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COMPARISON OF THERMAL DIFFUSIVITY
MEASUREMENT TECHNIQUES

presented by

XIN HUANG

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6/01 c:/CIRC/DateDue.p65—p.15

COMPARISON OF THERMAL DIFFUSIVITY
MEASUREMENT TECHNIQUES

By

Xin Huang

A THESIS
Submitted to
Michigan State University
in partial fulﬁllment of requirements
for the degree of
MASTER OF SCIENCE

Department of Mechanical Engineering

2004

ABSTRACT

COMPARISON OF THERMAL DIFFUSIVITY MEASUREMENT TECHNIQUES
By

Xin Huang

Accurate knowledge of thermal diffusivity is important for thermal analysis
design. Although oﬁen assumed to respond isotropically, many polymers and composite
materials exhibit thermally anisotropic responses. Materials with unknown or changing
properties especially challenge the design of experiments to measure thermal diffusivities.
Three thermo—optical techniques that have been proposed to measure anisotropic thermal
diffusivities are the ﬂash, thermal wave, and ﬂash-ﬁeld methods. In this work, D-
optimum criteria are applied to optimize experimental parameters such as the number of
temperature sensors and sensor positions in the ﬂash method, or the number of
measurements and measurement times in the other two methods. The modiﬁcation of
these experimental parameters can be easily achieved after an experiment for which the
temperature field history of one of the specimen surfaces is measured, such as by an
infrared camera. Here, each of these methods is optimized using simulated experimental
temperature responses calculated for radially symmetric thermal diffusivity; noise has
been added to simulate experimental data. These optimizations are then compared to

determine the preferred technique for this material symmetry.

ACKNOWLEDGMENTS

I am grateful to my thesis advisor, Dr. Neil Wright, for his guidance and support
during my thesis study. In particular, I want to acknowledge the useful discussions with
Dr. James Beck and Sean Davis. I would like to thank Dr. James Beck and Dr. Craig
Somerton for serving on my thesis defense committee.

I owe a debt of thanks to all my friends at Michigan State University, especially to
Jian Wu, Xingkai Chi, Yanbing Li and Liping Jia, for their encouragement and help
during my time at MSU. Finally, special thanks are extended to my parents and the rest of

my family for their love and support.

TABLE OF CONTENTS

LIST OF FIGURES ....................................................................................................... v
LIST OF TABLES ...................................................................................................... viii
CHAPTER 1 INTRODUCTION ........................................................................... 1
CHAPTER 2 OPTIMIZATION METHOD ........................................................ 5
2.] Mathematical Model ............................................................................................... 5
2.2 Estimation Technique ............................................................................................. 8
2.3 Optimization Method ............................................................................................ 10
CHAPTER 3 OPTIMIZATION ........................................................................... 13
3.1 Flash Method ........................................................................................................ 13
3.1.1 Temperature Simulation ................................................................................ 13
3.1.2 Sensitivity Analysis ...................................................................................... 14
3.1.3 Optimization ................................................................................................. 21
3.2 Thermal Wave and Flash-field Methods ............................................................... 32
3.2.1 Temperature Simulation ................................................................................ 32
3.2.2 Sensitivity Analysis ...................................................................................... 34
3.2.3 Optimization ................................................................................................. 34
CHAPTER 4 COMPARISON OF METHODS ............................................... 45
CHAPTER 5 CONCLUSION .............................................................................. 49
REFERENCE ................................................................................................................ 51

LIST OF FIGURES

Figure 2.1 Schematic representation of the mathematical model. ...................................... 6

Figure 3.1 Schematic representation of the ﬂash method with an extended heating.

Heating occurs within the radius rp on the front face. Temperature sensors are positioned

at Tm] and Tmz. ................................................................................................................ 13

Figure 3.2 Simulation of temperature rise at various radii for the ﬂash method with a
short heat pulse (a) and an extended heating (b) ............................................................... 16

Figure 3.3 Normalized sensitivity coefﬁcients of axial thermal diffusivity (a2) for the
ﬂash method with a short heat pulse (a) and an extended heating (b). ............................. 17

Figure 3.4 Normalized sensitivity coefﬁcients of radial thermal difﬁisivity (arr) for the
ﬂash method with a short heat pulse (a) and an extended heating (b). ............................. 18

Figure 3.5 Normalized sensitivity coefﬁcients of Biot number (H ) for the ﬂash method
with a short heat pulse (a) and an extended heating (b) .................................................... 19

Figure 3.6 Normalized sensitivity coefﬁcients of ﬂash (Q) for the ﬂash method with a
short heat pulse (a) and of heat ﬂux ( q") for the ﬂash method with an extended heating
(b). ..................................................................................................................................... 20

Figure 3.7 Optimization of measurement time (to, if) for the ﬂash method with a short
heat pulse (a) and an extended heating (b). The heating time (t p = 30 s) and heating area
(rp =2mm)are ﬁxed. ..................................................................................................... 23

Figure 3.8 Optimization of sensor position (rm) and ﬁnal measurement time (t f ) for the

extended ﬂash method, with initial measurement time to =10 s . The heating time
(t p = 30 s) and heating area (rp = 2 mm) are ﬁxed ........................................................ 24

Figure 3.9 Optimization of the second sensor position (rm, 2) for the extended ﬂash

method, with the ﬁrst sensor at the center on the rear surface and initial measurement
time to =10 s . The heating time (tp = 30 s) and heating area (rp = 2 mm) are ﬁxed. .25

Figure 3.10 Optimization of positions of both sensors (rm, 1 , rm, 2) for the extended ﬂash

method. The ATZ of each point is obtained with optimized measurement time interval.
The heating time (t p = 30 s) and heating area (rp = 2 mm) are ﬁxed. .......................... 26

Figure 3.11 Optimization of positions of the second and third sensors (rm, 2, rm,3) for

the extended ﬂash method, with the ﬁrst sensor at the center on the rear surface. The AB
of each point is obtained with optimized measurement time interval. The heating time
(tp=3OS)andheatarea(rp=2mm)areﬁxed ............................................................. 27

Figure 3.12 Ratio of determinants with increasing numbers of sensors (M ) evenly
positioned between the center and the edge on the rear surface for the extended ﬂash
method. The measurements start at to =10 s. The heating time (tp = 30 s) and heat area

(rp = 2 mm) are ﬁxed. ..................................................................................................... 28

Figure 3.13 Optimization of heating time (t p) and ﬁnal measurement time (t f) for the

extended ﬂash method, based on a priori information about unknown parameters. The
heat area (rp = 2 mm) is given. ....................................................................................... 30

Figure 3.14 Optimization of heating area (rp) and second sensor position (rm, 2) for the
extended ﬂash method, based on a priori information about unknown parameters, with

one sensor at the center on the rear surface. The ATZ of each point is obtained with
optimized heating time (t p) and measurement interval (to , t f ). ................................... 31

Figure 3.15 Schematic representations of the thermal wave method (a), with temperature
measurements on the front surface, and of the ﬂash-ﬁeld method (b), with temperature

measurements on the rear surface. Heating occurs within the radius rp on the front face.

