A study of surface temperature- and heat flux estimations
by solving an Inverse Heat Conduction Problem
Patrik Wikström
Licentiate Thesis
Royal Institute of Technology
School of Industrial Engineering and Management
Department Of Materials Science and Engineering
Division of Energy and Furnace Technology
Se- 100 44 Stockholm
Sweden
Akademisk avhandling som med tillstånd av Kungliga Tekniska Högskolan i
Stockholm framlägges för offentlig granskning för avläggande av Teknologie
licentiatexamen tisdagen den 7 februari 2006, kl. 10:00 i sal B3, Brinellvägen 23,
Kungliga Tekniska Högskolan, Stockholm.
ISRN KTH/MSE--05/92--SE+ENERGY/AVH
ISBN 91-7178-244-3
Patrik Wikström,
A study of surface temperature- and heat flux estimations in heating processes by
solving an Inverse Heat Conduction Problem
Royal Institute of Technology
School of Industrial Engineering and Management
Department Of Materials Science and Engineering
Division of Energy- and Furnace Technology
Se- 100 44 Stockholm
Sweden
ISRN KTH/MSE--05/92--SE+ENERGY/AVH
ISBN 91-7178-244-3
 Patrik Wikström, January 2006
To my beloved Jenny and my son Hugo
i
ii
Abstract
The topic of this thesis is estimation of the dynamic changes of the surface temperatureand heat flux during heating processes by using an inverse method. The local transient
surface temperature and heat flux of a steel slab are calculated based on measurements in
the interior of the slab.
The motivations for using an inverse method may be manifold. Sometimes, especially in
the field of thermal engineering, one wants to calculate the transient temperature or heat
flux on the surface of a body. This body may be a slab, or billet in metallurgical
applications. However, it may be the case that the surface for some reason is inaccessible
to exterior measurements with the aid of some measurement device. Such a device could
be a thermocouple if contact with the surface in question is possible or a pyrometer if an
invasive method is preferred. Sometimes though, these kinds of devices may be an
inappropriate choice. It could be the case that the installation of any such device may
disturb the experiment in some way or that the environment is chemically destructive or
just that the instruments might give incorrect results. In these situations one is directed to
using an inverse method based on interior measurements in the body, and in which the
desired temperature is calculated by a numerical procedure.
The mathematical model used was applied to experimental data from a small scale
laboratory furnace as well as from a full scale industrial reheating furnace and the results
verified that the method can be successfully applied to high temperature thermal
applications.
iii
iv
Acknowledgements
First of all I would like to express my deepest appreciation to my supervisor Professor
Wlodzimierz Blasiak who gave me the possibility to work in the field of Inverse
Problems and advised me during the work.
I am most grateful to the Swedish Steel Producer’s Association (Jernkontoret) and to
STEM for financing this work.
Many thanks to Ph.D Fredrik Berntsson at Linköping University, Dept. of Mathematics,
Division of Numerical Analysis, for his generous support and for his contribution to this
work. He persistently answered my numerous questions in a very positive and serviceminded way.
I would also like to say Thank You to Mr. Jonas. B. Adolfi at AGA AB/Linde gas for the
excellent collaboration during the experimental work. My gratitude also goes to his boss
Tomas Ekman for allowing me to complete the experiments there in the first place. A lot
of thermocouples were wasted in the name of science.
A big thank to Jonas Engdahl who provided me with the industrial data and help in
understanding the topic of reheating furnaces.
