Consider Table 4
first. The effect of changing a1 is considerably greater upon
k1 and k, (the first two diagonals) than upon pcl and pcz (the
3rd and 4th diagonals). From Table 5 it can be observed that
_ the effects of (:3 are more pronounced upon pc than k. The
ratio of the diagonal elements in both tables reveals the same
effect. These observations are consistent with those found in
connection with Figures 4 and 5. The relative increases in
the diagonal components of A can also be observed in these
tables; for example in Table 4. the greatest change is in the
k: diagonal and in Table 5 the greatest is in the pc, diagonal.
Even though in Table 4 the an diagonal is over two orders of
magnitude greater than the ‘22 diagonal. the increase in the
an diagonal is less than a factor of 2; even so. the effect
of increasing (:1 is quite marked upon k2. as shown in Fig. 4.
An analogous comment regarding pc, can be made by using Table
5.

Based on the parameter estimates shown in Figures 4 and
5. a reasonable range of regularization parameter values was
chosen. The purpose of regularization is to restrict the
variability of the estimates for this first set of data.
Since the regularization is carried through to subsequent sets
of data. it is desired to use a slight amount of
regularization. so that the effect of the regularization is
transparent through a complete sequential analysis. By over-

regularizing. one would in fact defeat the purpose of the

48

sequential-over-experiments estimation method. and prevent the
method from obtaining accurate parameter estimates for the
complete set of data.

Thus. the regularization parameters were restricted from
the range of 572500 to 4000000. to illustrate the
effectiveness of the method over a relatively wide range of a
values without over-regularizing. Note that the density-
specific heat estimates did not converge for an a, value of

less than 572500 (See Figure 5).

49

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51

Table 4

Diagonal Componenets of A for First Data Set (27%”
(varying (:1 values)

In

chm-750 (“LC/J)

Units of elm are (mm-C/W)

Elements of Diagonal Components

Ratios

ll11/“22 ‘JI/W

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

c‘1 ‘11 ‘22 ‘33 ‘u

0 .445310 .158338 .116489 .893436 281.2 130.3
1000 .472310 .179638 .117739 .901386 262.6 130.6
2000 .492310 .218588 .118739 .907286 225.3 131.5
3000 .501310 .272188 .119039 .909686 184.1 130.8

«,1’3-2ooo (n3-c/a)
Elements of Diagonal Components Ratios

«11” an ‘2: an ‘u , '11"22 ‘n/‘u

0 .443810 .157138 .119489 .432937 281.7 27.6
1000 .469310 .178488 .120639 .433687 262.7 27.8
2000 1.489810 .217388 .121539 1.434287 225.2 28.0
3000 .498310 .270938 .121839 .223387_ 183.8 28.0

 

 

 

 

 

 

 

 

 

 

 

Diagonal Components of A for First Data Set

Units of a,

52

Table 5

(varying a, values)

“1112.750 (mm-CIR)

I”

are (mmJ-C/J)

Elements of Diagonal Components

(27°C)

Ratios

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

“3m ‘11 '2: '33 ‘u '11":: ‘n/‘u
750 .464610 .171434 .117339 .496966 269.8 130.5
1000 .462E10 . .171239 .117559 .133557 270.0 - 88.01
1250 .462810 .170924 .117969 .169727 270.0 62.15
1500 .461510 .170438 .114569 .2564s7 270.0 45.9

elm-2000 (mm-CIR)
Elements of Diagonal Components Ratios

“3m ‘11 ‘12 ‘31 ‘u‘ ‘11"12 ‘33"44
750 .492510 .214524 .114729 .907286 225.3 130.4
1000 . 491210 . 217924 . 114459 . 134327 225 . 2 44 . 5
1250 .490510 .217624 .119229 .190527 225.2 62.6
1500 .490:10 .217424 .119439 .259227 225.3 46.2

 

 

 

 

 

 

 

 

 

Chapter 5
Estimation of Thermal Properties
5.1 Introduction/Motivation

The purpose of this chapter is to describe. the results of
the complete sequential analysis in detail. Two sets of
regularization parameters are used. in an effort to determine
the effects of relatively large and small amounts of
regularization in a complete sequential analysis. The
confidence intervals of the estimates are discussed. as well
as a detailed look at the transfer of prior information (See
eqs. (3.32b) and (3.33)) through the analysis. Further
evidence of the certainty of the estimates is given in the
residuals. sensitivity coefficients (Beck and Arnold. 1977).
and in comparison to another commonly used method of parameter
estimation. Similar work was performed by (Scott. 1990) and
(Loh. 1989).

