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. ITWM Activities in GlassGlassmaking Shape optimization of thermal-electrical flanges Gob temperature (Spectral remote sensing) Temperature(Impedance Tomography)PATENT Coupling of glass tank with electrical network Form of the gob(FPM)

3. ITWM Activities in GlassGlassprocessing I PressingTV panels Lenses Interface Glass-Mould (Radiation) Identification of the heat transfer coefficient Floatglasswindow glasses display glasses High precision forming Blowing Bottles Wavyness of thin display glasses . . . Foaming Minimization of thermal stresses Optimal shape of the furnace Fiberproduction Fluid-Fiber-Interaction

4. ITWM Activities in GlassGlassprocessing II Simulation of temperature field Free cooling Tempering of glass Control of furnace temperature to minimize the thermal stress Cooling in a furnace

5. 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

6. Models for fast radiative heat transfer simulations N. Siedow Fraunhofer-Institute for Industrial Mathematics, Kaiserslautern, Germany Montecatini, 15. – 19. October 2008

7. Models for fast radiative heat transfer simulationsOutline • Introduction • Numerical methods for radiative heat transfer • Grey Absorption • Application to flat glass tempering • Conclusions

8. 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

9. Heat transfer on a microscale nm Conductivity in W/(Km) With Radiation mm - cm Without Radiation Temperature in °C Heat radiation on a macroscale Models for fast radiative heat transfer simulations 1. Introduction Radiation is for high temperatures the dominant process

10. Heat transfer on a microscale nm mm - cm Heat radiation on a macroscale Models for fast radiative heat transfer simulations 1. Introduction + boundary conditions

11. Heat transfer on a microscale nm mm - cm Heat radiation on a macroscale Models for fast radiative heat transfer simulations 2. Numerical methods for radiative heat transfer • Rosseland-Approximation • Radiation = Correction of Conductivity • PN-Approximation • Discrete-Ordinate-Method (FLUENT) • ITWM-Approximation-Method

12. Models for fast radiative heat transfer simulations 2. Numerical methods for radiative heat transfer Klar: We study the optically thick case. To obtain the dimensionless form of the rte we introduce and define the non-dimensional parameter which is small in the optically thick – diffusion – regime.

13. Models for fast radiative heat transfer simulations 2. Numerical methods for radiative heat transfer We rewrite the equation And apply Neumann‘s series to (formally) invert the operator Rosseland-Approximation

14. BUT • Standard method in glass industry Models for fast radiative heat transfer simulations 2. Numerical methods for radiative heat transfer Rosseland-Approximation • Treats radiation as a correction of heat conductivity • Very fast and easy to implement into commercial software packages • Only for optically thick glasses • Problems near the boundary

15. Heat transfer on a microscale nm mm - cm Heat radiation on a macroscale Models for fast radiative heat transfer simulations 2. Numerical methods for radiative heat transfer • Rosseland-Approximation • Radiation = Correction of Conductivity • PN-Approximation • Spherical Harmonic Expansion • Discrete-Ordinate-Method (FLUENT) • ITWM-Approximation-Method

16. e optical thickness (small parameter) Neumann series Models for fast radiative heat transfer simulations 2. Numerical methods for radiative heat transfer • Larsen, E., Thömmes, G. and Klar, A., , Seaid, M. and Götz, T., J. Comp. Physics 183, p. 652-675 (2002). • Thömmes,G., Radiative Heat Transfer Equations for Glass Cooling Problems: Analysis and Numerics. PhD, University Kaiserslautern, 2002

17. identical to P1-Approximation • SP3-Approximation O(e8) coupled system of equations Models for fast radiative heat transfer simulations 2. Numerical methods for radiative heat transfer • SP1-Approximation O(e4)

18. Models for fast radiative heat transfer simulations 2. Numerical methods for radiative heat transfer Example: Cooling of a glass plate Parameters: Density 2200 kg/m3Specific heat 900 J/kgKConductivity 1 W/KmThickness 1.0 mSurroundings 300 Kgray mediumAbsorption coefficient: 1/m