........................................................................................................................................... 33

Figure 3.16 Simulation of temperature rise for the thermal wave method (a) and the ﬂash-
ﬁeld method (b) ................................................................................................................. 36

Figure 3.17 Normalized sensitivity coefﬁcients for the thermal wave method (a) and the
ﬂash-ﬁeld method (b). ...................................................................................................... 37

Figure 3.18 Ratio of determinants with two measurements for the thermal wave method
(a) and the ﬂash-ﬁeld method (b). The heating time (t p =30s) and heating area

(rp =2mm)are ﬁxed. ..................................................................................................... 38

vi

Figure 3.19 Ratio of determinants with a third measurement at various times for the
thermal wave method (a) and the ﬂash-ﬁeld method (b). The heating time (t p = 30 s)

and heating area (rp = 2 mm) are ﬁxed. .......................................................................... 39

Figure 3.20 Ratio of determinants with a fourth measurement at various times for the
thermal wave method (a) and the ﬂash-ﬁeld method (b). The heating time (t p = 30 s)

and heating area ( rp = 2 mm) are ﬁxed. .......................................................................... 40

Figure 3.21 Ratio of determinants with a ﬁfth measurrnent at various times for the termal
wave method (a) and the ﬂash-ﬁeld method (b). The heating time (II) = 30 s) and heating

area (rp = 2 mm) are ﬁxed. ............................................................................................. 41

Figure 3.22 Optimization of heating time (t p) and heating area (rp) for the thermal wave

method (a) and the ﬂash-ﬁeld method (b), based on a priori information about unknown
parameters, with four optimized measurement times. ...................................................... 43

vii

LIST OF TABLES

Table 3.1 Optimal position of the second sensor, rm,2/rs , for changing thermal

diffusivity, with rm] /r, = 0. (a0 =1.ox10‘7 mzs“) ................................................. 29

Table 3.2 Maximum ratio of determinants ATZ , from samples of different sizes with the

ﬂash method, for materials with (12 =1.5x10—7 mzs’1 and a, = 1.0x10—7 mzs-l.... 32

Table 3.3 Maximum ratio of determinants Aiz , from samples of different sizes with the
thermal wave method based on four measurements, for materials with

a2 =1.5x10—7 mzs‘l and a, =1.Ox10—7 mzs’l. ....................................................... 44

Table 3.4 Maximum ratio of determinants ATZ , from samples of different sizes with the
ﬂash-ﬁeld method based on four measurements, for materials with

a, =1.5><10”7 mstl and a, =l.0x10T7 mzs’l. ....................................................... 44

Table 4.1 Estimation results with measurement errors of standard deviation
0‘ = 0001me. ................................................................................................................. 47

Table 4.2 Estimation results with measurement errors of standard deviation
0 = 0.01Tmax. ................................................................................................................... 47

Table 4.3 Estimation results with measurement errors of standard deviation
(7:0.027’max .................................................................................................................... 48

viii

CHAPTER 1 INTRODUCTION

The standard ﬂash method, proposed by Parker et a1. [1], is a widely used
approach to measure thermal diffusivities of isotropic materials. The top surface of a
cylindrical sample is subjected to a short heat pulse from a laser or a ﬂash lamp and the
temperature rise at the center of the rear surface is measured. As originally proposed, the
time at which the temperature rise reaches one-half of its maximum is inversely
proportional to the thermal diffusivity. An ASTM standard now exists for the
implementation of the ﬂash method [2].

Maillet et a1. [3] extended the standard ﬂash method to measure simultaneously
the axial and radial thermal diffusivities of anisotropic materials. Only the central area of
the upper surface of a disk—shaped specimen was exposed to a heat pulse. The
temperature histories at two locations on the opposite face were measured with
thermocouples. One thermocouple was placed at the center and the other was placed at an
optimal position, as determined by calculation, outside of the projected illumination. The
geometric parameters of heated area and specimen size were also adjusted to improve the
quality of measurements.

The high temperature rise at the heated face of the sample after the very short
ﬂash limits the utility of these techniques in the measurements of materials with low
thermal diffusivities or in large samples. Vozar et al. [4] replaced the heat ﬂash with an
extended heating to reduce heating intensity and thus the maximum temperature rise at
the heated area. Instead of using two thermocouples, they demonstrated that one
temperature sensor at the center of rear surface was enough to identify both axial and

radial thermal diffusivities, heat ﬂux, and heat transfer coefﬁcients at the boundaries. To

improve the estimation quality, they applied a sensitivity analysis, the D-optimization
method, to optimize heating duration and measurement time. Their analysis revealed
several important design considerations, but the number and positions of thermocouples
used to measure the temperature rise of the specimen were ﬁxed, leaving unconsidered
adjustment of temperature sensors.

Applying D-optimality to set the positions of thermocouples requires assuming
values for thermal diffusivities. The quality of optimal design then depends on the values
initially guessed for thermal diffusivities. Materials with unknown or changing thermal
diffusivities challenge the design of experiments intended for optimal measurements of
the components of thermal diffusivity tensors. Thermal diffusivities of some isotropic
materials, such as polymers, have been shown to become anisotropic under deformation
[5]. Wright et a1. [6] proposed to expand the method of Maillet et a1. [3] to measure the
three principal components of thermal diffusivity of elastomer subject to biaxial ﬁnite
stretching. Doss and Wright [7] used the expanded ﬂash method to measure the thermal
diffusivity of polyvinylchloride. LeGall [8] then substituted an extended laser pulse of
about four seconds for the ﬂash and measured the thermal diffusivities of several
elastomers subject to biaxial stretching. LeGall demonstrated that the thermal
diffusivities of elastomers change under biaxial deformation. Highly accurate estimation
of thermal diffusivities is especially important for this kind of materials because the
differences in the components of thermal diffusivity tensor caused by deformation may
be small, especially for small deformation, but could yield insight into thermomechanical
behavior. To improve the estimation accuracy, either additional experiments with

improved design of sensor positions, based on the estimated thermal diffusivity from the

ﬁrst experiment, may be needed or the number and positions of temperature sensors must
be adjustable a postori in order to enhance the quality of the measurements that may
result from material property change. Such movable sensor positions are easily
implemented if the temperature ﬁeld history is measured, as with an infrared camera.

An advantage of substituting an infrared camera for the thermocouples used, for
example, by Doss and Wright [7], is that the entire temperature at one surface can be
recorded and the choice of pixels, or sets of pixels, used to determine the thermal
diffusivity may be adjusted optimally. Applying infrared cameras to measure temperature
is also a non-contact method, which is preferred for easily deformed materials, such as
rubber and biological soft tissue. The analysis presented in this thesis seeks to optimize
the number and positions of temperature sensors in the extended ﬂash method of LeGall
[8] for given parameters of heating duration and geometry, which are ﬁxed prior to the
experiment. This is a step to improve the quality in measuring anisotropic thermal
diffusivity.

Two other methods of thermal diffusivity measurement will be examined in
addition to the extended ﬂash method examined above. With the use of infrared cameras
becoming common, thermal imaging method is an emerging nondestructive technology
for thermal property characterization. Philippi et al. [9] and Kulkarni and Brady [10]
applied a heat pulse on one surface of the sample and recorded the entire transient
temperature images on the opposite surface. This method is here called as the ﬂash-ﬁeld
method. Telenkov et al. [11], for example, applied the thermal wave method to measure
the thermal diffusivity of tissue; thermal wave method records the temperature ﬁeld on

the surface subjected to heat at two times after heating. Thus, the goal here is to compare

the ﬂash-ﬁeld and thermal wave methods with the extended ﬂash method using D-
optimality. Results that are shown are representatives of elastomers with radially

symmetric thermal diffusivity under deformation.

CHAPTER 2 OPTIMIZATION METHOD

2.1 Mathematical Model
Similar to the work of Maillet et a1. [3], the physical model considered here

consists of a cylindrical specimen of radius rs and thickness e , with uniform and

constant mass density p , heat capacity c, and axial and radial components of thermal

diffusivity az , a,. The sample is initially at uniform temperature T; , equal to the

ambient temperature. The central part of the upper face (2 = O , r S rp) is exposed to a

constant heat ﬂux q" for time t S 1,, (Figure 2.1). By deﬁning the temperature excess

T(z,r,t)=r*(z,r,r)—Tg, (1)
the diffusion equation may be written as

«2 ~2
“ﬂan—0 Tug—0 H191), (2a)
3 " arz rar

0t 522
with the initial condition
T = O at t = O , (2b)
where k, and k, are the axial and radial components of thermal conductivity,
respectively.
The sample has convective heat loss through its upper, lower and lateral surfaces.
Assuming that the upper ho , lower he and lateral h, heat transfer coefﬁcients are

constant and uniform on each surface, then the boundary conditions may be written

_%Z:_(HO/e)T+g/kz at 2:0, (7—0)
2

6T

——a—Z—=(He/e)T at 226, (2d)
-Z—:=(Hr/rs)T at r=rS. (26)

The surface heat ﬂux may be treated as a source term

" OSrSr,OStSt ,z=0
g = q p p , (2f)
0 others
where H 0 , H e and H r are the upper, lower and side Biot numbers, respectively

h i, h
H0: 06, He: "6, H,= ”S. (3a-c)

 

 

 

k2

 

 

 

 

 

 

 

 

Figure 2.1 Schematic representation of the mathematical model.

