Glass 1 Radiative Heat transfer and Applications for Glass Production Processes Axel Klar and Norbert Siedow Department of Mathematics, TU Kaiserslautern Fraunhofer ITWM Abteilung Transport processes Montecatini, 15. – 19. October 2008

Glass 2 Radiative Heat transfer and Applications for Glass Production Processes Planning of the Lectures 1.Models for fast radiative heat transfer simulation 2.Indirect Temperature Measurement of Hot Glasses 3.Parameter Identification Problems

Glass 3 Indirect Temperature Measurement of Hot Glasses N. Siedow Fraunhofer-Institute for Industrial Mathematics, Kaiserslautern, Germany Montecatini, 15. – 19. October 2008

Glass 4 Indirect Temperature Measurement of Hot Glasses Outline 1.Introduction 2.Some Basics of Inverse Problems 3.Spectral Remote Sensing 4.Reconstruction of the Initial Temperature 5.Impedance Tomography 6.Conclusions

Glass 5 Models for fast radiative heat transfer simulations 1. Introduction Temperature is the most important parameter in all stages of glass production  Homogeneity of glass melt  Drop temperature  Thermal stress To determine the temperature:  Measurement  Simulation

Glass 6 Indirect Temperature Measurement of Hot Glasses 1. Introduction With Radiation Without Radiation Temperature in °C Conductivity in W/(Km) Radiation is for high temperatures the dominant process Heat transfer on a microscale Heat radiation on a macroscale mm - cm nm

Glass 7 Indirect Temperature Measurement of Hot Glasses 1. Introduction Heat transfer on a microscale Heat radiation on a macroscale mm - cm nm + boundary conditions

Glass 8 Indirect Temperature Measurement of Hot Glasses 1. Introduction Direct Measurement Thermocouples Indirect Measurement Pyrometer (surface temperature) Spectral Remote Sensing

Glass 9 Indirect Temperature Measurement of Hot Glasses 1. Introduction Glass is semitransparent Inverse Problem Spectrometer T(z) [µm] Emissivity [°C] Depth [mm] Spectral Remote Sensing

Glass 10 Inverse Problems are concerned with finding causes for an observed or a desired effect.  Control or Design, if one looks for a cause for an desired effect.  Identification or Reconstruction, if one looks for the cause for an observed effect. Indirect Temperature Measurement of Hot Glasses 2. Some Basics of Inverse Problems

Glass 11 Example 1: Black Box Input Signal Output Signal - Measurement Assume: If:Continuous differentiable Solution: Indirect Temperature Measurement of Hot Glasses 2. Some Basics of Inverse Problems

Glass 12 Example 1: We find:Given is: analytically exact Indirect Temperature Measurement of Hot Glasses 2. Some Basics of Inverse Problems

Glass 13 Example 1: A small error in the measurement causes a big error in the reconstruction! Indirect Temperature Measurement of Hot Glasses 2. Some Basics of Inverse Problems

Glass 14 Example 1:Numerical Differentiation  In praxis the measured data are finite and not smooth Indirect Temperature Measurement of Hot Glasses 2. Some Basics of Inverse Problems

Glass 15 Example 1:Numerical Differentiation A finer discretization leads to a bigger error Indirect Temperature Measurement of Hot Glasses 2. Some Basics of Inverse Problems

Glass 16 Example 2:Parameter Identification Practical meaning:  Heat transfer equation Temperature Thermal conductivity  Diffusion equation Concentration Diffusivity  „Black-Scholes“ equation Option price Stock price Indirect Temperature Measurement of Hot Glasses 2. Some Basics of Inverse Problems

Glass 17 Example 2:Parameter Identification Practical meaning:  Electrical potential equation Electrical potential Electrical conductivity Knowing the potential find the conductivity  Elasticity equation displacement Youngs Modulus Indirect Temperature Measurement of Hot Glasses 2. Some Basics of Inverse Problems

Glass 18 Example 2:Parameter Identification Exact Measurement Indirect Temperature Measurement of Hot Glasses 2. Some Basics of Inverse Problems

