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International Journal of Heat and Mass Transfer 108 (2017) 2514–2522
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International Journal of Heat and Mass Transfer
journal homepage: www.elsevier.com/locate/ijhmt
Determination of thermal diffusivity of fibrous insulating materials
at high temperatures by thermal wave analysis
H. Brendel ⇑, G. Seifert, F. Raether
Fraunhofer Institute for Silicate Research, Center for High Temperature Materials and Design, Bayreuth 95448, Germany
a r t i c l e
i n f o
Article history:
Received 15 September 2016
Received in revised form 4 January 2017
Accepted 6 January 2017
Keywords:
Thermal diffusivity
Fibrous insulation
Thermal wave analysis
High temperature
a b s t r a c t
This work investigates the potential of measuring thermal diffusivity of materials in the high temperature
range up to 1580 °C by thermal wave analysis of cylindrical samples. In particular, aiming at the typical
situation of fiber insulation materials, the method is evaluated for investigations on heterogeneous media
with low thermal conductivity, low density and heat capacity, and significant internal radiative heat
transport (participating media). Furthermore, a model for combined conductive and radiative heat transfer is introduced. The model is adapted to a commercial insulation material using experimental scattering
and absorption spectra evaluated at 1000 °C. The predictions of this numerical model are in accordance
with the thermal diffusivity data obtained from the thermal wave analyses.
Ó 2017 Elsevier Ltd. All rights reserved.
1. Introduction
THERMAL conductivity and thermal diffusivity are the most
important parameters for describing the heat transport properties
of a material. Fibrous insulations are a common solution for high
temperature processes. They are applied in all kind of furnaces
due to their favorable properties such as low thermal conductivity,
low heat capacity and high thermal shock resistance. Conductive
and radiative heat transfer are the predominant heat transport
mechanisms at high temperatures. The established methods for
measuring thermal conductivity and thermal diffusivity differ from
one another primarily by recommended temperature range,
required sample volume and achievable range of thermal conductivity and diffusivity values. The methods can be distinguished by
their operation mode: Heat-Flow meters or Guarded-Hot-Plates
are measuring in steadystate, whereas the Hot-Wire technique is
a transient method [1]. Due to their capability for investigating
large samples, Guarded-Hot-Plate [2,3] and Hot-Wire technique
[4,5] are the methods primarily used for high temperature
measurements on heterogeneous sample materials, but only the
Hot-Wire technique is applicable for measuring above 1000 °C.
One of the main drawbacks of the Hot-Wire method is that sample
preparation requires a great deal of effort to provide good thermal
contact between the sample and the heater wire [5,6]. Furthermore, one has to ensure that test material and heater wire show
⇑ Corresponding author.
E-mail addresses: [email protected] (H. Brendel), gerhard.seifert@
isc.fraunhofer.de (G. Seifert), [email protected] (F. Raether).
http://dx.doi.org/10.1016/j.ijheatmasstransfer.2017.01.063
0017-9310/Ó 2017 Elsevier Ltd. All rights reserved.
no chemical reaction [4,6]. To overcome the disadvantage of thermal contact uncertainty we investigated the less widespread
Ångstrom’s method [7], nowadays mostly called thermal wave
analysis (TWA) to determine thermal diffusivity of high temperature insulation materials. For TWA, an oscillating temperature
boundary condition is applied to the sample. The thermal
diffusivity is then obtained from an analysis of phase and amplitude of the temperature oscillation measured inside the sample
specimen [8–11].
In previous publications, TWA was realized at temperatures up
to 1000 °C by applying the oscillating temperatures to the faces of
the sample. This paper presents a high temperature application
(T >¼ 1000 C) of TWA, where the temperature oscillation is
applied uniformly at the lateral area of a long cylindrical sample
of ceramic-fiber insulation material. Furthermore a theoretical
model for combined conductive and radiative heat transfer is
developed to predict the thermal diffusivity of the material. The
model employs radiative transport parameters experimentally
derived at 1000 °C.