The temperature excess may be expressed in the form of a Fourier series [4]

 

 

 

 

 

 

 

 

2 2
u a- w a
1_ex _ n z. + m rt
T(z,r,t)=— Z ZA)1(HO’Hea-)Bm(Hra—a_) 2 2
pcen=lm=l e rs rs u" a, Wm (1r
‘ +
e2 rs2
OStStp, (4a)
or m (X) z r r
T(z,r,t) =—‘-’— z 2An<H0,H..—>B,,,(H..—p.—)
pcen=1m=l e rs rs
2 2
u" a: Wm al’
uza, w 2a CXp[( e2 + r2 )t]
exp[( I1 .. + m I‘)t ]_1 S
2 2 p 2
e rs u” a- wm a,
e2 rs2
t>tp, (4b)
Here
2. k-
a2 :L a}: :4“, (53,b)
PC PC
with
2 2 2 Z HO . z
7 21.4,, (un +He )[cos(u,,;)+Z—-sm(u,, ;)]
An(H0,He,:)= ” (6a)

(14,,2 + H02)(u,,2 + H} ) + (H0 + He)(u,,2 + HOHe)

r r r
2Wm iJMWm ’rﬂ)‘]0(wm 7—)

rP r rs s s
3,, (H,,—,—) = 2 2 2 , (6b)
rs rs JO (wm)(wm +Hr )

 

and u" and wm are the positive roots of equations

(u2 —- H0H8)tan(u) = (HO + He)u, (7a)

mm) = 11,100.», (7b)
where J 0 and J1 are Bessel functions of the ﬁrst kind, of order 0 and 1.

If the sample is subjected to an instantaneous heat pulse (i.e., a ﬂash) instead of an
extended heating, an expression for the temperature excess can be obtained by

differentiating Equation 4a, for the extended heating, with respect to time [12]. The result

may be written as

 

2 2
0° 0° 2 r r u a
T(errt): Q 2 ZAn(HO’He’_—)Bm(Hr’ P,_)exp[_( n .. + m r)t], (8)
pcen=lm=l e r5 r3 3 ’3

where Q is the energy absorbed per unit area from the ﬂash.

2.2 Estimation Technique

The goal here is to estimate the thermal diffusivity components 612 and a,. In
this problem, H 0 , H e , H r and q" (or Q) are difﬁcult to be identiﬁed a priori, and thus
are taken as additional (or nuisance) parameters to be estimated. We will assume that the
heat transfer coefﬁcients ho, he and h, are equal, then the parameters to be estimated
are thermal diffusivity components a: and a, , Biot number
H 2 H0 = He = Hrarre/az /rs and heat ﬂux q" (or energy Q).

Following the development in Beck and Arnold [l3], assume that M sensors are
used to record the transient temperature rise Y(z,r,t) at I different times, and that the

error in temperature measurement is random, additive, uncorrelated and normally
distributed with a zero mean and a known constant standard deviation 0'. The four

unknown parameters may be obtained by minimizing the ordinary least-squares norm

5m = [Y -T(P)]T[Y -T(P)1, (9)
where P is the vector of unknown parameters

PT =[P/. P2 ----- P~1=taz. ar. H. q"], (10)
and

[Y -T(P)IT = [Y1 —T1(P),Y2 —T2(P>....,Yi 4m]. (11)
Each element [Yi — Ti (P) ] is a row vector such that

[Yi 'Ti (1’)] = [Yil — 731(1’), Yi2 ‘ 712(P),--»Y1M — TIM (P)], (12)
where Yim is the temperature rise measured by sensor m at time ti, for m =1,...,M and
i = 1,...,1 . The estimated temperature rise Tim(P) is obtained from Equation 4 or 8 by

using the current estimate of P.
The Levenberg-Marquardt method is applied to calculate the unknown parameter

vector P and the iterative procedure is given by [14]
Pk” =P" +((J")TJ" +yknk)‘1JT[Y—T(P")], (13)
where k is the iteration superscript, Jk is the sensitivity matrix, 52" is a diagonal matrix

and ,uk is a positive scalar damping parameter [14]. Matrix ,ukﬂk is set to damp the

oscillations and instabilities caused by the ill-conditioned character of this problem [15].

The sensitivity matrix is deﬁned as

 

 

 

 

- QZL ﬂ .5__T1 1
5P1 6P2 6m
.52. 37:2. 9L2
T 6‘31 6’37- 6p!"
Jan-simi’: = = = (14)
61) 8T5 5T5 6T5 ’
51.91 5132 BEN
arm arm arm
[ 6171 6P2 61w -

 

 

where s=(i—1)M+m for i=1, 2,...,I and m=1,2,...,M . The elements of the

sensitivity matrix may be denoted as sensitivity coefﬁcients

5T,

an :52, (15)

where n =1, 2, ..., N.

2.3 Optimization Method

Recall that the error in temperature measurement has been assumed to be random,
additive, uncorrelated and normally distributed with a zero mean and a known constant
standard deviation 0'. It is also assumed that there are no errors in the measurement of

sensor location r , z and that the parameters to be estimated are constant. Several
optimal criteria have been proposed based on the information of matrix J TJ , deﬁned as
Fisher Matrix F [13]: (1) maximization of the determinant of JTJ [15] [13]; (2)
maximization of the minimum eigenvalue of J TJ [16]; (3) maximization of the trace of

J TJ [13]; (4) minimization of the condition number of .1 TJ [16].
The criterion applied here is D-optimality, similar to [15], namely

max A = max det(F). (16)

10

Maximizing the determinant of F is equivalent to minimizing the hypervolume of the
conﬁdence region. A small conﬁdence region ensures a small variance for the estimation,
which means high precision of the measurement [15].

Since more than one parameter are measured, it is desirable to write the D-

optimum design of experiment in the dimensionless form as [15]

max N =max det(J+TJ+), (17)

+ . . . . . . . . . . . .
where J IS the dmrensronless sensrtrvrty matrix composed of dimensronless sensrtrvrty

coefﬁcients, which are deﬁned as

P 5T.
an+ = n _—S" (18)
Tmax apt:

 

where Tmax is the maximum temperature rise within the sample. This allows the
different experimental techniques and parameters, with different T max , to be compared.

If a large but ﬁxed number of equally spaced measurements are recorded, it is
more convenient to design the experiment by maximizing the determinant of G , instead

of F [13]. In the extended ﬂash method, the elements in G are given by

1 M’Zf’ 57

[G]x,y= Wmélt 11:0(JraP—T-ﬂ)(ygpy Tmax

 

     

) 2(It for x,y=l, 2, ...,n, (19a)

where to and t f are the start and end times of measurement, respectively. In the thermal

wave and ﬂash-ﬁeld methods, the elements in G are given by

l’zrm (357‘

Zl(P:x—;i-Py-S-)(-)(

”mi: Ir: 0 yap), Tmax

 

 

[G]x,y= ) 2dr for x, yzl, 2. (19b)

where rm is the radius of temperature ﬁeld measured.

11

For cases where only some of the estimated parameters are of interest, and thus
need to be estimated accurately, the estimated parameters can be partitioned into two
parts, P] and P2 , as

T T T
P =[P1 P2 ]- (20)
Here, the vector P1 consists of the unknown parameters of primary interest (az , a,) and

P2 contains the other unknown parameters (H , q" ). Then, the sensitivity matrix may be

partitioned as

J“ =1JiJ‘2‘1 (21)
Accordingly, the optimal design is obtained by maximizing the ratio of det(J+T J +) to
det(JgTig) [13], which is

det(J+TJ+)
det(JJZ'T .13)

 

max ATZ = max (22)

12

CHAPTER 3 OPTIMIZATION

3.1 Flash Method

3.1.1 Temperature Simulation

In the ﬂash method as expanded to 2D, the transient temperature rise at either one
position [4] or two [3] on the rear surface is recorded after the front surface has been
subjected to heating over part of its area (Figure 3.1). Figure 3.2 presents the simulated
temperature rise at different positions, for the ﬂash method with a short heat pulse

(Equation 8) and for the ﬂash method with an extended heating (Equation 4).