The outline of the remainder of Chapter 5 is now given.
Section 5.2 gives the sequential-over-experiments results.
Section 5.3 presents the confidence intervals of the
estimates. Section 5.4 discusses the transferral of prior
information through a complete sequential analysis. Section
5.5 discusses a verification of the method-namely a sequential

analysis in reverse order. entitled a 'backwards run.‘ The
A 53

54

residuals. sensitivity coefficients. and sequential-over-time
parameter estimates are discussed in Sections 5.6-5.8.
Finally. the sequential-over-experiments method is compared to

another commonly used method called the 'direct' method.
, 5.2 Sequential-Over—Experiments Results

See Table 6 which shows sequential results of estimation
of two conductivity components and two density-specific heat
components for the cases of alm=750 (mm-C/W). 0.31/28750 (mm3-
C/J) and all/Q2000 (mm-C/W). a31’2=2000 (mm3-C/J). Important
results are for using all the data: see the 96°C rows in Table
6. In comparing the two sequential runs. the first k
components (corresponding to 27°C) are different by
approximately .38. while the first component of pcs are
different by approximately 1.08. The second components of
both k and pc for both cases. corresponding to the 96 degree
temperature level. are the same for both sets of a1 and a,
values. Consequently. the choice of the regularization
parameters (:1 and a3 is not sensitive for the final parameter
estimates for (111” and (131’: from 750 to 2000. Notice also the
large overlap of the confidence regions for the 96°C values.

See Figure 6 which shows the parameter estimates at each
temperature level as the data is processed sequentially.
Since the data is analyzed from the lower level temperature
first. the lower temperature estimates are established

immediately. while the upper temperature estimates will vary.

55

Table 6

Sequential Analysis. Including Estimates
and Confidence Regions

Units for k are Ulm C
Units for pc are J/mJ C

elm-750 and (231/28750

11°02 nus k(27) k(96) 2:527) 3:596)

27 .083 .4841 2.175 .1627E7 .0783987
(.0367)* (.404) (.04087) (.014E7)
55 .097 .5724 .6357 .158537 .153237
(.0215) (.0635) (.01287) (.04737)
72 .082 .5657 .6542 .155187 .192387
(.0211) (.0427) (.01287) (.02937)
96 .095 .5812 .6137 .158437 .177487

(.0200) (.0273) (.01137) (.02037)

elm-2000 and 6,1“s2000

1:121 8L5 1:127) 5961 25(27) 2c(96)
27 .088 .5492 1.171 .154537 .139127
(.0219) (.079) (.01437) (.01427)

55 .097 .5726 .6356 .158587 .153187
(.0202) (.0555) (.01287) (.03087)
72 .082 .5630 .6567 .158187 .189657

(.0197) (.0392) (.012E7) (.023E7)
96 .095 .5794 .6138 .1602E7 .1773E7
(.0192) (.0261) (.01287) (.018E7)
* Confidence intervals are listed in parenthesis directly
under corresponding estimates

Thermo) Conductivity (J/m—C)

Dens. Spec. Heot (J/m’—C)x10"

56

 

L3
L2-

1.0-
0.9-
0.8-)
0.7-
0.5-)
0.5-

 

 

 

0A»

 

C122

(IZOJ'

0J8q
(116-
(114‘

(112‘

 

 

27 55 27.55.72 All Data
1 I

 

 

(110

Figure 6-

2 _ 3 4

Analysis number

Sequential parameteroestimates
and confidence regions

 

 

57

depending on the amount of regularization used. Notice that
the density-specific heat estimates begin in a decreasing
fashion. then eventually become increasing as the third set of
data is analyzed. This inversion occurs because more
information is known at this stage about the upper level
density-specific heat estimate. as compared to the information

acquired at the previous stage.