2. Analytical models
This section describes the analytical model for an infinitely long,
periodically heated cylinder. The problem has an analytical solution, which is first used to calculate the general correlation
between the measuring quantities phase and amplitude (of temperature oscillation inside the sample) and the transfer properties
of the material. Thereafter the theory for radiative transfer in a
radially symmetric participating medium is presented. Next, a
H. Brendel et al. / International Journal of Heat and Mass Transfer 108 (2017) 2514–2522
2515
Nomenclature
velocity of light
specific heat
1
2
Rð
pffiffiffix=jÞ , dimensionless parameter
2Rarel , dimensionless parameter
connection parameter for fiber-fiber contact
ratio between the thermal conductivity of gas and the
solid material
rext =q, specific extinction
oscillation frequency
anisotropy parameter
Planck
p
ffiffiffiffiffiffiffi constant
1
Planck-Intensity distribution
Bessel-function (order n) of the first kind
Boltzmann constant
length of fibrous probe
index of refraction
pressure
radiative heat flux
outer radius of sample
radial coordinate
time
radial temperature
external temperature amplitude
mean temperature of oscillation
wall temperature
thermal conductivity
ratio of the volumes of fiber and gas
Young’s modulus
fiber orientation parameter
relative heat transfer coefficient
height of a fiber-gas cell in units of the contact area
emissivity
amplitude loss factor
tc=ðcp qÞ, thermal diffusivity
wavelength
phase shift
cosð#0 Þ, cosine of mean fiber orientation angle
cl
cp
c1
c2
C sb
Cr
e
f
g
h
i
ipk
Jn
kB
l
n
p0
qrad
R
r
t
TðrÞ
T ex
Tm
Tw
tc
Vr
Y
Z
arel
d
g
j
k
m
l
oþ1
#0
q
r
rext
rsca
rabs
1
s
sR
x0
x
area of a fiber-gas cell in units of the contact area
mean fiber orientation angle
density
Stefan-Boltzmann constant
extinction coefficient
scattering coefficient
absorption coefficient
Poisson’s number
r rext , optical depth
Rrext , total optical depth
albedo
2pf , angular frequency
Superscripts
low,high low, high quantity
lit
literature value
exp
experimentally derived quantity
num
numerically derived quantity
Subscripts
k
indicates a spectral quantity
m
index of spectral-band
bulk
attribute of the solid bulk-material
fs
attribute of the sample
g
attribute of the gas
b
refers to fiber-fiber contact
ext,sca,abs extinction, scattering and absorbtion
eff
refers to effective quantities
w
refers to quantities of the wall boundary
^
refers to scaled quantities
Abbreviations
RT
ambient temperature
Im
imaginary part
Re
real part
TWA
thermal-wave analysis
description of heat conduction by solid components and the
enclosed gas in the material is presented; this combined model
also accounts for the heat transfer due to fiber contacts. Finally,
the inhomogeneous diffusion equation for combined gaseous, solid
and radiative heat transfer is introduced. A brief summary of the
numerical implementation of the models concludes this section.
2.1. The periodically heated cylinder
An infinite cylinder heate periodically by a harmonic cosinetype temperature profile is considered (Fig. 1). Heat transfer is
assumed to be radiative between cylinder lateral surface and its
surroundings. In the case of small temperature differences the
radiative heat transfer coefficient can be approximated by a convection term; the corresponding relative heat transfer coefficient
arel reads:
arel ¼ 4
Here
eff rT 3
the
ð1Þ
tc
effective
emissivity is
defined as
eff
Fig. 1. Schematic diagram of the periodically heated cylinder.
¼ ð1=fs þ
ð1=w 1Þ R=Rw Þ1 where fs and w are the spectrally integrated
emissivities of sample and the surrounding wall, respectively. R
and Rw are the radii of the inner medium (sample) and the surrounding wall. tc is the thermal conductivity and r the Stefan-
Boltzmann constant. In [12] an analytical solution of the problem
is given for a cylinder at zero temperature with sinusoidal periodic
radiation at its surface. For experimental realization a cosine-type
modulation is more beneficial because it facilitates a slope zero
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H. Brendel et al. / International Journal of Heat and Mass Transfer 108 (2017) 2514–2522
for the temperature variation at starting time instead of a technically unrealizable kink. The equivalent solution for a cylindrical
medium of radius R heated by a cosine-type temperature profile
is given by the equation
T ðr; tÞ ¼ Re
h
c2 T
ffiffiffi
p=2ÞÞ
i
2ic1 J 1 ðc1 wÞþic2 J 0 ðc1 wÞ
1
pffiffiffi
P
2 c2 T ex
þ
J ðc wr=RÞ exp ðiðxtþ
ex
p0 1
n¼1
J 0 ðbn r=RÞ exp ðb 2n xt=c12 Þ
ð2Þ
ðb4n =c 41 þ1Þðb2n þc22 =2ÞJ0 ðbn Þ
the dimensionless parameters c1 ¼ Rðx=jÞ 1 = 2 and
pffiffiffiffi
2 R arel . Further parameters are r: radial coordinate; j: therc2 ¼
mal diffusivity; x: angular velocity of the harmonic oscillation; T ex :
external amplitude of excitation; Jn: Bessel-function of first kind of
order n; w ¼ expð3i=4pÞ. Finally, bn are the roots of the following
equation [12]:
with
pffiffiffi
2bJ 1 ðbÞ ¼ c2 J 0 ðbÞ
ð3Þ
The first part of Eq. (2) describes the steady-state periodic oscillations, while the second part of Eq. (2) accounts for the transient
oscillation whose contribution converges to zero with progressing
time. A proof for the validity of Eq. (2) is given in the Appendix. If
one is primarily interested in the stationary temperature profile at
radial coordinate r, the first part of Eq. (2) can be converted to the
following form:
Tðr; tÞ ¼ T ex gðrÞ cosðxt mðrÞÞ
ð4Þ
In Eq. (4), gðrÞ is the amplitude loss factor and mðrÞ refers to the
phase-shift at radial coordinate r. By using the abbreviations
f1 ðc1 ; c2 Þ ¼
c1
c2
½ReðJ 1 ðc1 wÞÞ Im ðJ 1 ðc1 wÞÞ ImðJ 0 ðc1 wÞÞ
ð5Þ
f2 ðc1 ; c2 Þ ¼
c1
c2
½Re ðJ 1 ðc1 wÞÞ þ Im ðJ 1 ðc1 w ÞÞ þ Re ðJ 0 ðc1 w ÞÞ
ð6Þ
was chosen as control parameter, because it is proportional to the
thermal diffusivity j. Experimentally, phase shift and amplitude
decay are the measured quantities from which thermal diffusivity
is deduced by the help of Eqs. (5)–(14). Insofar Fig. 2 may be used
as a guide to design a TWA experiment, e.g., for large g and m (to
increase signal-to-noise ratio). However, besides practical limits
of an existing setup (maximum sample size, heating rate and precision, etc.) the inverse relation of amplitude and phase effects (as
clearly seen in Fig. 2) requires always a compromise. The same
statement holds for the parameter c2 . If it is increased (meaning
a higher heat transfer coefficient and/or a higher cylinder radius
R), the observed phase shift decreases while the observed amplitude decay increases.
