Radiative Heat transfer and Applications for Glass Production Processes

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

2. 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

3. Parameter Identification Problems N. Siedow Fraunhofer-Institute for Industrial Mathematics, Kaiserslautern, Germany Montecatini, 15. – 19. October 2008

4. Parameter Identification ProblemsOutline • Introduction • Some Basics • Shape Optimization of Pipe Flanges • Impedance Tomography • Further Examples • Optimization of Thermal Stresses • Conclusions

5. Formally we can write or We have to calculate derivatives! Parameter Identification Problems 1. Some Basics Example 2: Parameter Identification Conductivity is unknown Measurement Additional information:

6. (Partial Differential Equation) subject to Lagrangian: Parameter Identification Problems 1. Some Basics A very convenient way of calculating derivatives is the Adjoint Method Derivatives: State Equation Adjoint Equation Parameter Equation

7. (Partial Differential Equation) subject to Parameter Identification Problems 1. Some Basics Example 2: Lagrangian: Derivatives: + b. c. State Equation + b. c. Adjoint Equation Parameter Equation

8. Parameter Identification Problems 1. Some Basics Example 2: (Partial Differential Equation) subject to

9. Parameter Identification Problems 1. Some Basics Example 2: subject to (Partial Differential Equation)

10. Parameter Identification Problems 2. Shape Optimization of Pipe Flanges • Objective: • Electric heating to keep the glass at desired temperature • Constraints: • Control temperature (e.g. to avoid solidification) • Control the input current (hot spots, cold shocks) • Shape Optimization

11. Parameter Identification Problems 2. Shape Optimization of Pipe Flanges Find a surface shape of the flange so, that in a small section near the pipe boundary: Under certain constraints: • Equation of electrical potential (1) • Heat transfer equation (2) • . . .

12. Parameter Identification Problems 2. Shape Optimization of Pipe Flanges 3D Model (1) (2)

13. - small parameter electrical potential: Parameter Identification Problems 2. Shape Optimization of Pipe Flanges Asymptotic Approach • Flange is very thin compared to the other dimensions • Assume: the shape of the flange is symmetric with respect to z

14. Electrical potential: + boundary conditions Heat transfer: + boundary conditions Parameter Identification Problems 2. Shape Optimization of Pipe Flanges Asymptotic Approach (3) (4)

15. Parameter Identification Problems 2. Shape Optimization of Pipe Flanges Parameter Optimization Find of the flange so, that in near the pipe Under certain constraints: • Equation of electrical potential (3) • Heat transfer equation (4) • Minimal material • . . .

16. Parameter Identification Problems 2. Shape Optimization of Pipe Flanges Lagrangian method: • Total Lagrangian • Necessity condition • Potential equation (3) • Adjoint potential equation • Heat transfer equation (4) • Adjoint heat transfer equation

17. Parameter Identification Problems 2. Shape Optimization of Pipe Flanges Example Thickness Before optimization After optimization

18. Parameter Identification Problems 2. Shape Optimization of Pipe Flanges Example Temperature Before optimization After optimization

19. Parameter Identification Problems 2. Shape Optimization of Pipe Flanges Example Heat Flux Before optimization After optimization

20. Parameter Identification Problems 3. Impedance Tomography Glass melting in a glass tank The knowledge of the temperature of the glass melt is important to control the homogeneity of the glass

21. Parameter Identification Problems 3. 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

22. Parameter Identification Problems 3. Impedance Tomography Determine the temperature of the glass melt during the melting process Experiment electrode Glass melt measure Voltage applyElectric current Neutral wire

23. Parameter Identification Problems 3. Impedance Tomography The forward Problem

24. Parameter Identification Problems 3. Impedance Tomography The forward Problem Electric potential Electric current density

25. Find so that • is solution of the potential equation • is solution of the (so-called) form equation Parameter Identification Problems 3. Impedance Tomography The inverse Problem under the constraints that Looking for a smooth solution

26. Parameter Identification Problems 3. Impedance Tomography The inverse Problem • Adjoint Form Equation • Potential Equation • Adjoint Potential Equation • New Form Function

27. Parameter Identification Problems 3. Impedance Tomography Example original Reconstruction

28. Parameter Identification Problems 4. Further Examples • Heat Transfer Coefficient Dip Experiment

29. Parameter Identification Problems 4. Further Examples • Heat Transfer Coefficient • Brinkmann, Siedow. „Heat Transfer between Glass and Mold During Hot Forming.“ In Krause, Loch: Mathematical Simulation in Glass Technology; Springer 2002

30. Initial condition • Boundary condition Control Problem Parameter Identification Problems 4. Further Examples

31. Parameter Identification Problems 6. Optimization of Thermal Stresses Wrong cooling of glass and glass products causes large thermal stresses Undesired crack

32. Parameter Identification Problems 6. Optimization of Thermal Stresses Heating of the glass at a temperature higher the transition temperature • Thermal tempering consists of: Very rapid cooling by an air jet Better mechanical and thermal strengthening to the glass by way of the residual stresses generated along the thickness • N. Siedow, D. Lochegnies, T. Grosan, E. Romero, J. Am. Ceram. Soc., 88 [8] 2181-2187 (2005)

33. Parameter Identification Problems 6. Optimization of Thermal Stresses How to control the heating to achieve a predefined temperature profile inside the glass products?

34. Parameter Identification Problems 6. Optimization of Thermal Stresses Minimize thermal stresses during the cooling process Linear Cooling Optimized Cooling

35. Parameter Identification Problems 6. Optimization of Thermal Stresses Minimize the cooling time with constraint that permanent thermal stress < 3.5 Mpa Used Cooling Thermal Stress

36. 691s -24% 915s Parameter Identification Problems 6. Optimization of Thermal Stresses Minimize the cooling time with constraint that permanent thermal stress < 3.5 Mpa Used Cooling Thermal Stress

37. Parameter Identification Problems 6. Optimization of Thermal Stresses Optimal Mould Design during Pressing How to design the mould that after cooling the glass lens has the desired shape? • Sellier, Breitbach, Loch, Siedow.“An iterative algorithm for optimal mould design in high-precision compression moulding.“ JEM606, IMechE Vol.221,2007, 25-33

38. Parameter Identification Problems 6. Optimization of Thermal Stresses Deviation of the upper glass side from the desired shape Deviation of the lower glass side from the desired shape

39. Parameter Identification Problems are inverse problems • Ill-posed Regularization Parameter Identification Problems 7. Conclusions • We have discussed different Parameter Identification Problems • For constraint optimization problems the Lagrangian approach is very convenient for calculating derivatives