Last but not least I would like to thank all the members of the Division for contributing to
a stimulating and creative workplace in which seriousness and humor are blended in a
perfectly balanced harmony.
v
vi
Supplements
This licentiate thesis comprises an introduction to the Inverse Heat Conduction
Problem, and the theoretical model used. At the end of this thesis, the following
papers are appended:
Supplement 1:
“Estimation of the transient surface temperature- and heat flux of a
slab using an inverse method”
P. Wikström, F. Berntsson, W. Blasiak
Submitted to Int. J Heat Mass Transfer, Sept. 2005
Supplement 2:
“Estimation of the transient surface temperature, heat flux of and
effective heat transfer coefficient of a slab in a full scale industrial
reheating furnace by using an inverse method”
P. Wikström, F. Berntsson, W. Blasiak
Submitted to Scandinavian Journal of Metallurgy, Dec. 2005
vii
viii
CONTENTS
Abstract ............................................................................................ iii
Acknowledgements.......................................................................... v
Supplements ................................................................................... vii
1. Introduction................................................................................... 1
1.1 Literature review........................................................................................2
1.2 Objectives ..................................................................................................3
2. Methodology ................................................................................. 3
2.1 Theoretical part .........................................................................................3
2.1 .1 The concept of an ill-posed problem ...............................................3
2.1.2 The Inverse Heat Conduction Problem .............................................4
2.1.3 The mathematical model ....................................................................5
2.1.4 Way of solving the inverse problem..................................................7
2.2 Experimental part ......................................................................................9
2.2.1 An application to a laboratory scale heating process .....................9
2.2.2 An application to industrial conditions...........................................11
3. Results and discussion ............................................................. 13
4. Concluding remarks................................................................... 18
Future work ..................................................................................... 19
References ...................................................................................... 19
ix
1. Introduction
Controlling the temperatures and heating rates at several stages during the production line
is very important in order to achieve good quality of the products as well as being
equivalent to good production economics.
The usage of inverse methods has gained more interest in recent years. Applications are
especially useful for cases where the target of investigation for some reason is
inaccessible to exterior measurements with the aid of some measurement device. Such a
device could be a thermocouple if contact with the surface in question is possible or a
pyrometer if an invasive method is preferred. Sometimes though, these kinds of devices
may be an inappropriate choice. It could be the case that the installation of any such
device may disturb the experiment in some way or that the environment is chemically
destructive or just that such instruments might give incorrect results. Pyrometers would
not measure the correct value of the surface temperature since oxide scale may be formed
on the slab surface due to the furnace atmosphere. Likewise, inaccurate readings would
occur when using thermal sensors directly attached to the surface since radiant energy
from the furnace refractory and burner flames will dominate the heat transfer mechanism
to the slab and thus the thermal sensor will not only read the desired contribution from
heat conduction on the surface. In these situations, it is more accurate to measure the
temperature history inside the slab.
In this work the transient surface temperature and heat flux of a steel slab is calculated
using a model for inverse heat conduction. That is, the time dependent local surface
temperature and heat flux of a slab is calculated on the basis of temperature
measurements in selected points of its interior.
This thesis is divided into two parts. The first part is intended to introduce the reader to
the concept of an inverse heat conduction problem. A summary of the solution technique
used is also entailed. The second part of the thesis comprises the applications and
verification of the theoretical model on experimental data which extends from a small
laboratory scale heating experiment to a full scale industrial heating application.
1
1.1 Literature review
Most inverse problems belong to a family of problems that have inherited the property of
being ill-posed in the sense of Hadamard [1, 2]. Since the interest in these methods begun
with one of the first published papers [3] in the 60’s, the applications nowadays range
over many scientific fields. Those fields include medicine, fluid dynamics and heat
transfer to name only a few.
Of special interest in this work are inverse methods connected to heat transfer analysis. In
the literature [2, 3], this family of problems is most often referred to as inverse heat
conduction problems (IHCP). A common family of methods for solving the inverse heat
conduction problems transforms the problem into an integral equation of first kind [6, 7].
The drawback of these methods is that often the kernel in the corresponding integral
equation is not known explicitly. This is the case, for instance, if the properties of the
material, e.g. thermal conductivity, specific heat and density, are dependent on the
temperature, i.e. the problem is non-linear. In metallurgical applications, such as the
experiments described in this thesis, such methods cannot easily be used since the
material properties of steel change considerably in the large temperature range present in
the experiments. The method developed in [8], which is applied in this thesis, allow for
problems in which the material properties depend on the temperature, i.e. the Fourier’s
heat conduction equation with non-constant coefficients. A method for solving the IHCP
using wavelets was proposed in [9].