Glass 19 Example 2:Parameter Identification Noisy Measurement Indirect Temperature Measurement of Hot Glasses 2. Some Basics of Inverse Problems

Glass 20 Example 2:Parameter Identification Noisy Measurement Indirect Temperature Measurement of Hot Glasses 2. Some Basics of Inverse Problems

Glass 21 Example 4: Reconstruction: Indirect Temperature Measurement of Hot Glasses 2. Some Basics of Inverse Problems

Glass 22 A common property of a vast majority of Inverse Problems is their ill-posedness A mathematical problem is well-posed, if Hadamard ( ) 1.For all data, there exists a solution of the problem. 2.For all data, the solution is unique. 3.The solution depends continuously on the data. A problem is ill-posed if one of these three conditions is violated. Indirect Temperature Measurement of Hot Glasses 2. Some Basics of Inverse Problems

Glass 23 What is the reason for the ill-posedness? Example 1: A small error in measurement causes a big error in reconstruction Indirect Temperature Measurement of Hot Glasses 2. Some Basics of Inverse Problems

Glass 24 What is the reason for the ill-posedness? Example 1: Step size must be taken with respect to the measurement error Indirect Temperature Measurement of Hot Glasses 2. Some Basics of Inverse Problems

Glass 25 What is the reason for the ill-posedness? Example 2: Numerical differentiation of noisy data Indirect Temperature Measurement of Hot Glasses 2. Some Basics of Inverse Problems

Glass 26 What is the reason for the ill-posedness? Example 4: Eigenvalues: Condition number: Indirect Temperature Measurement of Hot Glasses 2. Some Basics of Inverse Problems

Glass 27 What is the reason for the ill-posedness? Example 4: Eigenvalues: Let be the eigenvectors The solution can be written as: A small error in Indirect Temperature Measurement of Hot Glasses 2. Some Basics of Inverse Problems

Glass 28 What can be done to overcome the ill-posedness?Regularization Regularization Methods 1.Truncated Singular Value Decomposition  We skip the small eigenvalue (singular values) identical to the minimization problem and take the solution with minimum norm  Replace the ill-posed problem by a family of neighboring well-posed problems Indirect Temperature Measurement of Hot Glasses 2. Some Basics of Inverse Problems

Glass 29 Regularization Methods 2.Tichonov (Lavrentiev) Regularization  We look for a problem which is near by the original and well-posed We increase the eigenvalues How to choose ? Indirect Temperature Measurement of Hot Glasses 2. Some Basics of Inverse Problems

Glass 30 Regularization Methods 2.Tichonov (Lavrentiev) Regularization Take  L-curve method Indirect Temperature Measurement of Hot Glasses 2. Some Basics of Inverse Problems

Glass 31 Regularization Methods 3.Landweber Iteration We consider the normal equation given Use a fixed point iteration to solve Iteration number plays as regularization parameter Stopping rule = discrepancy principle Solution after 4 iterations: Indirect Temperature Measurement of Hot Glasses 2. Some Basics of Inverse Problems

Glass 32 Regularization Methods 4.Classical Tichonov Regularization Regularization of the normal equation Equivalent to the minimization problem Tichonov ( )  Dealing with an ill-posed problem means to find the right balance between stability and accuracy Indirect Temperature Measurement of Hot Glasses 2. Some Basics of Inverse Problems

Glass 33 Regularization Methods 4.Classical Tichonov Regularization Regularization of the normal equation Equivalent to the minimization problem  L-curve  Discrepancy principle Indirect Temperature Measurement of Hot Glasses 2. Some Basics of Inverse Problems

Glass 34 Regularization Methods 4.Classical Tichonov Regularization Regularization of the normal equation Equivalent to the minimization problem To get a better solution we need to include more information! Assume: Indirect Temperature Measurement of Hot Glasses 2. Some Basics of Inverse Problems

Glass 35 Indirect Temperature Measurement of Hot Glasses 3. Spectral Remote Sensing formal solution: Non-linear, ill-posed integral equation of 1. kind One-dimensional radiative transfer equation

Glass 36 Indirect Temperature Measurement of Hot Glasses 3. Indirect temperature measurement Formal solution: Linearization: Regularization:

Glass 37 Indirect Temperature Measurement of Hot Glasses 3. Indirect temperature measurement (Rosseland-Approximation) is FDA of Temperature satisfies the radiative heat transfer equation Radiative Flux Iteratively regularized Gauss – Newton method: How to choose ? ?