 

 

 

 

 

e
i r.
r Tm2
p
q” —>i 2
(t5 tp) _—’ """""" ' -------- ->-
Tml
—>
< 2>
no he

 

 

 

<—-.---.-

Figure 3.1 Schematic representation of the ﬂash method with an extended heating.
Heating occurs within the radius rp on the front face. Temperature sensors are positioned

at Tm] and Tmz-

13

The results in Figure 3.2 are based on a disk-shaped sample with thickness

e = 3 mm and radius r5 =10 mm. The central area (rp = 0.2r3) of the front surface is

subjected to heating. The axial and radial thermal diffusivities are 1.5 x 10'7 mzs‘l and 1.0
x 10'7 mzs‘l, respectively; these are known for the forward problem. These diffusivities
are representatives of elastomers and biological soft tissue, which may exhibit directional

thermal response. The heat transfer coefﬁcients ho, he and h, are assumed to be the
same, yielding Biot numbers of H = H0 = He = Hrare/(zz /rs = 0.05 . Figure 3.2a
shows the temperature response for an instantaneous heating. The extended heating time
in Figure 3.2b is 30 s, which corresponds to a Fourier number F0 = aver/e2 = 0.5. The
energy absorbed per unit area Q from the ﬂash and the heat ﬂux absorbed per unit area
q" from the extended heating have been adjusted to make the maximum temperature rise
Tmax (i.e., the temperature at the center of the front surface) the same in each sample

[13]. Random noise following a Gaussian distribution with a zero mean and a standard

deviation 0 = 0.0017"max has been added to the calculated value to simulate experimental

noise. Although the curves in Panels a and b have similar shapes, the temperature rise
recorded on the rear surface is about one order greater in Panel b than in Panel a,

corresponding to the difference in the total energy added to the specimen.

3.1.2 Sensitivity Analysis

Figures 3.3 through 3.6 show the dimensionless sensitivity coefﬁcients J; ,

J; , J I, and J (7 (or Jé ), corresponding to the temperature rise simulated in Figure
r

14

3.2. It can be observed that the sensitivity curves of J; , J; , J L and J J" (or Jé) at
r

2
each point have different shapes and thus are uncorrelated. Therefore, it is possible to
estimate all of the unknown parameters using one point measurement [15].

The sensitivity coefﬁcients obtained by applying a short pulse and an extended

heating have similar shapes. The sensitivities from extended heating are, however, about

one order greater than those from pulse. In both cases, the magnitudes of J; , J H and
J (7 (or J 5) decrease as the sensor position is changed from the center to the edge; J g
,

ﬁrst decreases and then increases. Furthermore, J; , J; , J}; and J 3"(or J5) each
2 r

have their maximum magnitudes with the sensor at the center of the rear surface. The
sensitivity coefﬁcients change dramatically at the beginning of the measurements and
then reach asyrnptotes as time progresses. Therefore, the temperature recorded at the

beginning contains more information than it does during other times.

15

0.03 I TI I I r I I I r

 

 

 

 

 

 

 

 

 

 

 

0.025 " 1' ff : ‘
m S
002 r r 1:0.225 (a) ‘
M m S
E
t—« 0.015- ' .. _ -
)2, 1‘ 1:04.50
H
"1 0.01 - =11 .. . ._ -- '1
o 1' 06 5
v
I" I, .
13.005- “It' . .3 § .
. "hf... Tl" 15,1, I1"... "h. LEI: ~ ‘i'_|.
. ”1 --_ If“: :- .H‘,l'rfll' i1“. z-h‘jw. $511} L
Di; ’:'3-
0005 I I I I l I I I l
0 05115 2 2.523 3.5 4 4.5 5
a We )
Z
03 I I I I r I I T T
0.25
02
>4
E
t_—1 0.15
is
"'1 0.1
4)
V
{—1
0.05
0
_005 I I I l I l I I I
0 05 1 1.5 2 2.523 35 4.5 5
0t t/(e )

Figure 3.2 Simulation of temperature rise at various radii for the ﬂash method with a
short heat pulse (a) and an extended heating (b).

16

16 I I I I I I I I I

.1 .

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(a)
12 - 5
10 - 4
-— rm/rs=0
:3 8" ........ rm/rs=0.225 *
L. 6~" _- r /r=0.450
:'=. m S ‘
_ r /r=0.675
4 : m S .1
..... r /r=0.900
m S
2 .4
D
-2 1 1 1 L 1 1
2 2.5 2 3 3.5 4 45 5
a L’(e )
z .
0.18 I I I ‘T l f
0.16 _
(b)
0.14 5
0.12 5
0.1 4
—— r /r =0
5N008 .mt.s_o 2‘); -1
t5 ........ lmus- . ..
= '§
0.06 _. rm/rs 0.4-0 -
_ r /r =0.675
0.04 m s .-
:' ..... rm/rs=0.900
0

 

 

 

 

 

0.02 I I I I I L J I I
0 0.5 1 15 2 2.5 2 3 . .
azt/(e )

Figure 3.3 Normalized sensitivity coefﬁcients of axial thermal diffusivity (az) for the
ﬂash method with a short heat pulse (a) and an extended heating (b).

17

 

 

 

 

t3”

   

 

 

 

 

 

 

 

 

 

 

  

 

   

 

 

 

 

 

 

__ r /r =0
m S
_5 _ ........ r /r =0.225 :
m S
-....- r it =O.450
m, S
-3 _ ..... rm/rs=0.675 _
r /r =0.900
m S
40 1 1 1 1 L 1 1 1 1
[J 0.5 1 1‘5 2 t2:52 3 3.5 4 4.5 5
(122(6 )
01 I I I I I I I I I
(b)
0.05 - ‘
U --------------------------------------------
-D.D5
L6
-U.1 __ r [1' :0
m S
........ rmfrs=0.225
015" ..... r /r=o.450 ‘
m S
..... 1‘ III. =O.675
02 _ m S _
_ r lr=0.900
m 5
_D25 1 J l l l I l l L
0 0.5 1 1.5 2 2.5 2 3 3.5 4 4.5 5
alt/(c )

Figure 3.4 Normalized sensitivity coefﬁcients of radial thermal diffusivity ((1,) for the
ﬂash method with a short heat pulse (a) and an extended heating (b).

18

 

 

x 1D-
2 I I I I I I I I I
— fr =0
m S
........ 1' /1' =0.225
m S
..... r /1' =0.450
U m S _

 
   
 

 

..... r fr =0.675
m S
1' fr =0.900
m S

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

LP -2 ~
_4 .. u.o...-......".u.-.:“
-6 l l l l l 1 l l l
D 0.5 1 1.5 2 2.5 2‘ 3 3.5 4 4.5 5
0. Me )
z
x 10'3
2 I I r I I r I I I
__ r fr =0
0‘ m s _
\ “x“ ..... lm/rs=0.225
‘. ~ — — 1’ ' = .' 5 ..
,2 _ 1 \ 5‘“ rm Is 0 4 0
~. \ xx ---- rm/i's=0.675
-4 " ' ‘ -1
. \ ‘~\ _ rmlrs=0.900
-8 ¥ 1 \ \“~‘ .1
ti“ ' \
\ ‘\ (b)
-8 .- \ 5“ _
x!“ ~~
x x s...”
-10 _ ‘1‘ K M “-5 -
‘ ------------------------------------------- :“THE
-12 - ':
44 l" T
45 1 1 1 1 1 1 1 1 1
D 0.5 1 1.5 2 3.5 4.5 5

2.5 2 3
aZt/(e )

Figure 3.5 Normalized sensitivity coefﬁcients of Biot number (H ) for the ﬂash method

with a short heat pulse (a) and an extended heating (b).