Applications of inverse methods span over many heat transfer related topics. Sometimes
the temperature- and heat flux data on the boundary are known and one wants to
determine the material properties of the material investigated. Those problems are often
referred to as parameter identification problems in the literature [10, 11]. An application
to the determination of thermal heat conductivity of thermo plastics under moulding
conditions was studied in [12] and a parameter identification problem for determination
of the temperature dependent heat capacity under a convection process was carried out by
[13]. If temperature- and heat flux data are known then heat transfer coefficients for the
boundary conditions may be determined. Some applications of these techniques are given
in [14, 15].
However, in the classical IHCP the temperature data themselves are to be identified by
means of using interior measurements. An example of this was given in [16] where the
heat flux on the surface of ablating materials was to be determined. Another investigation
of an inverse method for estimation of the outer-wall heat flux in a turbulent circular pipe
flow was conducted by [17] and the influence of a coating in wood machining from the
heat flux was carried out by [18].
Controlling the temperatures and heating rates at several stages during the production line
in the steelmaking industry is very important in order to achieve good quality of the end
products as well as being equivalent of good production economics. The hot-rolling is an
area where inverse methods have been applied. It has been reported that temperature
2
gradients in the order of one hundred Celsius can cause damage to the rolling mill and
therefore investigations in this area are important. Huang et.al [19] performed a study of
the thermal behaviour of the working rolling mill process and a further application on the
effect of high speed rolling on the surface heat flux was done by Keanini [20]. An
application to a blast furnace was performed by Fredman [21] where the thickness of the
accretation layer was estimated by an inverse method.
This thesis is focused on an application to reheating of slabs prior to the hot-rolling
process in the line of heat treatment processes. This paper sets out to use the method for a
heating problem in a temperature range relevant to reheating furnaces in steel industry
[22]. By using an inverse method [8] the aim of this thesis work is to determine the
transient surface temperature and heat flux of a steel slab in a large scale industrial
reheating furnace. Furthermore, the time dependent heat transfer coefficient at the surface
of the steel slab is determined. To the author’s knowledge, no applications have been
published directly in relation to the slab heating process in a reheating furnace. An
application to a cooling experiment of an aluminum block using this method was
performed in [23].
1.2 Objectives
The objectives of this thesis are:
1. to estimate the local transient surface temperature- and heat flux during slab heating
processes by applying an inverse method
2. to experimentally verify the mathematical model used against experimental data from a
small scale laboratory furnace as well as for a full scale industrial reheating furnace.
2. Methodology
2.1 Theoretical part
2.1 .1 The concept of an ill-posed problem
Most inverse problems are ill-posed in the sense of Hadamard [1, 2]. In this section, some
features of ill-posed problems are illustrated. This is most naturally done by defining, in
parlance of mathematical stringency, what it means for a problem to be well-posed.
Let X and Y be Hilbert spaces. Let x ∈ X be the desired unknown solution and y ∈ Y be
the available data. Define a mapping through the bounded operator K such that
K: DK → RK , where DK ⊆ X and RK ⊆ Y . Then, the operator equation
3
y = Kx
(1)
is well-posed in the sense of Hadamard if the following statements hold:
•
•
•
for every y ∈ Y there is at least one x ∈ X such that Kx = y
for every y ∈ Y there is at most one x ∈ X such that Kx = y
the solution x depends continuously on y such that for every
sequence {x n }∈ X: Kx n → Kx ⇒ x n → x when n → ∞.