5.3 Confidence Intervals

The confidence intervals (Dhattacharyya and Johnson.
1977) of the estimates are also illustrated in Table 6.
(Since the residuals. measured minus calculated T's. are
correlated the confidence intervals are correlated, the
measurement errors must be modeled (Beck and Arnold. 1977)).
The range of the parameter estimates is smaller for the case
of higher regularization. Examination of the confidence
intervals for each temperature level.reveal that the magnitude
of the interval decreases sequentially (except for pc: at
27°C). This trend was expected. since we have more
information about all four estimates as the data is processed
sequentiallyu The relative confidence intervals for k and pc
are smaller for k. indicating this experiment provides more
accurate k than pc values.

Calculation of confidence intervals is very important and

is recommended. Notice that although the k and pc estimates

58

increase with temperature. the confidence intervals permit
these values to be estimated as being independent of
temperature. In other words. it is possible to assume that
the thermal properties are constant. taking the confidence

intervals into consideration (See Figure 12).
5.4 Transferral of Prior Information

Table 7 illustrates how prior information is transferred
sequentially. The components of the 27°C A matrix are the
regularized output. and reduce the parameter variations for
this and subsequent cases. There is a sequential decrease in
the ratios of the diagonal components. until the final results
are attained. where these ratios approach unity. indicating
that about the same amount of information was available for k1
and k: and also for pcl and no}. Since the final values of
diagonal components for both sets of (2's are approximately the
same. a large range of initial a. values is possible. resulting

in nearly the same parameters in the complete sequential

analysis.
5.5 'Backwards Run'

It was verified that a sequential run in 'reverse' order
of initial runs resulted in the same parameter estimates.
This observation added credence to the method. The actual k

estimates differed by approximately 1.58. whereas the actual

59
Table 7

Diagonal Components of A for the Sequential Analysis

elm-750 (ma-cm). (hm-750 (”fl-cu)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Elements of Diagonal Components Ratios
T(°C) an ‘33 ")3 a“ an/a” 833/0“
27 1464810 .171888 .117339 .8989E6 269.8 130.5
55 .641310 .980989 .1517E9 .189488 6.53 8.00
72 .690E10 .263E10 .1620E9 .523938 2.61 3.09.
96 .690810 .655E10 .1621E9 .1247E9 1.05 1.30
elm-2000 (mm-CIR). 63m-2000 (“Le/.7)
Elements of Diagonal Components Ratios
'1- (°C) an ‘1: ‘33 ‘M 911/922 ‘13’ ‘u
27 .489E10 .217388 .121589 .4342E7 225.2 28.0
55 .666810 .9852E9 .155929 .2238E8 6.76 6.97
72 .715810 .264E10 .1662E9 .5586E9 2.71 2.98
96 .716E10 .655210 .166389 .1282E9 1.09 1.301L

 

 

 

 

 

 

 

 

 

60

pc estimates differed by approximately 5’4. Using
regularization on the 96°C data. and sequentially using the
prior information until the final results were acquired at
27°C. the parameter estimates fell within the range of the
final estimates in the 'forward' run. This range is
determined by the confidence intervals of the estimates. The
large overlap in possible parameter estimates for the forwards
and backwards run is further verification of the efficacy of

the method.
5.6 Residuals

The residuals are the difference between’experimental and
calculated temperatures. The magnitude of the error for the
27°C and 96°C sets of data averages to approximately 2 percent
of the total temperature rise (see Figure 7). Although it is
desired to match experimental and calculated values as closely
as possible. these residuals indicate that the data is a basis
for a viable study; The RHS value for the 27°C experiment was
.088. and for the 96°C experiment was .093. The errors appear
to be normally distributed. random. and uncorrelated.
However. the thermocouple at the 3rd interface displays some
awry. periodic behavior. which is disturbing. This
periodicity indicates that there is some correlation of errors

for this thermocouple. which is highly unusual.