2.2. Scaled diffusion approximation
For optically dense media the diffusion equation for radiative
transfer is an appropriate approximation for calculating the radiative flux qrad [14,15]. A medium can be considered as optically
dense when the optical depth s has at least a value of 1. The physical meaning of this value is that the mean free path of the photons
is much lower than the size of the medium. Using the scaling rules
given in [16] and the definition of the diffusion equation for an
absorbing and anisotropically scattering cylindrical enclosure
given in [17], the scaled diffusion equation transforms to
d2
db
s
2
k
qrad b
sk þ
1
d
bs k dbs k
b 0k þ
qrad b
sk 3 1 x
b 0k
¼ 4p 1 x
d
db
sk
p
ik
T b
sk
1
bs
2
k
qrad b
sk
ð15Þ
b 0k are defined as [16]:
bk ¼ r r
b extk and x
The scaled quantities s
b 0k ¼ x0k
b
s k ¼ sk ð1 x0k g k =3Þ; x
1 g k =3
1 x0k g k =3
ð16Þ
f3 ðr; c1 Þ ¼ ReðJ 0 ðc1 r=RwÞÞ
ð7Þ
f4 ðr; c1 Þ ¼ ImðJ 0 ðc1 r=RwÞÞ
ð8Þ
In Eq. (16), g k is the anisotropy parameter; g k =3 can be interp
preted as the factor of anisotropy [16]. Furthermore ik represents
the Planck-function
ð9Þ
ik ðT Þ ¼
and
CA ¼
CB ¼
f3 f1 þ f4 f2
f21
þ
f22
f1 f4 f3 f2
f21 þ f22
2
ð10Þ
the phase shift at r reads
tan mðrÞ ¼
CA
CB
p
n2eff;k 2hcl
k
5
exp
1
hcl
kkB T
1
ð17Þ
According to [18] it is necessary to consider the refractive index
n of the medium for calculating the radiative flux. For the case of
fibrous media an effective index of refraction neff has to be defined,
ð11Þ
and the expression for the amplitude decay g yields
gðrÞ2 ¼ C2A þ C2B
ð12Þ
For r ¼ 0, Eqs. (11) and (12) reduce to
tan m ¼ f1
; r¼0
f2
1=g2 ¼ f21 þ f22 ; r ¼ 0:
ð13Þ
ð14Þ
This means that the harmonic oscillation from the surroundings
remains harmonic inside the cylinder with a reduction in amplitude given by g and a lag in time corresponding to m. Eqs. (5)–
(14) in connection with Eq. (4) were derived already by [13], but
with a different definition of the parameters c1 and c2 . The definition in the current work was made such that, for the case of r ¼ 0,
both g and m can be described solely by the two parameters c1 and
c2 . Thus they can be displayed in one line chart as done in Fig. 2. c2
1
Fig. 2. Amplitude decay g and phase shift
parameter c2 at r ¼ 0.
m as a function of c2
for different
1
H. Brendel et al. / International Journal of Heat and Mass Transfer 108 (2017) 2514–2522
which typically shows a strong dependence on the porosity or the
fiber volume fraction, respectively [19,20]. For our numerical
model, neff of fibrous media is obtained from the index of refraction
of bulk-alumina according to the model introduced in [19]. The
temperature dependence of the bulk materials index of refraction
is implemented by usage of the data introduced in [21]. The
boundary condition at r ¼ R is the one given in [17] and is transformed by application of Eq. (16) to:
1
x 0k
1 b
1
d
bs k dbs k
b
s k qrad
^
R;k
þ2
2w;k
w;k
s
b R;k
qrad s
ð18Þ
Here, b
s R;k is the scaled total optical depth. For symmetry reasons
qrad ¼ 0 at r ¼ 0. A proof for the validity of Eqs. (15) and (18) is given
in the Appendix. The temperature dependent spectral emissivity
w;k of the wall is taken from [22] for Al2 O3 .