(2)
(3)
(4)
The above properties; existence (2), uniqueness (3) and stability (4) are assumed for a
well-posed problem. A problem is ill-posed if it does not satisfy one or more of these
properties. The existence and uniqueness can be ascribed to the algebraic properties of
the spaces whilst stability depends on topology; that is whether the inverse operator, K −1 ,
is continuous. For the case of the inverse heat conduction problem (IHCP) the
corresponding operator equation has an unbounded inverse. Hence the problem is illposed in the sense of Hadamard, as condition (4) is not satisfied. Also, since the
temperature measurements y will contain measurement errors making the stability of the
problem is a serious difficulty in applications.
2.1.2 The Inverse Heat Conduction Problem
This text is concerned with an application of a one-dimensional inverse heat conduction
problem (IHCP). The desired thermal data is the surface temperature- and heat flux of a
slab in a heating process. As mentioned in the introductory part of this text it may be
difficult to measure directly the temperature history on the surface of a body. In a
physical situation similar to those that arose during the experiments on which this thesis
work is based, it was unsuitable to directly measure the desired thermal properties by
means of sensors. In a furnace at high temperature, typically radiant energy from furnace
refractory and burner flames will dominate the heat transfer mechanism to the slab.
Contributions may also be given from convection due to circulating furnace gases to
some extent. Thus, the thermal sensor will not only read the desired contribution from
heat conduction on the surface but will be affected by flames, furnace refractory and from
convective flows. In these situations, it is more accurate to measure the temperature
history inside the slab.
The term temperature- and heat flux “estimation” is frequently used in this work, and
deliberately so as internal measurements are always associated with measurement errors
that will affect the accuracy of temperature- and heat flux calculations. The IHCP is a
difficult problem because it is extremely sensitive to measurement errors. One major
source of uncertainties when using an inverse method comes from internal temperature
measurements. The information gained from these measurements is incomplete in several
regards. Information is lost since there are only a limited number of sensors positioned
inside the heat conducting body, in this work only two or three. The measurements are
4
available only at discrete times, not continuously. Furthermore, the measurements are not
continuous errorless functions but will inherently contain random errors.
In this work, it is of interest to acquire the surface temperature which is easier done than
calculating the surface heat flux, as calculating the heat-flux requires an extra numerical
differentiation; which adds to the degree of ill-posedness as numerical differentiation in
itself is an ill-posed problem. Consequently, when one tries to get as much possible
information from the estimation as for example when an effective heat transfer
coefficient is calculated, where both of the above estimated properties are needed
together with additional information from one or more thermal measurements, the
difficulties are pronounced.
2.1.3 The mathematical model
The situation encountered in this thesis work is shown schematically in Fig. 1. The target
is to restore the boundary conditions on the indicated surface using internal
measurements with the aid of the thermocouple. Mathematically, the inverse problem to
solve is the following: determine the temperature distribution T ( x, t ) for 0 ≤ x < L1 from
measurements of temperature f m (t ) and heat flux q m (t ) along the line x = L1 where
T ( x, t ) satisfies:
∂ 
∂T 
∂T
,
 λ (T )
 = ρ (T ) ⋅ c(T )
∂x 
∂x 
∂t
T ( L1 , t ) = f m (t ) ,
∂T
λ (T ) ⋅
( L1 , t ) = q m (t ),
∂x
T ( x, 0) = 0,
0 < x < L1 , t ≥ 0
(5)
t ≥ 0,
(6)
t ≥ 0,
(7)
0 < x < L1 .
(8)
Here, λ (T ), ρ (T), and c(T) are the temperature dependent thermal conductivity, density
and specific heat capacity, respectively.
5
Figure 1. Schematic figure demonstrating the conditions for solving the IHCP.