19. Heat transfer on a microscale nm mm - cm Heat radiation on a macroscale Models for fast radiative heat transfer simulations 2. Numerical methods for radiative heat transfer • Rosseland-Approximation • Radiation = Correction of Conductivity • PN-Approximation • Spherical Harmonic Expansion • Discrete-Ordinate-Method (FLUENT) • Full-discretization method Klar • ITWM-Approximation-Method

20. Heat transfer on a microscale nm mm - cm Heat radiation on a macroscale Models for fast radiative heat transfer simulations 2. Numerical methods for radiative heat transfer • Rosseland-Approximation • Radiation = Correction of Conductivity • PN-Approximation • Spherical Harmonic Expansion • Discrete-Ordinate-Method (FLUENT) • Full-discretization method • ITWM-Approximation-Method

21. Taylor Approximation with respect to Models for fast radiative heat transfer simulations 2. Numerical methods for radiative heat transfer • ITWM-Approximation-Method Formal solution: with

22. Models for fast radiative heat transfer simulations 2. Numerical methods for radiative heat transfer • ITWM-Approximation-Method Formal solution: with

23. Models for fast radiative heat transfer simulations 2. Numerical methods for radiative heat transfer • ITWM-Approximation-Method Formal solution: with Rosseland:

24. Models for fast radiative heat transfer simulations 2. Numerical methods for radiative heat transfer • ITWM-Approximation-Method Formal solution: with

25. Models for fast radiative heat transfer simulations 2. Numerical methods for radiative heat transfer Improved Diffusion Approximation • In opposite to Rosseland-Approximation all geometrical information is conserved • Lentes, F. T., Siedow, N., Glastech. Ber. Glass Sci. Technol. 72 No.6 188-196 (1999).

26. Models for fast radiative heat transfer simulations 2. Numerical methods for radiative heat transfer Improved Diffusion Approximation • Correction to the heat conduction due to radiation with anisotropic diffusion tensor • Lentes, F. T., Siedow, N., Glastech. Ber. Glass Sci. Technol. 72 No.6 188-196 (1999).

27. Models for fast radiative heat transfer simulations 2. Numerical methods for radiative heat transfer Improved Diffusion Approximation • Boundary conditions • Convection term

28. Models for fast radiative heat transfer simulations 2. Numerical methods for radiative heat transfer Two Scale Asymptotic Analysis for the Improved Diffusion Approximation so that Introduce

29. Models for fast radiative heat transfer simulations 2. Numerical methods for radiative heat transfer Two Scale Asymptotic Analysis for the Improved Diffusion Approximation Ansatz: Comparing the coefficients one obtains the Improved Diffusion Approximation • F. Zingsheim. Numerical solution methods for radiative heat transfer in semitransparent media. PhD, University of Kaiserslautern, 1999

30. Alternatively we use the rte Formal Solution Approximation Models for fast radiative heat transfer simulations 2. Numerical methods for radiative heat transfer • N. Siedow, D. Lochegnies, T. Grosan, E. Romero, J. Am. Ceram. Soc., 88 [8] 2181-2187 (2005)

31. Models for fast radiative heat transfer simulations 2. Numerical methods for radiative heat transfer Example: Heating of a glass plate Parameters: Density 2500 kg/m3Specific heat 1250 J/kgKConductivity 1 W/KmThickness 0.005 mSemitransparent Region:0.01 µm – 7.0 µm Absorption coefficient:0.4 /m … 7136 /m (8 bands) Wall T=800°C Glass T0=200°C Wall T=600°C

32. Models for fast radiative heat transfer simulations 2. Numerical methods for radiative heat transfer Example: Heating of a glass plate Computational time for 3000 time steps Exact 81.61 s Ida 00.69 s Fsa 00.69 s