2.3. Solid and gas conductivity
For a dense fibrous insulation the thermal conductivity tcsgb of
solid fibers, gas and their coupling can be approximated according
to [23]:
1
d
oþ1
1
tcsgb ¼ ðd þ 1Þ
þ
tcsg
o tcg þ tcbulk C sb
ð19Þ
C sb is a dimensionless connection parameter representing fiber to
fiber contacts. It has to be derived experimentally. The parameters
d and o are calculated using the relations given in [23] (see Appendix). The thermal conductivity of the gas is denoted as tcg and the
thermal conductivity of the fiber bulk material as tcbulk . The expression tcsg is defined as
Cr 1
1 þ V r ð1 þ ZðC r 1Þ=ðC r þ 1ÞÞ
ð20Þ
where C r ¼ tcg =tcbulk is the ratio between the thermal conductivities
of the gas and the solid material. V r qfs =ðqbulk qfs Þ represents
the volume ratio of fiber material and gas. Z accounts for the fiber
fraction oriented perpendicular to the macroscopic heat flow. For
randomly oriented fibers, Z has a value of 0.66; this value is used
throughout the paper.
2.4. Coupling conduction and radiation
A comprehensive description of heat transfer in the considered
case requires including both radiative and conductive heat transfer. Therefore the appropriate description of the transient temperature evolution in the cylindrical sample is given by the following
inhomogeneous diffusion equation
qfs cp @t@ T ðr; tÞ tcsgb
where
the
1 d
ðrqrad ðr; tÞÞ
r dr
h
@2
@r 2
þ 1r
@
@r
i
T ðr; t Þ ¼ div qrad ðr; t Þ
divergence of the radiative
appears as source term.
flux
ð21Þ
divqrad ðr; tÞ ¼
2.5. Numerical implementation
Eqs. (21) and (15) are solved in a finite difference scheme. The
boundary condition at r ¼ R is updated after every time step by
the cosine-type wall intensity.
2jMt
ðMrÞ2
<1
The temperature profile is considered as converged if the maximal
difference in temperature between the preceding and the present
iteration is less than 106 K. The nonlinear equations describing
the temperature distribution in the periodically heated cylinder
are solved by the Scilab built-in nonlinear-solver which is based
on the Powell hybrid method [25].
3. Sample preparation and characterization
h i
p
s R;k ipw;k
¼ 4p ik b
tcsg ¼ tcbulk 1 þ
2517
ð22Þ
Equidistant time steps are chosen in accordance with the Neumann stability condition Eq. (22). The nonlinearity in Eq. (21) is
solved iteratively by the method introduced in [24] (Appendix).
The sample material is the commercial fiber insulation UltraBoard A99 of the M.E. SCHUPP Industriekeramik GmbH & Co. KG
[26]. According to manufacturer information the fiber-insulation
is made of Al2 O3 with 99% purity and average fiber diameters of
3–5 lm. SEM images recordings suggest a random orientation of
the fibers. Before sample preparation the insulation material was
tempered at its maximum application temperature of 1600 °C to
avoid excessive shrinkage during the actual measurements. The
cylindrically shaped sample specimens are manufactured using a
conventional lathe to provide geometrical accuracy. The cylindrical
samples for thermal diffusivity measurements had a diameter of
26:0 0:2 mm and a length of 95:0 0:5 mm. This corresponds
to a radius to length ratio of 0.14 which is sufficient to consider
the sample as being of infinite length (see Appendix). Furthermore
two drillings were set for the placement of thermocouples, one on
the cylinder axis and a parallel one off-axis. The on-axis drilling
was realized on the lathe; the off-axis one was drilled by help of
a gadget on a conventional drilling machine to provide parallelism
of the two drilling axes. The average sample density was determined geometrically yielding a value of 616 43 kg=m3 . In order
to evaluate density variations within a sample, one cylindrical
specimen was cut in small disks (1–3 mm); their average density
was measured to be 636 kg=m3 with a standard deviation of
77 kg=m3 .
3.1. Determination of albedo and extinction
Disk-shaped samples with diameter of 20:00 0:15 mm and
thickness of 1:00 0:15 mm have been prepared for the determination of the optical transport parameters extinction and albedo
b extk =qfs
defined in Eq. (16). The effective specific extinction b
ek ¼ r
b 0k were measured at 1000 °C by a black body boundand albedo x
ary conditions method in the wavelength interval of 2–8 lm. The
same quantities were determined at ambient temperature by the
integrating sphere method in the wavelength interval between
0.5 lm and 2 lm. The integrating sphere method as well as the
black body boundary conditions method are described in detail
in [27–29]. Both measurement techniques exhibit an experimental
uncertainty of about 10%. This explains the difference at the intersecting wavelength of 2 lm in Fig. 3 where the effective specific
extinction determined at ambient temperature is 8% above the corresponding high temperature quantity. For numerical reasons the
continuous variable k is replaced by discrete spectral bands
m ¼ ½km ; kmþ1 where the interval limits are selected such that
every spectral band contains identical black body radiative power
(dotted-line in Figs. 3 and 4). The Planck-averaged albedo and
extinction for every spectral band is defined by:
1
R kmþ1
dkX k k5 ðexp ðhcl =kkB T Þ 1Þ
X m ¼ kRmk
1
mþ1
dk k5 ðexp ðhcl =kkB T Þ 1Þ
km
ð23Þ
X represents either the albedo or the extinction. For simplification it is assumed that the experimentally derived extinction and
albedo remain constant for temperatures above 1000 °C.