It is however difficult in practice to satisfy condition (7) since heat flux measurements are
usually not available in the interior of the material. Instead the temperature is measured at
a second location, and the desired heat flux at x = L1 is computed by solving a wellposed problem in the interval L1 < x < L2 , using the measured temperatures along x = L1
and x = L2 as boundary data. This direct problem addresses that if the heat flux- or
temperature histories at the surface of the slab are known as functions of time, then the
temperature history can be found and consequently, the heat flux history as well. A nonlinear Crank Nicholson Implicit scheme is used for this purpose. As can be seen in Fig. 2,
the original problem have been divided into two computational domains, one for the
direct problem and one for the inverse reconstruction of the surface data.
6
Figure 2. Schematic figure of the heat conduction problem divided into zones of directand inverse problem regions.
2.1.4 Way of solving the inverse problem
The inverse problem (5-8) is severely ill-posed and needs special numerical methods, i.e.
regularization techniques [24], to be solved in a stable way. The ill-posedness of the
problem is revealed by using the Fourier transform for reformulating the problem in the
frequency domain.
Let
fˆm (ξ ) =
1
2π
∞
∫ exp(−i ⋅ ξ ⋅ t ) ⋅ f
m
(t ) dt,
- ∞ < ξ < ∞,
(9)
−∞
be the Fourier transform of the data function. The solution of the inverse problem (5-8)
can formally be written as
T ( x, t ) =
1
2π
∞
∫ exp(−i ⋅ ξ ⋅ t ) exp((L
1
− x) ⋅
i ⋅ξ
α
−∞
where α = λ /( ρ ⋅ c) is the thermal diffusivity.
7
) ⋅ fˆm (ξ ) dξ ,
(10)
Since the real part of i ⋅ ξ is non-negative, (10) represents an unbounded operator.
Small errors in high frequency components of the data function f m are may blow up and
totally destroy the solution in the interval 0 < x < L1 . This may happen since high
frequency components are magnified by the factor exp(( L1 − x) ⋅
i ⋅ξ
α
) which grows
exponentially as ξ → ∞ . The integral in (10) exists only for functions fˆm with
exponential decay in the frequency domain. This means that the problem is severely illposed and must be stabilized, for instance by removing the high frequency components
from the calculations. Elimination of the high frequencies from the solution is done by
introducing a cut-off frequency, ξ c . Thus we restrict the problem to frequencies such that
ξ ≤ ξ c . In this fashion, the regularized solution looks like:
Tc ( x, t ) =
1
2π
ξc
∫ξ exp(−i ⋅ ξ ⋅ t ) exp((L
1
−
− x) ⋅
c
i ⋅ξ
α
) ⋅ fˆm (ξ ) dξ .
(11)
With the high frequencies removed continuous dependence on the data is restored and
(11) represents the solution of a well-posed problem. The fact that the IHCP is severely
ill-posed is a consequence of the fact that the time derivative is an unbounded operator
while the spatial derivative does not cause any problems. By replacing the time derivative
∂ / ∂t by a bounded approximation a well-posed problem is obtained. In the preceding
paragraphs, the IHCP was regularized by introducing the cut-off level in the frequency
domain. In what follows, an approximation of the time derivative is calculated by using
the Discrete Fourier Transform. The time interval is discretisized using an equidistant
grid. By discretisizing the time variable we consider the unknown temperature T ( x, t ) ,
and the data f m and q m to be vectors representing discrete functions on the grid. We can
then reformulate the original problem as a system of differential equations,
∂  T   0
∂T = ∂
 
∂x  α
 ∂x   ∂t
α −1  T 
 ∂T ,
0  α


∂x 
0 < x < L1 , t > 0
(12)
with the initial- and boundary conditions:
T ( x, 0) = 0, 0 < x < L1 , T ( L1 , t ) = f m (t ) and λ
∂T
( L1 , t ) = q m (t ), t > 0 .
∂x
(13)
After replacing the time derivative ∂ / ∂t by a discrete approximation which is
implemented using the Fast Fourier Transform (FFT), the problem (12) is an initial value
problem for a system of ordinary differential equations that can be solved using standard
methods, e.g. Runge-Kutta Methods. The MATLAB routine ode45 was used in the
computations for solving the initial value problem
8
2.2 Experimental part
The inverse heat conduction problem was solved for two applications as described in
Supplements 1 and 2. The first application was performed in a laboratory scale test
furnace and the second was subject to a full scale industrial reheating furnace.
