1. 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

2. Glass 2 ITWM Activities in Glass Glassmaking Form of the gob (FPM) Shape optimization of thermal-electrical flanges Gob temperature (Spectral remote sensing) Coupling of glass tank with electrical network Temperature (Impedance Tomography) PATENT

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

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

5. Glass 5 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

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

7. Glass 7 Models for fast radiative heat transfer simulations Outline 1.Introduction 2.Numerical methods for radiative heat transfer 3.Grey Absorption 4.Application to flat glass tempering 5.Conclusions

8. Glass 8 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

9. Glass 9 Models for fast radiative heat transfer simulations 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

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

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

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

13. Glass 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. Glass 14 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 Models for fast radiative heat transfer simulations 2. Numerical methods for radiative heat transfer Rosseland-Approximation BUT Standard method in glass industry

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

16. Glass 16 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  optical thickness (small parameter) Neumann series

17. Glass 17 Models for fast radiative heat transfer simulations 2. Numerical methods for radiative heat transfer  SP 1 -ApproximationO(  4 )  SP 3 -ApproximationO(  8 ) identical to P 1 -Approximation coupled system of equations

18. Glass 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/m 3 Specific heat 900 J/kgK Conductivity 1 W/Km Thickness 1.0 m Surroundings300 K gray medium Absorption coefficient:1/m

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

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

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

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

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

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

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

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

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

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

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

31. Glass 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/m 3 Specific heat 1250 J/kgK Conductivity 1 W/Km Thickness 0.005 m Semitransparent Region: 0.01 µm – 7.0 µm Absorption coefficient: 0.4 /m … 7136 /m (8 bands) Wall T=800°C Wall T=600°C Glass T 0 =200°C

32. Glass 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 Exact81.61 s Ida00.69 s Fsa00.69 s

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

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

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

36. Glass 36 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 20,000 space points 12 million unknowns Not suitable for optimization  Development of fast numerical methods  Reduce the number of unknowns „Grey Kappa“ („Find a wavelength independend absorption coefficient?“)

37. Glass 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. Klar:Remark – Frequency averages Models for fast radiative heat transfer simulations 3. Grey Absorption

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

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

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

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

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

43. Glass 43 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 Models for fast radiative heat transfer simulations 3. Grey Absorption

44. Glass 44 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 How to approximate the intensity? How to get rid of the direction? AND Models for fast radiative heat transfer simulations 3. Grey Absorption

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

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

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

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

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

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

51. Glass 51 Models for fast radiative heat transfer simulations 4. Application to flat glass tempering Wrong cooling of glass and glass products causes large thermal stresses Undesired crack

52. Glass 52  Thermal tempering consists of: Models for fast radiative heat transfer simulations 4. Application to flat glass tempering  Heating of the glass at a temperature higher the transition temperature  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)

53. Glass 53  Cooling of the glass melt depends on the temperature distribution in time and space  Characteristically for glass: No fixed point where glass changes from fluid to solid state There exists a temperature range The essential property is the viscosity of the glass temperature low high high viscosity low viscosity Linear-elastic material Newtonian fluid Models for fast radiative heat transfer simulations 4. Application to flat glass tempering

54. Glass 54  Viscosity changes the density depending on the temperature  Change in density (structural relaxation) influences the stress inside the glass  A numerical model for the calculation of transient and residual stresses in glass during cooling, including both structural relaxation and viscous stress relaxation, has been developed by Narayanaswamy und Tool  Commercial software packages like ANSYS and ABAQUS have implemented this model Models for fast radiative heat transfer simulations 4. Application to flat glass tempering

55. Glass 55 Models for fast radiative heat transfer simulations 4. Application to flat glass tempering  N. Siedow, D. Lochegnies, T. Grosan, E. Romero, J. Am. Ceram. Soc., 88 [8] 2181-2187 (2005) ITWM model gives the closest result for temperature

56. Glass 56 Models for fast radiative heat transfer simulations 4. Application to flat glass tempering  N. Siedow, D. Lochegnies, T. Grosan, E. Romero, J. Am. Ceram. Soc., 88 [8] 2181-2187 (2005) Rosseland gives the worst surface and mid-plan temperature difference CPU time in s: ITWM model comparable with Rosseland and much faster than exact solution model

57. Glass 57 Models for fast radiative heat transfer simulations 4. Application to flat glass tempering  N. Siedow, D. Lochegnies, T. Grosan, E. Romero, J. Am. Ceram. Soc., 88 [8] 2181-2187 (2005) ITWM model gives the closest result for transient and residual stresses

58. Glass 58 Models for fast radiative heat transfer simulations 4. Application to flat glass tempering Production of bodies, like cubes, cylinders, angles („Kipferl“), …. Special products by post- processing (grinding) of these simple geometrical pieces Deformation after cooling

59. Glass 59 Models for fast radiative heat transfer simulations 5. Application to flat glass tempering

60. Glass 60 Models for fast radiative heat transfer simulations 5. Conclusions 1.Temperature is one of the main parameters to make „good“ glasses 2.To simulate the temperature behavior of glass radiation must be taken into account 3.One needs good numerics to solve practical relevant radiative transfer problems - Improved Diffusion Approximation methods are alternative approaches for simulating the temperature behavior in glass 4.A grey absorption coefficient can save CPU time 5.The right temperature profile is necessary to simulate stresses during glass cooling

61. Glass 61