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H. Brendel et al. / International Journal of Heat and Mass Transfer 108 (2017) 2514–2522
the unknown connection Parameter C sb . The average value
obtained is C sb ¼ 0:00391 0:00030.
4. Experimental setup and measurement procedure
4.1. Setup and temperature control
Fig. 3. Effective specific extinction ^
e (solid-line) measured at room temperature in
the short wavelength interval and at 1000 °C in the long wavelength interval and
their band approximation (dotted-line).
The cylindrical sample is fixed concentrically inside a ceramic
muffle by means of a gadget made of the same fiber insulation
material. The ceramic muffle is made of alumina with a purity of
99.7%. The setup was placed in the middle of the self-constructed
furnace TOM_air which is controlled by an Eurotherm Nanotac
unit. The sample temperature is analyzed by butt-welded type B
thermocouples of 0.75 mm diameter inside the on-axis and offaxis excentric drilling (positions A and B in Fig. 5). The distance
between these thermocouples is 8:0 0:5 mm. The muffle temperature is also analyzed by a thermocouple of type B which is located
midway between the outer and inner muffle wall. For fixation of
the thermocouples at the muffle a high temperature adhesive (Ceramobond 552) is used. The thermovoltage is logged by the Eurotherm Nanotac controller unit and by an external data
acquisition device.
To provide time synchronicity the temperature of the guidance
thermocouple is logged simultaneously by the controller and the
external data acquisition device. Data acquisition is done with a
sampling-rate of 1 Hz; the signal is filtered by a time constant of
1.6 s. To realize the cosine-type muffle wall temperature a script
for calculating the time dependent target values with an update
rate of 2 Hz was written. To enable accurate temperature control
the controller parameter setup has to be adapted for every frequency and furnace-temperature before the actual thermal diffusivity measurements.
4.2. Temperature homogenity and limitation of operating parameters
^ (solid-line) measured at room temperature in the short
Fig. 4. Effective albedo x
wavelength interval and at 1000 °C in the long wavelength interval and their band
approximation (dotted-line).
3.2. Determination of the connection parameter C sb
The connection parameter C sb used in Eq. (19) to describe the
conductive heat transfer due to fiber-to-fiber contacts has to be
derived experimentally. The idea is to measure the thermal conductivity tcsgb of the material at low temperatures, where radiative
contributions are negligible, and then extract C sb by help of Eq. (19)
using known values for the parameters of the components (gas and
fibres). In practice, thermal diffusivity of the samples was measured by laser-flash measurements (Netzsch LFA 457) at a temperature of 200 °C to avoid detrimental effects of air moisture. The
thermal conductivity tcsgb was calculated using the measured density of the fibrous samples and the heat capacity taken from [30].
The other material parameters required were obtained as follows:
due to the observed random orientation of the fibers, a value of
0.66 for the orientation parameter Z was used. The thermal conductivity of bulk-alumina was assumed to be similar to the values
given in [31]. A value of 3950 kg=m3 is used for the bulk-density of
alumina [32]. The gaseous properties for dry air are taken from
[33]. The literature data used for calculation at 200 °C are summarized in Table 3. With these input data Eq. (19) could be solved for
The temperature homogeneity over the inner muffle wall during periodic heating was evaluated at six positions of the inner
muffle wall. For this, the muffle after each complete measurement
cycle was rotated such that the position of thermocouple C takes
the previous position of the auxiliary thermocouple D. Comparing
the phase shifts obtained for a frequency of f ¼ 0:025/min in the
six different positions, a maximum difference of approximately
0.15 rad was observed at 1000 °C, whereas a value of only
0.012 rad was measured at 1580 °C. The obvious improvement of
temperature homogeneity towards higher temperature is attribu-
Fig. 5. Experimental setup for measurement of the thermal diffusivity via thermal
wave analysis (TWA).
H. Brendel et al. / International Journal of Heat and Mass Transfer 108 (2017) 2514–2522
ted to some cracks in the furnace insulation, which are closed by
thermal expansion at higher temperature.
The maximal frequency being achievable for a given amplitude
(T ex ¼ 10 K) is limited for two reasons: first, the furnace must have
a self-cooling rate high enough to follow the specified temperature
profile. Second, the homogenity of the temperature at the inner
muffle wall is deteriorated at higher frequencies. We found that
the maximum achievable frequency in our setup is f ¼ 0:05/min
at 1000 °C. Above 1200 °C the limit is f ¼ 0:15/min.