Glass 38 Indirect Temperature Measurement of Hot Glasses 3. Indirect temperature measurement Discrepancy principle Iteratively regularized Gauss – Newton method: How to choose ? ? Stopping rule for k? ?

Glass 39 Indirect Temperature Measurement of Hot Glasses 3. Indirect temperature measurement Furnace Experiment Furnace Glass slabThermocouples

Glass 40 Indirect Temperature Measurement of Hot Glasses 3. Indirect temperature measurement Drop Temperature

Glass 41 Indirect Temperature Measurement of Hot Glasses 3. Indirect temperature measurement The Improved Eddington-Barbier-Approximation If we assume that

Glass 42 Indirect Temperature Measurement of Hot Glasses 3. Indirect temperature measurement The Improved Eddington-Barbier-Approximation If we assume that

Glass 43 Indirect Temperature Measurement of Hot Glasses 3. Indirect temperature measurement A fast iterative solution of the integral equation i.For Calculate using the IEB-method iii.For Use to calculate ii.Using some additional information continue to

Glass 44 Indirect Temperature Measurement of Hot Glasses 3. Indirect temperature measurement A fast iterative solution of the integral equation v.Calculate a new temperature profile in the IEB points using vi.Using the additional information continue to and go back iii. iv.If then STOP else continue with v.

Glass 45 Indirect Temperature Measurement of Hot Glasses 3. Indirect temperature measurement Iteration 1

Glass 46 Indirect Temperature Measurement of Hot Glasses 3. Indirect temperature measurement Iteration 2

Glass 47 Indirect Temperature Measurement of Hot Glasses 3. Indirect temperature measurement Iteration 3

Glass 48 Indirect Temperature Measurement of Hot Glasses 3. Indirect temperature measurement Iteration 4

Glass 49 Indirect Temperature Measurement of Hot Glasses 3. Indirect temperature measurement Iteration 5

Glass 50 Indirect Temperature Measurement of Hot Glasses 3. Indirect temperature measurement Iteration 10

Glass 51 Indirect Temperature Measurement of Hot Glasses 3. Indirect temperature measurement Error 1%

Glass 52 Indirect Temperature Measurement of Hot Glasses 3. Indirect temperature measurement Error 1%

Glass 53 Indirect Temperature Measurement of Hot Glasses 4. Reconstruction of Initial Condition Initial condition Boundary condition Temperature + Additional Measurements of boundary temperature / heat flux Is it possible to reconstruct the initial temperature distribution from boundary measurements? ?

Glass 54 Indirect Temperature Measurement of Hot Glasses 4. Reconstruction of Initial Condition Additional measurement at boundary Problem is ill-posed: Solution exists Solution is unique Continuous dependence on right hand side is violated

Glass 55 Indirect Temperature Measurement of Hot Glasses 4. Reconstruction of Initial Condition Backward Heat Sideways Heat New Heat Problem Given: 2 conditions at y 2 Additional bc at yTemperature at T Looking for: bc at y 1 Initial condition

Glass 56 Indirect Temperature Measurement of Hot Glasses 4. Reconstruction of Initial Condition Problem is ill-posed: Solution exists Solution is unique Continuous dependence on right hand side is violated For One obtains

Glass 57 Indirect Temperature Measurement of Hot Glasses 4. Reconstruction of Initial Condition Problem is ill-posed: Solution exists Solution is unique Continuous dependence on right hand side is violated For One obtains

Glass 58 Given data Indirect Temperature Measurement of Hot Glasses 4. Reconstruction of Initial Condition Solve the system by Tichonov regularization While searching for the correct Solve Adjust according to the parameter choice rule with by some method is the regularized solution