In both tests, the same material was used for the test slab. The composition of the steel is
shown in Table 1 below.
Table 1. The composition of the steel used for the investigations.
Element C
Wt-%
0.06
Si
0.01
Mn
0.38
S
0.035
P
0.017
Cr
0.022
Ni
0.055
Mo
0.030
Cu
0.08
Al
0.001
2.2.1 An application to a laboratory scale heating process
The first application of the IHCP was performed in a small scale laboratory test furnace.
A test slab was heated in small scale test furnace and the task was to estimate transient
surface temperature- and heat flux. A constant temperature of about 1250 οC was aimed
for in the test furnace and the temperature of the test slab before the heating started was
about 25 οC. The dimensions of the furnace and the test slab are seen in Fig. 3.
9
Figure 3. A photograph of the test furnace (above) and a schematic figure showing the
dimensions of the furnace and the slab (below). Dimensions are given in mm.
Thermocouples were positioned in the interior of the slab. Measured temperature
histories inside the slab were taken at 5, 11, and 17 mm respectively from the top of the
slab surface at x = 0 mm as shown in Fig. 4. The original data vectors were sampled only
at 0.25 Hz . This was insufficient for the purposes since more data will give more
accurate results, and therefore the data vectors; originally of length 358 were resampled
to a larger grid using a smoothing cubic spline. The data vectors used in the actual
computations were of length 2048.
10
Figure 4. The dimensions of the test slab equipped with thermocouples.
2.2.2 An application to industrial conditions
The investigation in Supplement 1 was appreciable since the constant furnace
temperature created a rather simple heating curve. The same estimations were revisited in
Supplement 2 only this time a full scale industrial reheating furnace was studied.
The temperature data was collected from a so called “pig-test” in which a slab is
equipped with a data logger that follows through the furnace. A steel tube containing the
data logger is immersed in a water bath inside a cavity in the slab in order to withstand
the high temperatures. The data logger collects data from thermocouples that are
positioned at selected points in the interior of the slab.
“Pig-tests” tests are occasionally performed by the furnace operators in the industry to
verify that the calculated temperatures are in agreement with the measured data.
The dimensions of the test slab were 11000 × 1500 × 220 mm and thermocouples were
positioned at locations A, B, and C as shown in Fig. 5.
11
Figure 5. The positions A, B, and C of the thermocouples in the test slab. The letters U
and L are denoting upper- and lower surfaces respectively.
Mathematical models that recreate the heating process in order to investigate the
temperature distribution and to perform heat transfer analysis to see how changes in
different parameters will affect the model have been created, one such program is
®
STEELTEMP 2D [25] which is based on two-dimensional finite difference
calculations. It should be emphasized that the temperature data interior of the slab at
®
points A, B and C are calculated by STEELTEMP 2D from furnace’s gas- and wall
®
temperatures as boundary conditions. The software STEELTEMP 2D was used only as
an aid in helping to confirm the calculations made by the inverse method which is
referred to as SHESOLV. When using SHESOLV, temperature data calculated by
®
STEELTEMP 2D were used as in data for the sake of verification. Thus a numerical
®
test problem was created were SHESOLV was verified against STEELTEMP 2D.
®
The data vectors of the calculations generated by STEELTEMP 2D were sampled at
only 1 60 Hz resulting in data vectors of length 163. The solution of the inverse problem
works better with more data available. With a higher sample rate an averaging filter
would more effectively remove the random noise from the data, and also the need for the
initial re-sampling. Therefore, the data vector was re-sampled to a size of 1024. The cutoff frequency was set to ξ c = 100 .