19

 

0.03 I If ﬂ I I I ﬁr I I

 

   
    

__ r if =0
m s
_ 00000000 - . = c 25 "'
0.025 im/iS 0 2
..-o- I III. =0-450
m S
002- ..... r /1'=0.675 -
m S
__ 1' 1'1' =O.9OO
m S

 

 

  
   
    

 

0.015 -
LO '

 

 

 

 

 

 

 

 

 

  

 

 

 

 

 

_0005 1 l 1 1 1 l 1 l l
0 0.5 1 15 2 2.52 3 35 4 45 5
0.. (43(6 )
03 I I I I I I I I I
— fr =0
m S
. ........ ' ."-= . 2.5 _
0.25 lmis OZ
-.-.- r ,’1'=0.450
m S
0.2— ..... r /1'=0.675 —
m S
_ r ;‘1'=0.9OO
m S
0.15
150*
0.1
0.05
U.
-005 1 1 l 1 l 1 l 1 1
0 0.5 1 15 2 2.5 2 3 35 4 45 5
aztx'(e )

Figure 3.6 Normalized sensitivity coefﬁcients of ﬂash (Q) for the ﬂash method with a
short heat pulse (a) and of heat ﬂux ( q") for the ﬂash method with an extended heating
(b)-

20

3.1.3 Optimization

Experiments can be optimally designed by maximizing the ratio of determinants

of sensitivity matrices of all unknown parameters and of the nuisance

parameters A12 (Equation 22), calculation of which requires assuming values for the
unknown parameters being sought. Some of the independent parameters, such as sample
size, heating area and heating time are ﬁxed for a given experiment. Other parameters,
however, such as measurement time, number of temperature sensors and sensor positions,
can be optimized and adjusted after the test by recording the temperature ﬁeld with the

use of an infrared camera.

Figure 3.7 shows A12 obtained by measuring the temperature rise at the center of

the rear surface during different measurement time intervals for the ﬂash and long
duration heating. The curves in each ﬁgure represent measurements starting at different

times to. Observe that each curve ﬁrst increases and then decreases with the increasing

of ﬁnal measurement time t f. The optimal measurement time can be obtained at the

highest point, which represents the maximum ratio of determinants A12. Although the

temperature rise measured during heating contains more information than after heating, it
is not necessary to start measuring temperature at the beginning of heating. For the
temperature history of the ﬂash method with an instantaneous heat pulse in Figure 3.7a,
the optimal time to start measurement is approximately 3 s after the start of heating and
the corresponding optimal ﬁnal time is 300 s. The optimal measurement time for the ﬂash

method with an extended heating of 30 s is between 8 and 320 s (Figure 3.7b). The

maximum Asz obtained by using the ﬂash method with an extended heating is about ﬁve

21

orders greater than that with an instantaneous heat pulse, again with the constraint of the

same T max . Therefore, it is better to apply an extended heating to the sample instead of a
short pulse.

An advantage of measuring temperature with an infrared camera is that the
temperature history on the entire surface can be recorded, which means that the transient
temperature history at all positions is available. Figure 3.8 presents A12 obtained at

different points on the rear surface with measurements starting at to 2105. The

maximum A12 clearly decreases as the location of the temperature history is moved from

the center to the edge on the rear surface. The best position is then at the center of the

surface, if only one temperature sensor is used.

22

 

 

 

 

 

 

 

l

0 L l 1 1 L l
50 100 150 200 250 300 350 400 450
t.<s>

 

 

 

 

1.8

1.6

 

 

1,4

 

A12

1.2

[18

 

. 'u
I

 

 

0'6 1 l l 1 l 1
200 250 300 350 400 450 500 550

tf (S)

 

Figure 3.7 Optimization of measurement time (to, If) for the ﬂash method with a short
heat pulse (a) and an extended heating (b). The heating time ( t p = 30 s) and heating area

(rp 2 2mm) are ﬁxed.

23

x10-9

 

 

   
  

 

 

 

 

 

 

 

20 r I I f I
— r fr:
m S
__ — r fr =02
15 _ m S A
..... r fr =04
m S
__ r /r =06
m S
10 e
+ ﬂ
«<1
5 ..
0L --------------------------------------------
_5 1 1 1 1 L
0 1 00 200 300 400 500 600

t.<s>

Figure 3.8 Optimization of sensor position (rm) and ﬁnal measurement time (if) for the

extended ﬂash method, with initial measurement time to =10 s . The heating time
(IP = 30 s) and heating area (rp = 2 mm) are ﬁxed.
In the work of Maillet et al. [3], two sensors were used to measure temperature,

one at the center and the other outside of the heating area. Figure 3.9 shows A12 obtained

by placing one sensor at the center and the second sensor at different positions on the rear
surface. If the second sensor is also at the center, it is identical to the case with one sensor

placed at the center. With the second sensor moved from the center to the edge, the
maximum A12 ﬁrst increases and then decreases. The maximum A12 is obtained with
the second sensor at rm, 2 /rS = 0.4 and is about 30 times greater than the maximum A12

with one sensor at the center.

24

 

 

 

 

 

 

 

 

 

1 1 l 1 1
0 1 00 200 300 400 500 500
t. (s)

Figure 3.9 Optimization of the second sensor position (rm,2) for the extended ﬂash

method, with the ﬁrst sensor at the center on the rear surface and initial measurement
time to = 10 s . The heating time (t p = 30 s) and heating area (rp = 2 mm) are ﬁxed.

Figure 3.10 shows the maximum A12 obtained with both sensors movable. If the

sensors are placed at the same place, it represents the case in which one sensor is used.

The A12 of each point is obtained with optimized measurement time interval. The results

demonstrate that it is better to measure the transient temperature at two points than at one.
The optimal choice is to place one sensor at the center and the second one outside of the

heating area.

25

rm, 2/rS

 

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
rum/rS
Cl 0.00E+00-1.50E—07 l.50E—07-3.00E-07

I 3.00E—07-4.50E—07 I 4.50E—07—6.00E-07

Figure 3.10 Optimization of positions of both sensors (rm,1, rm, 2) for the extended ﬂash

method. The A112 of each point is obtained with optimized measurement time interval.
The heating time (t p = 30 s) and heating area ( rp = 2 mm) are ﬁxed.

Figure 3.11 shows the maximum A12 obtained by measuring the transient
temperature at three points during optimized measurement time, with the same heating
time and heating radius in Figure 3.9. One sensor is placed at the center of rear surface

and the other two sensors are moved from the center to the edge. It can be observed that

the A12 reaches a maximum with both sensors placed at r/rs :0.4 and thus the
maximum A12 obtained with three sensors (Figure 3.11) is smaller than the maximum

A12 with two sensors (Figure 3.9).

26

 

0.1 0.3 0.5 0.7 0.9 '

rm, z/rs

E 0.00E+00—1.70E—07 I 1.7OE-07-3.40E-O7 I 3.40E—O7—5.10E—07

Figure 3.11 Optimization of positions of the second and third sensors (rm, 2 , rm, 3) for

the extended ﬂash method, with the ﬁrst sensor at the center on the rear surface. The A12
of each point is obtained with optimized measurement time interval. The heating time
(tp = 30 s) and heat area ( rp = 2 mm) are ﬁxed.

Figure 3.12 illustrates the relative utility of using more sensors. The temperature
sensors are evenly positioned between the center (r = 0 mm) and the edge (r = 9 mm).
For example, if three sensors are used, they are place radially at 0, 4.5 and 9 mm. It can
be observed that the maximum ratio of determinants A12 decreases if more sensors are

used.

27

 

 

 

 

 

 

 

 

 

0
50 100 150 200 250 300 350 400 450 500 550 500
tf<s>

Figure 3.12 Ratio of determinants with increasing numbers of sensors (M ) evenly
positioned between the center and the edge on the rear surface for the extended ﬂash
method. The measurements start at to = 10 s. The heating time (t p = 30 s) and heat area

(rp = 2 mm) are ﬁxed.