4.3. Determination of phase shift, amplitude and thermal diffusivity
Fig. 6 shows the temperature response signal in a fibrous sample for an excitation frequency of 0:05= min at a temperature of
1000 °C. To obtain phase shift and amplitude of the sample signal,
only the stationary part of the temperature curve is fitted by the
equation
TðtÞ ¼ A1 sin xt þ A2 cos xt þ A3 þ A4 t
2519
Convergence is attained when the variation of the evaluated thermal diffusivities is lower than 0:75% around the final value in
each interval. In addition the deviation of the averages and the
starting values of both intervals must be lower than 0.5%. Since
the described evaluation was done during the experiment the measurements are abandoned thereafter. In general, the analysis
between two and three stationary periods is sufficient. Note that
identical values have been obtained when determining phase and
amplitude by fast-Fourier analysis of the entire temperature profile
by subtracting the characteristic background spectrum of the transient part. The verification by fast-Fourier analysis was done by
using the Scilab built-in algorithm. The reproducibility was analyzed by replication of the measurements at distinct temperatures
and frequencies after a holding time of 30 min. The deviation
between the obtained thermal diffusivities was below 2% in all
cases. Investigations on the evaluation of four to five stationary
periods lead to a non-essential improvement of measurement
accuracy.
ð24Þ
Here, x ¼ 2pf is the angular velocity of excitation, A3 represents the
mean temperature of oscillation and the expression A4 t considers a
drift in furnace temperature. Phase and amplitude are determined
by the fit parameters A1 and A2 through the relations
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
T fs ¼ A21 þ A22 and tan mfs ¼ A2 =A1 .
As seen in the magnified section on the right-hand side of Fig. 6,
the actual value of the muffle temperature is slightly delayed in
comparison to the target temperature. To obtain the true phase
shift of the sample with respect to the actual muffle temperature
one has to subtract the muffle temperature phase shift from the
sample temperature phase shift. Therefore phase mex and amplitude T ex of the present excitation temperature curve are also
derived via fit-approximation by Eq. (24). Finally, the amplitude
decay factor g ¼ T fs =T ex is calculated. The stationary part is separated by comparing the analyzed thermal diffusivity of individual
periods of the sample temperature. The onset of the stationary part
is indicated when two separately evaluated immediately consecutive periods show a difference in the analyzed thermal diffusivity
which is lower than 1.5% and subsequent addition of 0.25 periods
(rightarrow in Fig. 6). This criterion is more stringent like that used
in [11]. The asymptotic behavior of the thermal diffusivity is investigated by stepwise increasement (1/60 period) of the analysed stationary temperature data. This is done in the direction of
increasing time and in the reversed direction whereas the corresponding second halves are the intervals taken for further analysis.
Fig. 6. Experimental data derived at 1000 °C with an excitation frequency of 0.05/
min.
4.4. Experimental procedure
Initially the setup was heated linearly at a rate of 5 °C/min to
the set temperature, following a holding time of about 30 min to
enable thermal equilibrium of the furnace. Thereafter the measurements for different excitation frequencies were done over three to
four oscillation periods until the mentioned convergence criterion
for the evaluated thermal diffusivity is fulfilled. The amplitude of
temperature excitation was always 10 K. Measurements were done
at temperatures of 1000 °C, 1200 °C, 1400 °C and 1580 °C. The frequencies at 1000 °C are f ¼ 0:025/min and f ¼ 0:05/min. At all
other temperatures measurements were done at the frequencies
f ¼ 0:05/min, f ¼ 0:1/min and f ¼ 0:15/min.
5. Results and discussion
In Fig. 7 experimental and numerical results obtained for a representative measurement at 1580 °C and an oscillation frequency
of f ¼ 0:05/min are compared. The measured temperature is represented by a thick solid line; the thin solid curve is a representation
of the analytical solution Eq. (4) using the parameters c1 and c2 as
obtained from the solution of Eqs. (13) and (14) with the experimentally derived phase delay and amplitude decay as input
parameter. The dotted line is the result of the numerical solution
of (Eq. (21)). Apparently experimental and numerical results are
very close to each other, but not identical: phase shift and
Fig. 7. Calculated and measured temperature profile at 1580 °C.
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H. Brendel et al. / International Journal of Heat and Mass Transfer 108 (2017) 2514–2522
amplitude decay obtained from the measurement are
mexp ¼ 0:82 rad and gexp ¼ 0:83, leading to a thermal diffusivity
of 0:39 106 m2 =s. From the numerical calculation one derives a
phase shift of 0:86 rad and an amplitude decay of 0.82,
which
yield
a
slightly
smaller
thermal
diffusivity
(jnum ¼ 0:35 106 m2 =s).
The temperature curves shown in Fig. 7 as well as the thermal
diffusivity values derived from it via Eqs. (13) and (14) refer to a
position at the symmetry axis of the sample (thermocouple A in
Fig. 5). Using the temperature profiles measured by the off-axis
thermocouple, a second value for the thermal diffusivity jB can
be deduced by help of the general Eqs. (11) and (12). All thermal
diffusivities derived from the on-axis and off-axis temperature
measurements are shown in Table 1. The error is the standard deviation of the average values.
The values in Table 1 suggest that jB is lower than jA by 10–
15% in all cases, but the difference lies within experimental accuracy. It therefore appears highly desirable to improve the precision
of the measurements. Guidelines to achieve this can be found by
reconsideration of Fig. 2 which shows the general dependencies
of phase and amplitude decay as a function of the control parameter c2
1 for fixed parameters c 2 , respectively. In the current situation, the parameter c2 takes values between 3 and 5 for all
temperatures and frequencies, the corresponding c2
1 lies approximately between 0:15 and 0:45. In this parameter range the curves
for constant c2 are relatively flat. Therefore, already small errors in
phase and amplitude lead to comparably large errors for the
deduced thermal diffusivity. An obvious way to improve the
signal-to-noise ratio would be to move into ranges of steeper variation of the curves. Practically, this would mean to increase the
sample diameter or the frequency f of the excitation temperature
profile. For the current study it was not possible to implement
these ideas due to technical limitations.