61

 

Oe4’ Y I 1 l W 7 1 I V I

27°C Data Set

  

 

 

 

1200

A
0 _
b
l) ..
o u
3 -
.‘9
m I
0 -
L e
. fresi
-5: fresZ '-
fresIS
-'4-: fres4 -
fresS
" -5 ' I . I I I r I 1 I
0 200 400 600 800 1 000

time (seconds)

 

.‘r v I r I t I I I I 6

96‘C Data Set

  

residuals (°C)

H fre51
H fresZ
«o—i» fres3
x—x (res4
H fresS

 

 

 

" -5 j I I I fl I ' f ' I
O 200 400 600 800 1 000

time (seconds)

Figure 7- Residuals for 27°C and 96°C
data sets

62

5.7 Sensitivity Coefficients

Examination of Figures 8 and 9 reveals that the
sensitivity coefficients are linearly independent. which is
desirable (Beck and.Arnold. 1977). That is. the shapes of the
curves are quite different for the four parameters. This is
an indication that the parameters k1. k2 and pcl. pcz can be
simultaneously estimated. In examination of Figure 8. the
sensitivity coefficients for k; and pcz appear ‘correlated'
indicating that there is some uncertainty in these estimates.
This is expected. since very little information is known about
the 96°C parameters from the 27°C data. There is a similar
correlation in Figures 9 between the sensitivity coefficients
for k1 and pcl. indicating that little is known about the 27°C
data. However. taking both sets of sensitivity coefficients
into consideration. it is safe to say that all parameters can

be simultaneously estimated.
5.8 Sequential-Over-Time Parameter Estimates

Examination of Figure 10 (the sequential parameter
estimates for the 27°C experiment) reveals that certain
observations can be made. First. in comparing the relative
deviation between k1 and k; and pcl and pea. it appears that the
pc estimates were more highly regularized than the k
estimates. This statement is based on the fact that the pc

estimates are more highly restricted. whereas the k estimates

((th/dk

flea-dT/d(pc)

 

0.00

 

 

 

 

 

 

 

 

200

400 ‘ 600 ' 800 ' 1000 ‘1200

time (seconds)

 

0.50 .
0.00
-.so-f
-1 .001:
-..s..j

-2.00-j

 

B
O

I.
I.

IN.

AllLllllJJJllllll

lllll

'
q

 

 

-2.50 ‘
0

200

V

1 *7

400 600 800 f 1000 1200

time (seconds)

Figure 8- Sensitivity coefficients ksdT/dk
and pc'dT/dmc) for 27°C data set

k*dT/dl<

pC’dT/d(f)c)

64

 

-.50-§
- (.005
- (.50-5

-2.00-:

 

r

.‘.llln

 

 

-2.50
0

200

1 r '

400 600
time (seconds)

800

1000

1 200

 

0.50

0.00 j

-.50-:
-1.00-:
-...o.3

-2.00-

 

I»,

III

R .

 

JJIJIJAJALIIJllllll.llll'J

 

-2.50 ‘
0

200

V '

400 600 860
time (seconds)

1200

Figure 9- Sensitivity coefficients k‘dT/dk
and pcsdT/dwc) for 96°C data set

65

 

1.300 I. u x I I I ' ' ' '
1.200-
1.100-
' 1.000-
0.900-
0.800-

0.700- , .
J k! "'
0.5004 . A .

____
H+ +4
___‘

f - fl

' I
Q5000 j 3507 460 ' 600fi 800 1000 ‘1200

thermal conductivity (W/m‘C)

 

 

 

time (seconds)

 

0.180 1 I I I ‘ l ‘ I ' I

0.1701

0.160: pc,

I'
1
I)
/

i

. pca
0.150-

0.140:

.
aknlaaaaln

1
1
e

 

Dens.-Spec. Heat (J/m’ °C)x10“’
)

 

 

0.130‘ - 4 ' ' ‘ '
0 200 450 ° 55° 55° ‘°°° 120°

time (seconds)

Figure 10- Sequential-over-time thermal conductivity
and density-specific heat estimates for

27°C data set

66

are given some degree of freedom to wander. However. due to
the flexibility of the sequential-over-experiments method of
parameter estimation. there is a wide range of regularization
that will result in nearly the same final parameter estimates.
In other words. even though there is some uncertainty in the
estimate of k} in this first set of data. the basic criteria
of convergence was satisfied. and reasonable parameter
estimates could be acquired through a complete sequential
analysis.

