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

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To my beloved Jenny and my son Hugo

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Abstract The topic of this thesis is estimation of the dynamic changes of the surface temperature- and 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.

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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 service-minded 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.

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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

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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

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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.

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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

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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 Xx∈ be the desired unknown solution and Yy∈ be the available data. Define a mapping through the bounded operator K such that

KK RDK: → , where XDK ⊆ and YRK ⊆ . Then, the operator equation

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Kxy = (1)

is well-posed in the sense of Hadamard if the following statements hold:

• for every Yy∈ there is at least one Xx∈ such that yKx = (2) • for every Yy∈ there is at most one Xx∈ such that y Kx = (3) • the solution x depends continuously on y such that for every

sequence { } .nx xKx X: Kxx nnn ∞→→⇒→∈ when (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, 1−K , is continuous. For the case of the inverse heat conduction problem (IHCP) the corresponding operator equation has an unbounded inverse. Hence the problem is ill-posed 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

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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 ) ,( txT for 10 Lx

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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 1Lx = is computed by solving a well-posed problem in the interval ,21 LxL

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Figure 2. Schematic figure of the heat conduction problem divided into zones of direct- and 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

,- dt, )()exp(21)(ˆ ∞

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Since the real part of ξ⋅i is non-negative, (10) represents an unbounded operator. Small errors in high frequency components of the data function mf are may blow up and totally destroy the solution in the interval 10 Lx

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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 Si Mn S P Cr Ni Mo Cu Al Wt-% 0.06 0.01 0.38 0.035 0.017 0.022 0.055 0.030 0.08 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.

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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 mm 0=x as shown in Fig. 4. The original data vectors were sampled only at Hz 25.0 . 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.

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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 mm 220150011000 ×× and thermocouples were positioned at locations A, B, and C as shown in Fig. 5.

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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 Hz 601 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 cut-off frequency was set to 100=cξ . 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 70=cξ , 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

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shows the calculated temperature in points A, B, and C of the slab and the furnace gas- and 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 1xx = , referring to Fig. 4. That is, an inverse heat conduction problem was solved in the interval 21 xxx

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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 100=cξ 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 C 24 o . Since the temperature ranged from C 125025 o− this is a very good result.

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Figure 7. The calculated temperature (solid) and the measured temperature history (dashed) at mm 51 =x 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.

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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 (20-

C1350o ). On average the difference was of order 5°C .The conclusion drawn is that the verification of SHESOLV against STEELTEMP® 2D was successful.

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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.

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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 s 5000t = 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.

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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.

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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

References [1]. J. Hadamard. Lectures on Cauchy’s problem in linear partial differential equations.

Dover Publications, New Yourk, 1953. [2]. V. Maz'ya and T. Shaposhnikova, Jacques Hadamard, a universal mathematician,

American Mathematical Society, 1998. [3]. G. Jr., Stolz, Numerical Solutions to an Inverse Problem of Heat Conduction for

Simple Shapes, J. Heat Transfer (1960), 82:20-26. [4]. J. V. Beck and B. Blackwell, Inverse Heat Conduction. Ill-Posed Problems, Wiley,

New York, 1985. [5]. D. A. Murio, The Mollification Method and the Numerical Solution of Ill-posed

Problems, Wiley, New York, 1993. [6]. P. Lamm, Approximation of ill-posed Volterra problems via predictor-corrector

regularization methods. SIAM J. Appl. Math. (1996), 56(2):524-54. [7]. P. Lamm and L. Elden, Numerical Solution of First-Kind Volterra Equations by

Sequential Tikhonov Regularization, SIAM J. Numer. Anal. (1997), 34:1432-1450. [8]. F. Berntsson. A spectral method for solving the sideways heat equation, Inverse

Problems (1999), 15:891-906. [9]. Lars Elden, F. Berntsson, T. Reginska, Wavelet and Fourier methods for solving the

sideways heat equation, SIAM J. Sci. Comput. (2000), 21(6):2187-2205.

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[10]. J.V. Beck and K.J.Arnold, Parameter Estimation in Engineering and Science, Wiley, New York, 1977.

[11]. R. Luce and D. Perez, Parameter identification for an elliptic partial differential

equation with distributed noisy data, Inverse Problems (1995), 15(1):291-307. [12]. T. Jurkowski, Y. Jarny and D.Delaunay, Estimation of thermal conductivity of

thermoplastics under moulding conditions: an apparatus and an inverse algorithm, Int.J.Heat Mass Transfer (1997), 40:4169-4181.

[13]. J. Zueco, F. Alhama, C. F. G. Fernandez, An inverse problem to estimate

temperature dependent heat capacity under convection processes, Heat and Nass Transfer (2003), 39:599-604.

[14]. T. Kim and Z. Lee, Time-varying heat transfer coefficients between tube-shaped

casting and metal mold, Int. J. Heat Mass Transfer (1997), 40(15):3513-3525. [15]. J. Su and G. F. Hewitt, Inverse heat conduction problem of estimating time-varying

heat transfer coefficient, Numerical Heat Transfer, Part A (2004), 45: 777-789. [16]. A. P. Oliveira and H. R. B. Orlande, Estimation of the heat flux at the surface of

ablating materials by using temperature and surface position measurements, 4th International Conference on Inverse Problems in Engineering, Rio de Janeiro, Brazil, 2002.

[17]. C. Chen, L. Wu and Y. Yang, Estimation of unknown outer-wall heat flux in

turbulent circular pipe flow with conduction in the pipe wall, Int. J. Mass Heat Transfer (2005), 48:3971-3981.

[18]. A. Kusiak, J. Battaglia, R. Marchal, Influence of CrN coating in wood machining

from heat flux estimation in the tool, Int. J. Thermal Sciences (2005), 44:289-301. [19]. C. H. Huang, T. M. Ju, A.A. Tseng, The estimation of surface thermal behaviour of

the working roll in hot rolling process, International Journal of Heat and Mass Transfer, 1997 (Submitted).

[20]. Russel G.Keanini, Inverse estimation of surface heat flux distributions during high

speed rolling using remote thermal measurements, Int. J. Heat Mass Transfer (1998), 41(2):275-285.

[21]. T. P.Fredman, A boundary identification method for an inverse heat conduction

problem with an application in ironmaking, Heat and Mass Transfer (2004), 41: 95-103.

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[22]. M. Honner, Z. Vesely, M. Svantner, Temperature and heat transfer measurement in continuous reheating furnaces, Scandinavian Journal of Metallurgy (2003), 32:225-232.

[23]. F.Berntsson, L.Elden, An Inverse Heat Conduction Problem and an Application to

Heat Treatment of Aluminium, in: M.Tanaka, G.S. Dulikravich (Ed.), Inverse Problems in Engineering Mechanics II. International Symposium on Inverse Problem Engineering Mechanics 2000 (ISIP 2000), Elsevier, Nagano, Japan, (2000), pp. 99-106.

[24]. H.Engl, M.Hanke, A.Neubauer, Regularization of Inverse Problems, Kluwer

Academic Publishers, Netherlands, 1996. [25]. B.Leden, STEELTEMP®- A program for temperature analysis in steel plants,

Scand. J. Metallurgy(1986), 15:215-223.