The same problem was solved using SHESOLV directly applied to the raw data fro the
“pig-test”. The authentic temperature raw data of 917 sampling points, however, did not
need any re-sampling. The cut-off frequency was set to ξ c = 70 , a little lower than used
®
in the verification with STEELTEMP 2D since these data are authentic.
A schematic side view of the furnace and its partition into zones subject to this
®
investigation is shown in Fig. 6. The figure is generated by STEELTEMP 2D and
12
shows the calculated temperature in points A, B, and C of the slab and the furnace gasand wall temperatures.
Figure 6. A schematic figure of the reheating furnace and the variations of the
®
temperature curves in the different zones. Figure from STEELTEMP 2D. The
uppermost red curves are the furnace gas temperatures above (solid) and below (striped)
the slab, the blue curves are the furnace wall temperature above (solid) and below
(striped) the slab. The tree curves that starts from essentially the same temperature are the
temperatures in the slab at positions A (black), B (red), and C (green).
3. Results and discussion
A test problem was created by comparing the calculated and measured temperatures at
the position of the thermocouple TC1 at x = x1 , referring to Fig. 4. That is, an inverse
heat conduction problem was solved in the interval x1 < x < x 2 . To solve this problem,
the heat-flux at x = x 2 was desired as initial data. In Supplement 1, it was shown that
some initial smoothing of the measured data is needed for calculating the heat flux at
x = x 2 with sufficient accuracy. A moving average of 20 data points was used. Likewise,
the regularization parameter, the cut-off frequency ( ξ c ), affects the calculations at x = x1 .
This parameter has a significant influence on the computational results. Theoretically, a
too high cut-off level will result in a solution in which a significant part of the highfrequency noise remains, making the resulting curves very cusp, and unphysical. If, on
the other hand, a low cut-off frequency is used then relevant parts of the data are removed
13
along with the noise, with loss of actual information, and resulting a solution that is very
smooth, but false. This value was chosen to be set to ξ c = 100 in order to capture the
essential behavior of the actual heat flux curve, and without the solution being too
cluttered with noise. The results of the estimations are shown in Fig. 7. The maximum
difference between the measured and the calculated value was 24 o C . Since the
temperature ranged from 25 − 1250 o C this is a very good result.
300
1400
1200
250
2
Heat flux [kW/m ]
o
Temperature [ C]
1000
800
600
200
150
100
400
50
200
0
0
200
400
600
800
0
1000
0
200
400
600
800
1000
Time [s]
Time [s]
Figure 7. The calculated temperature (solid) and the measured temperature history (dashed) at
x1 = 5 mm below the surface (left) and the corresponding calculated heat flux (right).
The rest of this section refers to the test as described in Section 2.2.2.
The slab was heated from above as well as from below. In Fig. 8, the calculated
temperatures in points A, B, and C (TA, TB, TC) as well as the furnace gas- and wall
temperatures above and under the slab (TgU, TgL, TwU, TwL) as a function of time as
®
calculated by STEELTEMP 2D, are shown. The partition into zones is shown by the
vertical tiles and the numbers of the zones are denoted z1, z2 and so on.
14
1600
1400
o
Temperature [ C]
1200
TwU
TwL
TgU
TgL
TA
TB
TC
1000
800
600
400
200
z1
0
0
z2
2000
z3
z4
z5
4000
z6
6000
z7
8000
z8
10000
Time [s]
Figure 8. The slab temperatures at points A, B, and C and the furnace gas- and wall
®
temperatures above and under the slab, as calculated by STEELTEMP 2D, are shown.
In order to distinguish the calculations done by the inverse method to those performed by
®
the software STEELTEMP 2D, we will for simplicity refer to these as SHESOLV.
In Fig.11 a-b; the estimated surface temperatures of the upper- and lower side of the slab,
respectively, are shown. The data agree very well in the whole temperature range (201350 o C ). On average the difference was of order 5°C .The conclusion drawn is that the
®
verification of SHESOLV against STEELTEMP 2D was successful.