There is a disturbing jump at the time when the power
supply was turned off (at 1060 seconds). This had an effect
on.the parameter estimates for this experiment. but the effect
on the entire sequential analysis was negligible. Perhaps
more data should have been sampled after the power supply was
turned off. to determine the behavior of the sequential
parameter estimates after the disturbance.

Examination of Figure 11 reveals that the sequential-
over-time parameter estimates for thermal conductivity for the
96°C data are quite steady. indicating that these are good
estimates. The density-specific heat sequential estimates are
less steady. but stabilize as the last half of the data is
reached. This trend is as expected. since the estimates

include information from all previous temperature levels.

thermal conductivity (W/m’C)

Dens.-Spec. Heat (.l/m’ °C)x10"

67

 

0.800

 

 

 

 

 

 

 

 

 

 

 

 

 

0.600-

. g kfi --L 'fi I

kl

0.500 7 I I I - I I j . 1 I

0 ‘ 200 400 600 800 ' 1000 1200

time (seconds)

0.200 ' I I l ‘
0.190- -
0.180- -
0.170- P“: -
0160-qu . r , .
0.150- -
0.140- -
0.130 - .

0 200 460 600 800 1000 1200

time (seconds)

Figure 11- Sequential-over-time thermal conductivity

and density-specific heat estimates for
96°C data set

68

5.9 'Direct' vs. Sequential-Over-Experiments Parameter

Estimates

One common method of determining a thermal property
versus temperature curve is to analyze each experiment
separately. estimating two parameters (one thermal
conductivity and one density-specific heat) . rather than four.
Effectively. this method assumes that the thermal properties
are constant over the temperature range of the particular
experiment. This 'direct' method is a commonly used approach.
and is a. good basis for comparing the sequential-over-
experiments approach. Figure 12 illustrates that the
sequential-over-experiments approach is generally agreeable
with the direct approach. The confidence regions are slightly
smaller for the direct approach. but not as smoothly varying
with temperature. Perhaps the most important conclusion that
can be made from this figure is that there is a considerable
overlap in both sets of confidence regions. indicating that
both methods are satisfactory for this data.

It would seem that if the change in temperature for each
experiment were much greater. so that the temperature ranges
for consecutive experiments were to possibly overlap.
information would be acquired over the complete temperature
range, resulting in better parameter estimates for the
sequential-over-experiments approach. In a situation like
this. it would be more difficult to analyze the data directly.

since there would be some uncertainty as to which temperature

69

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70

to assign the estimates. Theoretically. the confidence
intervals would be greater for a larger temperature variation
using the direct method. However. a sequential-over-
experiments analysis of the same data.may be able to pinpoint
the estimates more effectively (possibly a topic of further
work).

One aspect of the experimental design that could be
considered a weakness would be that the heaters were not
capable of supplying enough heat to create a large temperature
variation. However. the sequential-over-experiments method

worked very well with small temperature variations.

Chapter 6
Conclusions and Recommendations

This chapter offers some conclusions about the method and
some recommendations for further possible areas of research.
Initially. the basic theory behind the linear formulation of
sequential-over-experiments estimation was described. The
method was proven viable for this case.

Some transient heat transfer experiments were run in
order to prepare data to be applied to the nonlinear
sequential-over-experiments analysis. A thorough analysis was
performed on the data to determine suitable ranges for
regularization.parameters. and to confirm the validity of the

method.
Conclusions

1, The use of regularization is shown to be very effective in
reducing the variability of parameter estimates for the first

set of data in a sequential-over-experiments analysis.

2. The magnitudes of the regularization parameters a1 and
a1 can be independently made. with “d influencing mainly

the conductivity components and my the pc components.

71

72

3. The final estimates after analyzing several sets of

data are quite independent of the magnitudes of the

regularizing constants.

4. A method is given for combining experiments
(simultaneousLy. This can be done even if the first

experiment contains only limited information about one or

more parameters .

5. Although the method is applied to the sequential-over-
experiments estimation of thermal properties. it is a
general technique and can.be widely used when a series Of

experiments is performed.