Glass 59 Indirect Temperature Measurement of Hot Glasses 4. Reconstruction of Initial Condition  L. Justen. An Inverse Heat Conduction Problem with Unknown Initial Condition. Diploma Thesis, TU KL, 2002 Exact solution: Tichonov regularization with Morozov‘s Discrepancy stopping rule

Glass 60 Indirect Temperature Measurement of Hot Glasses 4. Reconstruction of Initial Condition 2D Example:Parabolic Profile – Four-Sided Measurement

Glass 61 Indirect Temperature Measurement of Hot Glasses 4. Reconstruction of Initial Condition

Glass 62 Indirect Temperature Measurement of Hot Glasses 4. Reconstruction of Initial Condition  S.S. Pereverzyev, R. Pinnau, N. Siedow. Proceedings of 5th Conf. on Inverse Problems in Engineering, 2005  S.S. Pereverzyev, R. Pinnau, N. Siedow. Inverse Problems, 22 (2006), 1-22 Initial Temperature Reconstruction for a Nonlinear Heat Equation: Application to Radiative Heat TransferPhD S.S. Pereverzyev

Glass 63 Indirect Temperature Measurement of Hot Glasses 4. Reconstruction of Initial Condition  S.S. Pereverzyev, R. Pinnau, N. Siedow. Proceedings of 5th Conf. on Inverse Problems in Engineering, 2005  S.S. Pereverzyev, R. Pinnau, N. Siedow. Inverse Problems, 22 (2006), 1-22 Decomposition of the non-linear equation: Linearpartnon-linear measurement Fixed-point iteration: Tichonov regularization:

Glass 64 Indirect Temperature Measurement of Hot Glasses 4. Reconstruction of Initial Condition Parameter choice rule: Quasi-optimality criterion It does not depend on the noise level 1. Select a finite number of regularization parameters which are part of a geometric sequence, i.e. 2. For each solveand obtain 3. Among choosesuch that

Glass 65 Indirect Temperature Measurement of Hot Glasses 4. Reconstruction of Initial Condition

Glass 66 Indirect Temperature Measurement of Hot Glasses 4. Reconstruction of Initial Condition Reconstruction is dependent on noise level discretization How to choice the right discretization parameters depending on the noise level?

Glass 67 Indirect Temperature Measurement of Hot Glasses 5. Impedance Tomography The knowledge of the temperature of the glass melt is important to control the homogeneity of the glass Glass melting in a glass tank

Glass 68 Indirect Temperature Measurement of Hot Glasses 5. Impedance Tomography Thermocouples at the bottom and the sides of the furnace Use of pyrometers is limited due to the atmosphere above the glass melt

Glass 69 Glass melt Determine the temperature of the glass melt during the melting process using an impedance tomography approach apply Electric current measure Voltage Neutral wire Experiment electrode Indirect Temperature Measurement of Hot Glasses 5. Impedance Tomography

Glass 70  Inverse Problems are concerned with finding causes for an observed or a desired effect.  A common property of a vast majority of Inverse Problems is their ill-posedness (Existence, Uniqueness, Stability)  To solve an ill-posed problem one has to use regularization techniques (Replace the ill-posed problem by a family of neighboring well-posed problems)  The regularization has to be taken in accordance with the problem one wants to solve Indirect Temperature Measurement of Hot Glasses 6. Conclusions

Glass 71 Further Examples of Inverse Problems:  Computerized Tomography  Inverse Scattering Indirect Temperature Measurement of Hot Glasses 2. Some Basics of Inverse Problems

Glass 72 Example 3:  If exists no solution  If exist an infinite number of solutions Gauge condition Integration Indirect Temperature Measurement of Hot Glasses 2. Some Basics of Inverse Problems

Glass 73 Example 4: Exact Solution: Indirect Temperature Measurement of Hot Glasses 2. Some Basics of Inverse Problems

Glass 74 What is the reason for the ill-posedness? Example 3: System is singular Indirect Temperature Measurement of Hot Glasses 2. Some Basics of Inverse Problems