The analysis above shows that the optimal number of temperature sensors is two

for the case of radial symmetry, and the optimized sensor positions are at rm,1 = 0 mm

and rm“? =4mm when the heated region is rp =2mm with the sample size of

rs =10 mm and e=3 mm. This is based on the material with az = 1.5 x10—7 mzs’l

and a, = 1.0x10’7 "123—1. For materials with different values of thermal diffusivities,

Table 3.1 lists the optimal position for the second sensor corresponding to the change of

thermal diffusivities while the ﬁrst sensor is placed at the center. It can be observed that

28

the optimal point for the second sensor moves to the edge with the increasing of radial

thermal diffusivity and towards the heating radius, here rp /rS = 0.2 , with the increasing

of axial thermal diffusivity.

Table 3.1 Optimal position of the second sensor, rm,2 /rs , for changing thermal

diffusivity, with rm] n», = 0. (a0 =1.0x10‘7 mzs‘l)

 

 

 

 

 

 

 

a2 /a0
0.5 1.0 1.5 2.0 2.5
0.5 0.4 0.3 0.3 0.3 0.3
1.0 0.5 0.4 0.4 0.3 0.3
“r / a0 1.5 0.85 0.5 0.4 0.4 0.4
2.0 0.9 0.7 0.5 0.4 0.4
2.5 0.9 0.8 0.6 0.5 0.4

 

 

 

 

 

 

 

 

 

For materials with a priori information about the unknown parameters, it is also

possible to optimize other parameters, such as heating time, heating area and sample size.
In Figure 3.13, each curve traces A12 for different heating times while the other
parameters are given as the same as above. The temperature is recorded at rm] = 0 mm
and rm,2 =4mm , the optimal positions obtained above. The optimal heating time

corresponds to the curve with the highest point, which is obtained with a heating time of

200 s and a ﬁnal measurement time of approximately 300 s.

29

 

12

 

 

 

 

 

 

 

_2 J 1 1 1 1 1 1
0 50 100 150 200 250 300 350 400

tf<s>

 

Figure 3.13 Optimization of heating time (t p) and ﬁnal measurement time (t f) for the

extended ﬂash method, based on a priori information about unknown parameters. The
heat area (rp = 2 mm) is given.

The maximum A12 obtained with optimized heating time and measurement time

for different heating radii and sensor positions are compared in Figure 3.14. The optimal
heating radius is 7 mm with the ﬁrst sensor ﬁxed at the center and the second sensor

placed at r = 9 mm. The corresponding heating time for this case is 40 s and the ﬁnal

measurement time is 160 s.

30

0.9
0.8
0.7
0.6
0.5 I'm, 2/1'5
0.4
0.3
0.2
0.1

 

El 0.00E+00-2.50E—06 El 2.50E—06-5.00E—06
I 5.00E—06-7.50E-06 I 7.50E—06-1.00E-05
I 1.00E-05-1.25E—05

Figure 3.14 Optimization of heating area (rp) and second sensor position (rm, 2) for the
extended ﬂash method, based on a priori information about unknown parameters, with

one sensor at the center on the rear surface. The A12 of each point is obtained with
optimized heating time (t p) and measurement interval (to , t f ).

Finally, the maximum A12 obtained with samples of different sizes are shown in
Table 3.2. For samples with thickness 3 mm, 4 mm and 5 mm, the Optimal specimen
radius is 5 mm, 10 mm and 10 mm, respectively. The maximum ratio of determinants
13sz for the extended ﬂash method is about 10'5 if the axial thermal diffusivity is 1.5 x

10'7 mzs'land the radial thermal diffusivity is 1.0 x 10'7 mzs'l.

31

Table 3.2 Maximum ratio of determinants A12 , from samples of different sizes with the

ﬂash method, for materials with az ——-1.5x10"7 mzs_l and a, = 1.0x10—7 mzs—l.

 

 

 

 

 

 

 

 

 

A12 rS=5mm rS=IOmm rs=ISmm rs=20mm
e = 3 mm 2.61110" 1.2x10‘5 7.85110“ 4.551106
e = 4 mm 3.051106 9.75110r 1.15110r 7.6x10'6
e=5mm 1.2x10'6 7.11110" 1.15110" 9.9511045

 

 

3.2 Thermal Wave and Flash-ﬁeld Methods

3.2.1 Temperature Simulation

Instead of measuring the transient temperature at two points in the ﬂash method,
both the thermal wave and ﬂash-ﬁeld methods record the temperature of entire surface at
several discrete times (Figure 3.15). The temperature of the specimen was calculated
using the same equation as for the ﬂash method, namely Equation 4. Again, the heat ﬂux

absorbed per unit area q" from the extended heating has been adjusted to make the
maximum temperature rise T max the same in each sample. The error in the temperature

simulation is assumed to be random, additive, uncorrelated and normally distributed with
a zero mean and a known constant standard deviation 0. Figure 3.16 presents the
simulated temperature rise on the front surface for the thermal wave method (Panel a) and
on the rear surface for the ﬂash-ﬁeld method (Panel b). Random noise with a standard

deviation 0' = 0.001Tmax is added to the equation results. The experimental parameters,

such as sample size, heating area and heating time, are the same as those in Panel b of

Figure 3.2.

32

 

 

 

 

 

ﬂ

 

 

 

 

 

 

 

 

 

 

IR Camera \
(:21 2:)
Mirror ho he
a U 1..
1r
8 1
rs
,, (b)
q
(t5 tp) . r,,
‘1 2
_+1 ............. _ ............ D ...,
—>
IRCamera
(,2: :1)
ho he

 

 

 

‘u-m-m-w-

|"

Figure 3.15 Schematic representations of the thermal wave method (a), with temperature
measurements on the front surface, and of the ﬂash-ﬁeld method (b), with temperature

measurements on the rear surface. Heating occurs within the radius rp on the front face.

33

3.2.2 Sensitivity Analysis

Figure 3.17 presents the sensitivity coefﬁcients J; , J; , J Z, and J 2;" over the
" r

L

front and rear surfaces at 45 s, which is 15 s after heating has stopped. As may be seen,

the sensitivity curves J; , J; , J2, and Jilin have different shapes and are thus
Z r

uncorrelated, meaning it is possible to estimate the thermal diffusivities with one

measurement of the entire surface temperature [15].

3.2.3 Optimization

For the thermal wave and ﬂash-ﬁeld methods, the parameters that must be set
prior to the test are sample size, heating area and heating time. Those that can be adjusted

after the experiment are number of measurements and measurement times. Figure 3.18

shows A12 obtained by measuring temperature at two times. The ratio of determinants

A12 with two measurements being at the same time, i.e., rm] = tm, 2 , is equal to the A12
obtained with a single measurement. For both the thermal wave method (Panel a) and the

ﬂash-ﬁeld method (Panel b), the A12 based on two measurements are greater than the

A12 with one measurement. For the thermal wave method, the region of optimized

measurement suggests that one measurement time is between 15 and 25 s with a
corresponding second measurement between 30 and 40 s. For the ﬂash-ﬁeld method, the

optimized region suggests that the ﬁrst measurement time should be between 25 and 35 s

with the second between 40 and 65 s for the maximum A12. The maximum A12 for each

34

method is smaller than 108, which is smaller than the maximum A12 for the extended

ﬂash method (Figure 3.9).

Figure 3.19 shows variation in A12 obtained by adding a third measurement at

various times. For the thermal wave method (Panel a), the ﬁrst two measurements are

ﬁxed at 20 and 35 s, which are the optimal measurement times in Panel a of Figure 3.18.

If the third measurement is added after 150 s, the ratio of determinants A12 increases to
be greater than 107. If the three measurement times are 20, 35 and 270 5, A12 is about

2x10'7. In Panel b, with the ﬁrst two measurements at 30 and 40 s, the A12 increases

from 10‘8 (Panel b of Figure 3.18) to about 1.8x10'7 with the third measurement at 150 s.

Figure 3.20 shows the change of A12 after adding a fourth measurement. The ﬁrst three

measurements are ﬁxed at 20, 35 and 270 s in Panel a and at 30, 40 and 150 s in Panel b.