Recommendations

1. To further research the efficacy of the method. varying
temperature ranges could be tried for each experiment.

The results of a sequential-over-experiments analysis
with small temperature variations could be compared to
the results with large temperature variations. Also. a
series of experiments with non-uniform temperature
variations could be analyzed. which could lead to a

great variety of possible situations.

2. The method could be applied to a situation where the'

73

mathematical model changes with different temperature
ranges. One possible application is a situation where
conduction is modeled at low temperatures. and both
conduction and radiation are modeled at higher

temperatures.

APPENDICES

50

100

200

210

220

230

240

74

APPENDICES

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Bl-ADNH. 1) 'SY+ADN(1. 2) '52:
W (2. 1) «5an (2.219510:
1516:2112.» (11171.21.’ 02.1.22
1

8-1
11'“ 01.21119)
1811

210

79

76

81.52 11' (2 £0.31!”
1'16

EHD IF

sn-o

SY-O

SX-O

5105-0

00 210 14-1. 1+4
srx-sxx+¥ (L) 111(1)
82-51“! (L)
sx-sx+x (1.)
Sits-51642 (L) "2
CONTINUE

mu. 1) '5

mu. 2)-SX

m‘zo 1) -SX

X'I’X(2. 27.53

mm Ab m

RBI-Ru. 1) '81+A (1. 2) '82
832%(2. 1) '81+A (2. 2) '32

131' SUE-X1741»

Saul-SY‘tABl

SWISYX+ABZ

mm m A m

4(1. 1) 'DX (1. 1) +A(1. 1)

A(1o 2’ .m (10 2) +‘(1a22

‘62: 1)-X1'X(2. 1) 9" (20 1,
A(2.2) -1a'x(2.2) +A(2. 2) ,
DNA-Au. 1) '5 (2. 2) «(1. 2) '1(2. 1)

. mam 1) -‘(202) IDETA

m12.2)-A(1.l)/DETA
4mt1.2)-(-1.0)*A(1.21/nzu
m‘20 1) - (‘1 .0) 7‘62: 1’ IDETA
“(12. '1 'A- ' .30.. 1) .A(1.2)
MHZ. ') ’ ’.A(2.1).A(2.2)
s1-mvt1.1)'smn+m(1.2) W
B2-ADN (2. 1) 'sm+Antv (2. 2) '5”
1mm (12. ') '31“ .81. ' 32" .32
2-8+1

IP(2.EO.4)W

00 IO 79

m I?

0020 69

common

m

50

100

200
C

000

77

mm BY m HOLLISTER

MUCH M'O (£41.92)

DIMENSION X(20) .Y(20) .EIAHO) .225 (20)
DIMENSION mu. 2) .A(2. 2) .AIIW(2. 2) .HTH(2.2)
m 0.2

OPEN (11. FMI'RAND.OUT' . m3" 010')

OPEN (12.?HE"RAND.DM' .SMSI’ M')

00 50 47-1020

22110111. "05 (J)

our:

11103
no 100 1-1.5
tum-.5
xtI)-25.0
YtI)-E!A(I)+.02*EPS(I)
ETA(I+S)-.53333333333
X(I+5)-50.0
Y(I+5)-BTA(I+S)+.02*2Ps(1+5)
t:A(I+10)-.56666666666
xt1+10)-7s.0
Y(I+10)-BTA(I+10)+.02*EPS(I+101
BTA(I+15)-.6
x(1+15)-100.o
Y(I+15)-BTA(I+151+.02*EPS(I+15)

53-0

00 '200 K-1. 1

SYX-SYX+Y(K) 'X(K)

SYII3Y+Y(K)

SXISX+X(K)

SM3+XW "2

mm

sz-(zonm-srm I (20.'SB-SX"2)
31.81/20 . -B2 'SX/20 .

