Radiative Heat transfer and Applications for Glass Production Processes

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

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

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

Indirect Temperature Measurement of Hot GlassesOutline • Introduction • Some Basics of Inverse Problems • Spectral Remote Sensing • Reconstruction of the Initial Temperature • Impedance Tomography • Conclusions

To determine the temperature: 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 • Measurement • Simulation

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

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

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

1 1000 [°C] 0.8 Emissivity 0.6 950 0.4 0.2 0 900 0 1 2 3 4 0 1 2 3 4 Depth [mm] l [µm] Indirect Temperature Measurement of Hot Glasses 1. Introduction Glass is semitransparent Spectral Remote Sensing T(z) Spectrometer Inverse Problem

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

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

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

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

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

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

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

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

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

Hadamard (1865-1963) Indirect Temperature Measurement of Hot Glasses 2. Some Basics of Inverse Problems A common property of a vast majority of Inverse Problems is theirill-posedness A mathematical problem is well-posed, if • For all data, there exists a solution of the problem. • For all data, the solution is unique. • 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 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 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 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 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 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

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

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

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

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 Regularization Methods • Landweber Iteration We consider the normal equation

Regularization of the normal equation Tichonov (1906-1993) • 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 Regularization Methods • Classical Tichonov Regularization Equivalent to the minimization problem

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

Indirect Temperature Measurement of Hot Glasses 2. Some Basics of Inverse Problems Regularization Methods • 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 3. Spectral Remote Sensing One-dimensional radiative transfer equation formal solution: Non-linear, ill-posed integral equation of 1. kind

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

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

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

Indirect Temperature Measurement of Hot Glasses 3. Indirect temperature measurement Furnace Experiment Furnace Glass slab Thermocouples

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

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

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

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

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

Radiative Heat transfer and Applications for Glass Production Processes