The maximum A12 for the thermal wave and ﬂash-ﬁeld methods are about 5x10'7 and

3x107, respectively. Continuing this analysis, Figure 3.21 then shows the change of A12

by adding a ﬁfth measurement. It can be observed that adding a ﬁfth measurement does

not increase A12 as dramatically as does the third measurement in Figure 3.19 or the

fourth measurement in Figure 3.20.

35

 

 

 

 

 

12 I I j I I I ﬁ I
1 1 =15 \ _
m
t =30 s
m
>< 0.8 - —
E
[:4 (a)
A
*1 0.6 — ~
"'1 =45 s
8
m
4 1 =15 ..
m
0.2 ~ \\ .
~42
RE;—
0 1 1 1 —— —_ ._.
0 01 02 03 0.4 0.5 0.6 07 08 09
r/r
S
0.3 I I r I I j I I
1 =30 \
m
0.25 - _
. t =45 s
02- in 60 4
= s
,4 m (b)
E
a 0.15 ~ \ tm=>"5 S -
3 0.1 - «
1—1 /
/ \
0.06 - t =90 s \- -
m \
U 1 1 I L 1 1 -—-—2~

 

 

 

Figure 3.16 Simulation of temperature rise for the thermal wave method (a) and the ﬂash-
ﬁeld method (b).

 

03 I I I I I If I I

 

 

 

 

 

 

 

 

 

 

 

\
0.2 ~ . \ ‘ (a) d
\
\
0.1 — x ‘ q
\ §
0" ---/f~:‘—Ha_thrz;——z=_
"-3 1-- ..................... ./
143‘ /
01 - / J+ d
/ ...... 0L
/ Z
'02 :_’_ / — _- 1+ ‘
0L
J+r
-U.3 ~ 1:! d
— - — II
.014 l l l 1 1 1 1 1 q
0 0.1 0.2 0.3 0.4 I 0.5 0.5 0.7 0.8 [19
171'
S
025 a - 1 I I ' r r r I
\
0.2 - ‘ \ \ ...... I; q
\ z
0.15 - ‘ \ _ _ J; g

I
0.1” \\ 1+

 

 

 

 

 

\ II
005- \, q .
£1 """""""""""" \.
Lg ............ W'd—*:t:—:h_~
0_ 1%""----------.2- '*-—5":_;‘:
4" /
0051 -

\
E",

 

 

 

-0.1
-0.15 - / -
U2 /1 1 l L 1 I I i
I 01 02 03 04 0.5 05 07 08 09
11/18

Figure 3.17 Normalized sensitivity coefﬁcients for the thermal wave method (a) and the
ﬂash-ﬁeld method (b).

37

 

 

 

 

tm. 2 (S)

E] 0.00E+OO—2.50E-09 El 2.50E—09-5.00E—O9
I 5.00E-09-7.50E-09 I 7.50E—09-1.00E-08

 

 

‘ .1112; £13“: t 1&1 Lei ..ﬁ 1 :.
35 40 45 50 55 60 65

tm, 2 (5)

D 0.00E+00-2.00E—09 2.00E—09-4.00E—09
I 4.00E—09-6.00E—09 I 6.00E—09-8.00E-09

Figure 3.18 Ratio of determinants with two measurements for the thermal wave method
(a) and the ﬂash-ﬁeld method (b). The heating time (t p =30s) and heating area

(rp 2 2mm) are ﬁxed.

38

 

2.5- m1 ‘ m2 '

(a)

0.5 - ..

 

 

41_

0 50 100 150 200 250 300 350 400 450 500
t (S)
m, 3

 

 

A‘AA“A‘A‘A“
V'vvv7"""'

1.6-

1.4-

0.8 ~

0.6 ~

0.4 -

0.2b ‘

        

  

 

 

   

0 ‘ 1 1
0 20 40 60 80 100

tm. 3 (s)

 

120 140 150 180 200

Figure 3.19 Ratio of determinants with a third measurement at various times for the
thermal wave method (a) and the ﬂash-ﬁeld method (b). The heating time (t p = 30 s)

and heating area (rp = 2 mm) are ﬁxed.

39

 

 

 

 

 

1 1 l l l l

0 2O 40 60 80 100 120 140 150 180
t (s)
m 4

 

 

35 I I I I I I I I

2.5 *-

 

 

 

 

Figure 3.20 Ratio of determinants with a fourth measurement at various times for the
thermal wave method (a) and the ﬂash-ﬁeld method (b). The heating time (tp = 30 s)

and heating area (rp = 2 mm) are ﬁxed.

40

x10-

 

5.5 ~ _

3.5 ~

 

 

 

 

0 20 40 60 80 100 120 140 '

 

38 I I I I I I

3.6 ~

3.4 ~

3.2 ~

2.8 ~

2.6

I

2.4 ~

 

 

 

 

0 2o 40 60 I 80 100 120 140
tm 5 (b)

Figure 3.21 Ratio of determinants with a ﬁfth measurment at various times for the termal
wave method (a) and the flash-ﬁeld method (b). The heating time (t p = 30 s) and heating

area (rp = 2 mm) are ﬁxed.

41

 

Figure 3.22 illustrates the maximum ATZ with four measurements for different

heating area rp and heating time tp , based on a priori information about unknown

parameters. For both the thermal wave method (Panel a) and the ﬂash—ﬁeld method

(Panel b), the optimal heating radius rp / rs is between 0.5 and 0.8. The corresponding
heating time for the thermal wave method lies between 60 and 150 s and the maximum

AD is of the same order of the maximum Ah with the extended ﬂash method (Figure

3.14). For the ﬂash-ﬁeld method, the corresponding heating time is between 30 and 60 s

and the maximum ratio of determinants ATZ is about half of that of the thermal wave

method.

Finally, Tables 3.3 and 3.4 list Ab obtained with various sample sizes by

measuring the surface temperature ﬁeld at four times. The maximum AB is obtained by

using the thermal wave method (Table 3.3), which is about one and a half times of the

maximum ATZ of the extended ﬂash method (Table 3.2) and about twice of the
maximum Ah of the ﬂash-ﬁeld method (Table 3.4). It is still possible, however, to

increase the maximum ratio of determinants ATZ of the thermal wave and ﬂash-ﬁeld

methods by adding more measurements.

42

 

 

 

 

 

 

30 60 90 120 150 180'
t13(5)

El 0.00E+00-5.00E-O6 5.00E-06-l.00E—05
I l.00E—05-1.50E—05 I 1.50E—05-2.00E—05

$3“: ' (”1717); ‘KXA; g-Kfﬁ: 3“"

> R“ '1
7' .

 

 

 

 

 

 

1, (s)

El 0.00E+00-2.00E—06 2.00E—06-4.00E—06
I 4.00E—06-6.00E—06 I 6.00E-06-8.00E—06

Figure 3.22 Optimization of heating time (tp) and heating area (rp) for the thermal wave

method (a) and the ﬂash-ﬁeld method (b), based on a priori information about unknown

parameters, with four optimized measurement times.

43

Table 3.3 Maximum ratio of determinants Ab , from samples of different sizes with the

method based

on

four

measurements,

for materials with

 

 

 

 

thermal wave
a: =l.5x10’7 mzs-l and a, =l.Ox10“7 mzs’].