“In". ') 'mm' mm mum mm'
mo". ') mm

BTH(1.1)-0

813(2. 1)-0

H!!(1.2)-0

(113(2.2)-mm"2

17111.1) .1

mt2.1) 'SX

X'I'X(1.2)-SX

XTX(2.2) .m

A(1. 2)-X'1'X(1. 2) +811! (1.2)
042027-M(202) +I'1TH(2.2)
mum DEW G A m m
ans-Au. 1) 'A(2.2) -A(1. 2) 'A(2. 1)
mu. 1) -A(2. 2) /DE‘IA
W(2.2)-A(1. 1) /MA
m‘102).(-1.0) 'A(1.2)IDE'1'A
mu. 1) '(-1.0) ‘A(2. 1) IDBTA
m (12. ') Au. 1) .1111. 2)
mm. ') A(2. 1) .A(2. 2)
51-11mm. 1) 'SY+ADIV(1. 2) ’31!
”CW“. 1’ .SY+m (2. 2) m
m ‘12. ., '31., 031' ' 32-, '32
1-6

2.1

n(2.sa.2)rm

1'11

M00000

H
O

78

81.58 F(Z.BO.3)1'1EN

1'16

If

00 210 D-2.20

SIX-0

SY-O

SX-O

SXS-O

SIX-SIX“ (1.) 'X (1.)

SY-SY+Y(L)

SX-SX+X(L)

$XS-5XS+X(L) "2

CONTINUE

m(1.1)-1

XTX(1.2)-SX

XTX(2.1)-SX

XTX(2.2)-SXS

MING Ab mu

ADI-Au. 1) ‘31+A(1.2) '32
A32-A(2. 1) '81+A(2.2) '82

LB? SIM-X174)”

SM-SY-tfll

SMISYX+A32

CREATING NEW A MIX
A(1.1)-X'1'X(1. 1) +A(1.1)
A(1.2)-X'1'X(1.2) +A(102)
A(2.1)-m(2.1)+A(2.1)
A(2.2)-1G'X(2.2)+A(2.2)
011221-110. 1) *M2.2) -A(1. 2) 'M2. 1)
mt1.1)-A(2.2)/osn
AINV(2.2)-A(1.1)IDETA
AM(1.2)-(-1.0)'A(1.2)IDETA
1mm. 1) ' (-1.0) 'A(2. 1) IDer
10112202." '3' '.M1.1).A(1.2)
“(12." ' '.A(2.1).A(2.2)
Bl-ADNU. 1)tsm41+AmV(1.2)*smz
sz-anmz. 1) manna. 2) *SM
“(12.” 'Bld.31.' BZ-‘.B2
8-844

madman-m

GO ID 79

m I?

6070 69

cam-mas

1').

79

3

1..
3. .hv.

. 31
i

1

p" '
L

N

\.\f
2'.

Figure 13- Photograph of experimental apparatus and
hardware used to collect data

a .77.
A .1”, A)?
«iv

o
'i".
’r‘

1
'-
1»?

 

LIST 0? REFERENCES

80

LIST OF REFERENCES

Beck. J. V. and K. N. Arnold. 1977, Parameter Estimation in
Engineering and Science. John Wiley & Sons, New York.

Beck. J. V., L. Hollister, and A. Osman, 1990, "Sequential
Estimation of Temperature-Dependent Thermal Properties,"
Presented in Inverse Problems in Engineering Seminar,
East Lansing, Michigan.

Chang, M. N., T. M. Therneau, H. S. Wieand, and S. S. Cha,
1987. ”Designs for Group Sequential Phase II Clinical
Trials.” Biometrics 43, 865-874.

Dhattacharyya, G. K. and R. A. Johnson, 1977, Statistical
Concepts and MethodsI University of Wisconsin. John
Wiley & Sons.

Loh, M., 1989, M.S. Thesis, "Two-Dimensional Heat Transfer
Studies in Carbon Composite Materials, Department of
Mechanical Engineering. Michigan State University. East
Lansing. Michigan.

Rooke, S. P. and R. E. Taylor. 1988, "Transient Experimental
Technique for the Determination of the Thermal Ditfusivity
of Fibrous Insulation,” Journal of Heat Transfer, Volume
110, pg. 270.

Scott, E. P.. 1989, PhD Dissertation. "Parameter Estimation
During Curing of Composite Materials. Department of
Mechanical Engineering, Michigan State University.

East Lansing, Michigan.

81

Scott, E. P. and J. V. Beck. 1989, ”Analysis of Order of the
Sequential Regularization Solutions of Inverse Heat
Conduction Problems," Journal of Heat Transfer, Volume
111, pg. 218.