All values of thermal diffusivities obtained in this work by
either TWA measurements or the theoretical model are compared
in Fig. 8. The values from the TWA measurements shown there are
mean values of the results in Table 1. First we compare the thermal
diffusivity from TWA measurements with the theoretical predictions. At every temperature point considered the experimentally
derived values are above the theoretically derived values. Towards
higher temperatures, the experimental values for the thermal diffusivity show a steeper increase than the theoretical prediction,
though mostly still being in accordance within experimental error.
A possible explanation for the difference is that the absorption
coefficient measured at 1000 °C was used for the calculation also
at higher temperatures. According to literature, however, an
increase of the spectral absorption coefficient is expected with
increasing temperature for pure alumina [21]. Another possible
reason is sample imperfection like density fluctuation in the volume or slight surface damage during cylinder preparation. A third
point is the model for combined gaseous and conductive heat
transfer, which also depends on literature data. The model is calibrated at 200 °C and extrapolated to higher temperatures by
assuming that the temperature dependence of the solid and gaseous properties cp and tc is identical to the literature values of
Table 1
Measured thermal diffusivity through different thermocouple positions A and B.
T in °C
1000
1200
1400
1580
jB
jA
106 m2 =s
0:27 0:03
0:32 0:04
0:34 0:04
0:38 0:04
106 m2 =s
0:31 0:05
0:35 0:02
0:39 0:02
0:43 0:05
Fig. 8. Experimental and theoretical thermal diffusivities.
Table 2
Comparison of thermal conductivities.
T in °C
1000
1200
Thermal conductivity tc in W/(m K)
Manufacturer
TWA
Theory
0:24
0:26
0:22
0:26
0:21
0:23
[30,31,33]. Finally, our measurement results are compared to the
ones given by the manufacturer of the fibrous insulation material.
Latter were obtained by the hot-wire method at 1000 °C and
1200 °C. The manufacturer specified the thermal conductivity, so
we calculated the thermal conductivity from the measured and
calculated thermal diffusivities by the help of the measured density and the literature values for the specific heat (Table 3).
The results are shown in Table 2. We see that the experimental
values obtained by the TWA method and the theoretical results are
in accordance with the hot-wire measurements.
6. Conclusion
In this work, the feasibility of measuring thermal diffusivity of
fibrous insulation materials at high temperatures up to 1580 °C
by application of the thermal wave analysis (TWA) technique (periodic temperature excitation) was demonstrated. Furthermore, a
model has been developed which allows quantitative numerical
prediction of the experimental results, when high temperature
experimental input data, in particular spectral extinction and
albedo, are available. The thermal diffusivities experimentally
obtained are in reasonable agreement with the results from our
numerical calculations; as well, the values determined in this work
are in good agreement with the manufacturers specification for the
thermal conductivities at 1000 °C and 1200 °C. For an optimal
implementation of the technique a particular designed furnace
should be developed which enables large sample dimensions and
high temperature oscillation frequencies. Also, the dynamic temperature homogeneity of the sample environment (muffle) should
be improved. A properly sized vertical tube furnace with densely
located heating elements in form of a coil with an inner lining,
made of a material with a high thermal conductivity (for instance
dense alumina), is favorable. The outer thermal insulation has to be
rather thin so that the furnace could follow the target heating and
cooling curve also at temperatures below 1000 °C (which was the
lower limit achieved in this work).
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H. Brendel et al. / International Journal of Heat and Mass Transfer 108 (2017) 2514–2522
Summarizing our findings, we conclude that the TWA technique
appears to be very promising as an accurate method for high temperature measurements of heterogeneous samples, complementing the established, but temperature-restricted methods like the
guarded-hot-plate- or the hot-wire technique into the hightemperature regime.
Acknowledgments
The authors are indebted to J. Manara and M.C. ArduiniSchuster from the ZAE Bayern, Magdalene-Schoch-Straße 3,
D-97074 Würzburg for performing the extinction and albedo
measurements and for helpful discussions concerning radiative
heat transfer.
Appendix A
In this Appendix, some important mathematical details of the
model described in this paper are given. Furthermore, the validity
of the model is checked against finite element (FE) calculation of an
exemplary situation. We start with a 1-dimensional FE model to
prove the validity of the Eqs. (2)–(14). The FE calculation was done
using ANSYS 15 with the following parameters: T m ¼ 1190 C,
T ex ¼ 10 C,
eff ¼ 0:12; f ¼ 0:15=min; q ¼ 593kg=m3 ; j ¼ 0:52
106 m2 =s; r a ¼ 0:013 m. The results and a brief description of the
FE model are shown in Fig. 9. It is evident that the prediction of
Eq. (2) (solid line) is in perfect agreement with the FE calculation.