33. Models for fast radiative heat transfer simulations 2. Numerical methods for radiative heat transfer Example: Cooling of a glass plate

34. adiabatic T=1800 K 1 m T=1300 K adiabatic 5 m Models for fast radiative heat transfer simulations 2. Numerical methods for radiative heat transfer Radiation and natural convection (FLUENT) Example: gravity Radiation with diffusely reflecting gray walls in a gray material

35. Models for fast radiative heat transfer simulations 2. Numerical methods for radiative heat transfer Radiation and natural convection (FLUENT) Example: Diffusely reflecting gray walls in a gray material FLUENT-DOM ITWM-UDF >5000 Iterations 86 Iterations

36. „Grey Kappa“ • Reduce the number of unknowns („Find a wavelength independend absorption coefficient?“) Models for fast radiative heat transfer simulations 3. Grey Absorption The numerical solution of the radiative transfer equation is very complex Discretization: • 60 angular variables • 10 wavelength bands • 12 million unknowns • 20,000 space points Not suitable for optimization • Development of fast numerical methods

37. Problem: • many frequency bands yield many equations • Averaging the SPN equations over frequency is possible, yields nonlinear coefficients. • POD approaches are possible as well. Models for fast radiative heat transfer simulations 3. Grey Absorption Klar: Remark – Frequency averages

38. Models for fast radiative heat transfer simulations 3. Grey Absorption Typical absorption spectrum of glass

39. Models for fast radiative heat transfer simulations 3. Grey Absorption One-dimensional test example: • Thickness 0.1m • Refractive index 1.0001 Source term for heat transfer is the divergence of radiative flux vector

40. Models for fast radiative heat transfer simulations 3. Grey Absorption Values from literature: Rosseland-mean absorption coefficient Planck-mean absorption coefficient

41. Models for fast radiative heat transfer simulations 3. Grey Absorption Values from literature: Rosseland-mean absorption coefficient Planck-mean absorption coefficient

42. Models for fast radiative heat transfer simulations 3. Grey Absorption Comparison between Planck-mean and Rosseland-mean Good approximation for the boundary with Planck Good approximation for the interior with Rosseland

43. Models for fast radiative heat transfer simulations 3. Grey Absorption The existence of the exact “Grey Kappa” • We integrate the radiative transfer equation with respect to the wavelength • We define an ersatz (auxiliary) equation: • If then

44. Models for fast radiative heat transfer simulations 3. Grey Absorption The existence of the exact “Grey Kappa” • The “Grey Kappa” is not depending on wavelength BUT on position and direction • The “Grey Kappa” can be calculated, if we know the solution of the rte AND How to get rid of the direction? How to approximate the intensity?

45. Models for fast radiative heat transfer simulations 3. Grey Absorption How to approximate the intensity? We use once more the formal solution How to get rid of direction?

46. Models for fast radiative heat transfer simulations 3. Grey Absorption New (approximated) „grey kappa“ can be formulated as Planck-Rosseland-Superposition Planck-mean value Rosseland-mean value

47. Models for fast radiative heat transfer simulations 3. Grey Absorption Example of a 0.1m tick glass plate with initial temperature 1500°C

48. Models for fast radiative heat transfer simulations 3. Grey Absorption Example of a 0.1m tick glass plate with initial temperature 1500°C

49. Models for fast radiative heat transfer simulations 3. Grey Absorption Summary: • For the test examples the Planck-Rosseland-Superposition mean value gives the best results • For the optically thin case: PRS PlanckFor the optically thick case: PRS Rosseland Stored for different temperatures in a table Calculated in advanced

50. Models for fast radiative heat transfer simulations 3. Grey Absorption Summary: • For the test examples the Planck-Rosseland-Superposition mean value gives the best results • For the optically thin case: PRS PlanckFor the optically thick case: PRS Rosseland • These are ideas! – Further research is needed!