15
a)
1400
1200
SHESOLV TsU
Temperature [ oC]
1000
STEELTEMP Ts
U
800
600
400
200
z1
0
0
z2
z3
2000
z4
z5
4000
z6
6000
z7
z8
8000
10000
Time [s]
b)
1400
1200
SHESOLV Ts
L
STEELTEMP Ts
Temperature [ oC]
1000
L
800
600
400
200
z1
0
0
z2
2000
z3
z4
z5
4000
z6
6000
z7
8000
z8
10000
Time [s]
Figure 9. The comparison of surface temperatures for a) the upper slab surface and b) the
lower slab surface.
By using the inverse method, the boundary data can be calculated directly from the noisy
interior temperature measurements, i.e. the raw data. The calculations of the top and
bottom surface temperatures of the slab can be seen in Fig. 10, together with the authentic
temperature measurements at points A, B, and C.
16
1400
1200
TsU
Temperature [oC]
1000
TsL
GA
GB
GC
800
600
400
200
z1
z2
0
0
1000
2000
z3
3000
z4
4000
z6
z5
5000
6000
7000
z7
8000
9000
Time [s]
Figure 10. The calculated surface temperatures at the top (TsU) and bottom (TsL) of the
slab based on the measured temperatures at points A, B, and C (GA,GB,GC) are shown.
The heat fluxes corresponding to the investigations with temperature data from
®
STEELTEMP 2D and from raw data only are shown in Figs. 11 a and b, respectively.
In the latter case, the curves corresponding to Fig.11b, are naturally noisier because more
information are inherited from the authentic temperature measurements. The results are
expected by looking at Fig. 6 and Fig. 8. In Zone 1 the heat flux is increasing and even
more in Zone 2 as the temperature in the furnace is increasing more steeply before it falls
of to a local minimum just before entrance of Zone 3. In Zone 3, the heat flux is again
rising. At about t = 5000 s the maximum furnace gas temperature has been reached. As
the steel slab is getting hotter and more uniformly heated, the temperature gradients
inside the slab decrease and consequently the heat flux curves also decline.
17
200
250
qU
180
qU
q
q
L
L
160
200
2
Heat flux [kW/m ]
Heat flux [kW/m2]
140
120
100
80
60
150
100
40
50
20
0
−20
z1
0
z2
2000
z3
z4
z5
4000
6000
z6
z7
8000
z1
z8
10000
Time [s]
0
0
z3
z2
1000
2000
3000
z4
4000
z5
5000
6000
z6
z7
7000
8000
9000
Time [s]
Figure 11. The calculated heat fluxes at the top and bottom of the slab surfaces based on the
®
temperatures at points A, B, and C from left) STEELTEMP 2D and right) raw data only.
4. Concluding remarks
In this thesis, the transient surface temperature of a slab has been estimated by solving an
inverse heat conduction problem using interior temperature measurements.
The assumption of one-dimensional heat conduction is a fairly good assumption apart
from near the edges of the slab where end effects may occur. The measurements inside
the slab are by nature noisy and may introduce errors together with the diffusive nature of
heat conduction.
The sampling rate was too low in the laboratory test. With a higher sample rate an
averaging filter would more effectively remove the random noise from the data, and also
the need for the initial re-sampling. For future experiments a much higher sampling rate
will be used. However, the results were satisfactory even at the sampling rate used.
The results obtained for the estimation of the surface temperatures in the two applications
are satisfactory and this shows that the method can be successfully applied to thermal
applications for wide range of temperatures.
18
Future work
There is further need for:
•
•
•
•
investigations to give greater clarity to the surface phenomena observed
investigations of the surface heat flux and some practical way of measure it to
compare it with the calculations
applications were also the oxide scale formation of slabs is taken into account
application of this method to determine the wall temperatures in reheating
furnaces would be of interest since this is an area where direct measurements are
applied but the results are questioned
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