The same statement holds for the stationary case expressed by Eq.
(2) (dashed line) after typically one full cycle of the cosine-type
excitation. Next, a 2-dimensional FE model was used to test the
validity of assuming an infinite cylinder length (which is the basis
of Eqs. (4)–(14)) instead of the real radius to length ratio ð0:14Þ of
the fibrous samples used in our thermal diffusivity measurements.
The FE model for the finite cylinder features an additional blackboundary condition on the cross-sectional-areas: w ¼ 1. All other
parameters are identical to the previous calculations shown in
Fig. 9. The FEM results for the time dependent temperature profile
for an infinite cylinder and a finite cylinder with radius to length
ratio of 0:14 are compared in Fig. 10. The two curves are nearly
identical; very small differences, far below experimental accuracy,
are only seen in strongly magnified scaling (see insets in Fig. 10).
Apparently the additional heat flux through the cross-sectionalareas has almost no influence on the temperature profile at the
center of the cylinder.
Now we demonstrate by comparison to previous work (Ref.
[17]) that our formulation and implementation of the scaled diffu-
Fig. 9. Validation of Eqs. (2)–(14) by comparison with FEM-calculations.
Fig. 10. FEM-comparison of an infinite and a finite cylinder.
sion equation, in particular Eqs. (15) and (18), is correct. Results are
collected in Fig. 11. For the parameters g ¼ ð1; 0; 1Þ;
x0 ¼ 0:95; sR ¼ 1=ð1 x0 Þ; ¼ 1 and T fs ðrÞ ¼ T w = const the
results derived via Eqs. (15) and (18) by application of Eq. (16)
must be identical to the results given in Fig. 10 of [17]; there, the
unscaled form of Eq. (15) was used for calculation. Another comparison is possible for the radiative flux given in Fig. 6 (curve C)
of [17]; in our implementation, this corresponds to the parameters
and
T fs ðrÞ ¼ T w þ ðT fs ð0Þ T w Þ
g ¼ 0; x0 ¼ 0:5; sR ¼ 5; ¼ 1
h
i
2
1 ðr=RÞ with the relation T fs ð0Þ=T w ¼ 5. In all cases, the results
of our implementation are identical to the literature values within
the precision limits of graph digitalization, proving the validity of
Eqs. (15), (18) and (16).
At this point we give further information about the dimensionless parameters introduced in Eq. (19). According to [23], d is the
height of a fibre gas cell in units of the fiber diameter, and o þ 1
is the area of a fibre gas cell in units of the contact area.
dþ1¼
1=2
plqbulk
2qfs ð1 l2 Þ
qbulk =qfs
oþ1¼
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
l 1 l 1:5p 1 12bulk p0 =Y bulk 2
ð25Þ
!1=3
ð26Þ
Fig. 11. Comparison of the radiative flux calculated using Eqs. (15), (16) and (18)
with results given in (Fig. 6) and (Fig. 10) of [17].
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H. Brendel et al. / International Journal of Heat and Mass Transfer 108 (2017) 2514–2522
Table 3
Literature data according to [30,31,33,34] used for the theoretical model and further
calculations.
T in °C
200
1000
1200
1400
1580
Air
Al2 O3
cp
J/(kg K)
tcbulk
W/(m K)
tcg
W/(m K)
1022
1261
1295
1317
1341
22:38
6:04
5:70
5:72
6:13
0.038
0.082
0.092
0.101
0.110
l ¼ cos #0 is the mean fiber orientation angle with respect to the
external heat flow. p0 is an external pressure, 1bulk represents Poisson’s number and Y bulk is Young’s modulus. Assuming a random
fiber orientation (Z ¼ 0:66) and qfs ¼ 616 kg=m3 one finds
d ¼ 1:77. Using
1bulk ¼ 0:25; Y bulk ¼ 360 109 N=m2 from [32] and
5
p0 ¼ 10 Pa; o is calculated to be 2:11 103 . Therefore the expression ðo þ 1Þ=o in Eq. (19) is approximately 1.
Finally, the algorithm for solving the nonlinearity in the inhomogeneous diffusion Eq. (21) is introduced [24]. Within a spectral
p
band m ¼ ½km ; kmþ1 the Planck-function im ðTÞ is expressed as
p
im ðT Þ ¼ rp F 0-kmþ1 ðT Þ F 0-km ðT Þ T 4
ð27Þ
where F 0-k ðT Þ can be evaluated by a series expansion [18]. With
T 4 4T T~ 3 3T~ 4 one obtains
p
p
~
im ðT ji þ 1 Þ im ðT jþ1
¼ rp ðF 0kmþ1 ðT~ i Þ F 0km ðT~ i Þ Þ
i ; Ti Þ
ð4T ji þ 1 T~ i 3T~ i Þ
3
4
ð28Þ
where T ij :¼ T r i ; t j . For calculating T jþ1
; T~ i ¼ T i j is taken as initial
i
0
~
guess; then T i is calculated by Eq. (21). In the next step T~ i is
replaced by T~ 0 i . This is repeated until the condition
max jT~ 0 i T~ i j < 106 K is fulfilled. Then T jþ1
is set to T~ 0 i .
i
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