Ab rs=5mm rS=10mm rs=ISmm
e=3nm1 lelUS 1.6x10'S 13x105
e=4nmi 8.2x10'6 1.6x10‘5 15x105
e=5nmi 4.3x10'6 1.2x10‘S 12x105

 

 

 

 

 

 

Table 3.4 Maximum ratio of determinants A72 , from samples of different sizes with the

 

 

 

 

flash—ﬁeld method based on four measurements, for materials with
a: =l.5x10’7 mzs—l and a, ---1.Ox10’7 mzs’l.
A113 rs=5mm rs=10mm rs=15mm
e=3nmi 6Jx106 7.3x10'6 50x106
e=4nmi 2.6x10'6 74x106 7.1x10'6
e=5nm1 86x107 5.811106 72xlo°

 

 

 

 

 

 

44

 

CHAPTER 4 COMPARISON OF METHODS

Consider a sample with dimensions e=3mm and rs = 10 mm and axial and

radial thermal diffusivities of 1.5 x 10'7 mzs'l and 1.0 x 10‘7 mzs'l. The heat transfer

coefﬁcients on the front (ho ), rear (he) and side (hr) surfaces are assumed to be the

same (H = H0 = He = Hrare/az /r5 = 0.05 ). The heat ﬂux (1,,“ is set to produce the

same maximum temperature rise, for each measurement method.
Case 1 includes the optimal parameters for the extended flash method from Figure
3.14. The heating radius is 7 mm with the heating time of 60 5; one temperature sensor is

at the center and the other is at r = 9 mm on the rear surface. Each sensor is sampled for

80 measurements that are evenly spaced from 10 s to 168 s. In Case 2, only the
temperature at the center on the rear surface is recorded, but for 160 measurements, and
other parameters are the same as those in Case 1. Case 3 represents the flash method with
non-optimized parameters. The heating radius is 2 mm and the heating time is 20 3. One
sensor is positioned at the center and the second sensor is at r = 4 mm. Temperature is
recorded 80 times with each sensor for 5 s St S 84 5. Case 4 simulates the thermal wave
method with optimized parameters from Panel a of Figure 3.22. The heating time is 60 s
and the heating radius is 7 mm. The four measurements are at 30, 90, 120 and 270 s. In
Case 5, only the temperatures at 30 and 905 are measured, instead of the four
measurements in Case 4. Case 6 models the ﬂash-ﬁeld method with optimized

parameters from Panel b of Figure 3.22, with heating time t = 30 s and heating area

p

rp = 7 mm. The four measurement times are 25, 30, 60 and 180 s.

45

As before, the error in temperature simulation is assumed to be random, additive,
uncorrelated and normally distributed with a zero mean and a known constant standard
deviation 0. Tables 4.2 and 4.3 summarize the estimated parameters for three different
levels of measurement errors: a = 0.001Tmax , a = 0.01Tmﬂx and 0' = 002me , where

Tmx is the maximum temperature rise in the sample. In each case, a total of 160

measurements were considered. The parameters were accurately estimated if no errors

were considered and when the standard deviation of errors is as small as 0.001Tmax. For

Cases 1 and 4, with the maximum ATZ , all of the parameters were identiﬁed with high
accuracy even with measurement error as large as 0' = 0.02Tmax. In Cases 2, 3 and 6, the

identiﬁcation errors for thermal diffusivities were greater than those in Cases 1 and 4, but
were still smaller than 1%. In Case 5, the radial thermal diffusivity could be estimated

with an error less than 2% when the standard deviation of error is 0.02Tmax. The

identiﬁcation error of the axial thermal diffusivity, however, was as great as 20%.

Obviously, the thermal wave method by using two measurements, with AIZ two

orders smaller than when four measurements are used, is sensitive to measurement errors.
For the thermal wave and ﬂash-ﬁeld methods, it is necessary to measure the temperature
of entire surface more than twice for good accuracy. For the ﬂash method, it is better to

have two temperature sensors than one, if other parameters are the same. Cases with

greater AI), , especially those that are two orders greater, have better accuracy in the

estimation. In general, the optimized thermal wave and ﬂash-ﬁeld methods have the same

accuracy as the ﬂash method with optimized parameters.

46

 

 

 

 

 

 

 

 

Table 4.1 Estimation results with measurement errors of standard deviation
0' = 0001me .
az(mzs") a,(mzs") H (I'anmax
Case Aiz
1.500 x 10'7 1.000 x 10'7 5.000 x 10'2 1.000
1 1.2 x10'5 1.500 x 10'7 1.000 x 10‘7 5.008 x 10'2 1.000
2 1.0 x1045 1.499 x 10'7 1.001 x 10'7 4.991 x 10'2 1.000
3 3.2 x10'7 1.501 x 10'7 0.999 x 10'7 4.987 x 10'2 0.999
4 1.6 x10'5 1.500 x 10'7 1.000 x 10'7 4.999 x 10'2 1.000
5 7.8 1110'8 1.507 x 10'7 1.001 x 10'7 5.048 x 10'2 0.991
6 7.3 x10‘6 1.500 x 10'7 1.000 x 10'7 4.998 x 10‘2 1.000

 

 

 

 

 

 

Table 4.2 Estimation results with measurement errors of standard deviation

 

 

 

 

 

 

 

 

a = 0.01Tmax.
a: (mzs'l) a, (mzs'l) H (In/(Inmax
Case Ah 7

1.500 x 10‘7 1.000 x 10'7 5.000 x 10'“ 1.000
1 1.2 x10'5 1.501 x 10'7 1.000 x 10'7 5.028 x 10‘2 1.000
2 1.0 x106 1.494 x 10'7 1.003 x 10'7 4.987 x 10‘2 1.001
3 3.2 x107 1.510 x 10'7 1.003 x 10‘7 5.166x10'2 1.002
4 1.6 x105 1.501 x 10'7 1.000 x 10'7 5.001 x 10'2 1.000
5 7.8 x10'8 1.590 x 10'7 1.006 x 10'7 5.551 x 10'2 1.029
6 7.3 x10'6 1.503 x 10‘7 0.999 x 10'7 4.985 x 10'2 1.000

 

 

 

 

 

 

47

 

 

Table 4.3 Estimation

results with measurement

errors of standard deviation

 

 

 

 

 

 

 

 

o- = 0.027max.
a2 (mZS-l) (1,. (“1254) H qH/qnmax
Case Ab
1.500 x 10'7 1.000 x 10‘7 5.000 x 10'2 1.000
1 1.2 x10'5 1.504 x 10'7 1.002 x 10'7 5.057 x 10'2 1.003
2 1.0 x10'6 1.512 x 10'7 0.996 x 10'7 5.055 x 10'2 1.001
3 3.2 x107 1.513 x 10'7 0.993 x 10'7 4.986x 10'2 0.991
4 1.6 x105 1.502 x 10'7 1.002 x 10‘7 5.004 x 10‘2 1.002
5 7.8 x108 1.803 x 10'7 1.012 x 10'7 6.438 x 10'2 1.083
6 7.3 x106 1.498 x 10’7 1.005 x 10'7 4.996 x 10'2 1.003

 

 

 

 

 

 

48

 

 

CHAPTER 5 CONCLUSION

Three methods to measure simultaneously the axial and radial thermal
diffusivities of anisotropic materials have been optimized and compared based on the D-
optimal criteria: the extended ﬂash method, the thermal wave method and the ﬂash-ﬁeld
method. For the extended ﬂash method, the number of sensors and sensor positions were
optimized using the D-optimization method. The parameters that must be ﬁxed in
experiments, such as heating time, heating area and sample size, were then adjusted using
the information obtained about the unknown parameters. Although one temperature
sensor contains enough information to determine simultaneously the axial and radial
thermal diffusivities, a second temperature sensor improves the measurement accuracy.
For the thermal wave and ﬂash-ﬁeld methods, the adjustable parameters are number of
measurements, measurement times, heating time, heating area and sample size. It has
been demonstrated that using more than two measurements improves the accuracy of
estimation. Based on the D-optimal criteria, the thermal wave and ﬂash-ﬁeld methods

with optimal parameters are as accurate as the ﬂash method for materials with
0:2 =1.5x10—7 mzs‘1 and ar =1.Ox10'7 mzs—l.

This work has addressed estimating the axial and radial thermal diffusivities, heat
transfer coefﬁcients and heat ﬂux. The thermal diffusivities are the parameters of primary
interest. The heating transfer coefﬁcients and heat ﬂux are taken as additional unknown,
or nuisance, parameters. It is assumed that the inaccuracy of the sensor positions in the
ﬂash method and the inaccuracy of measurement times in the thermal wave and ﬂash-
ﬁeld methods can be neglected. To consider the effect of inaccuracy of sensor positions

and measurement times, the sensor positions in the ﬂash method and the measurement

49

times in the thermal wave and ﬂash-ﬁeld methods can be added into the additional

unknown parameters in future work.

50

[1]

[2]

[3]

[9]

[10]

[11]

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52