1. Simulation of radiative heat transfer in participating media with simplified spherical harmonics Ralf Rettig, University of Erlangen Ferienakademie Sarntal 18/09 – 30/09/2005

2. RADIATIVE HEAT TRANSFER WITH SIMPLIFIED SPHERICAL HARMONICS 18/09 – 30/09/2005Ralf Rettig – Ferienakademie Sarntal 2005 2 Contents 1.Introduction 2.Physics of radiative heat transfer 3.Mathematics of spherical harmonics (P N ) 4. P N in radiative heat transfer 5. Simplified spherical harmonics for RTE 6. Comparison of computational cost and precision 7. Conclusion

3. RADIATIVE HEAT TRANSFER WITH SIMPLIFIED SPHERICAL HARMONICS 18/09 – 30/09/2005Ralf Rettig – Ferienakademie Sarntal 2005 3 Introduction From: Larsen et al. (J Comp Phys 2002) 3D-simulation of the cooling a glass cube

4. RADIATIVE HEAT TRANSFER WITH SIMPLIFIED SPHERICAL HARMONICS 18/09 – 30/09/2005Ralf Rettig – Ferienakademie Sarntal 2005 4 Introduction Radiative heat transfer in participating media: –Glass industry –Crystal growth of semiconductors –Engines –Chemical engineering

5. RADIATIVE HEAT TRANSFER WITH SIMPLIFIED SPHERICAL HARMONICS 18/09 – 30/09/2005Ralf Rettig – Ferienakademie Sarntal 2005 5 Introduction Radiative transfer equations: seven variables (spatial (3), time, frequency, direction(2)) Approximations are needed for faster solving Spherical harmoncis: also complex in higher dimensions Simplified spherical harmonics: only five variables (no directional variables)

6. RADIATIVE HEAT TRANSFER WITH SIMPLIFIED SPHERICAL HARMONICS 18/09 – 30/09/2005Ralf Rettig – Ferienakademie Sarntal 2005 6 Contents 1.Introduction 2.Physics of radiative heat transfer 3.Mathematics of spherical harmonics (P N ) 4. P N in radiative heat transfer 5. Simplified spherical harmonics for RTE 6. Comparison of computational cost and precision 7. Conclusion

7. RADIATIVE HEAT TRANSFER WITH SIMPLIFIED SPHERICAL HARMONICS 18/09 – 30/09/2005Ralf Rettig – Ferienakademie Sarntal 2005 7 Physics of radiative heat transfer Energy balance equation Boundary condition :

8. RADIATIVE HEAT TRANSFER WITH SIMPLIFIED SPHERICAL HARMONICS 18/09 – 30/09/2005Ralf Rettig – Ferienakademie Sarntal 2005 8 Physics of radiative heat transfer Equation of transfer Boundary condition: Initial condition:

9. RADIATIVE HEAT TRANSFER WITH SIMPLIFIED SPHERICAL HARMONICS 18/09 – 30/09/2005Ralf Rettig – Ferienakademie Sarntal 2005 9 Physics of radiative heat transfer Planck‘s Law: Reflectivity: Hemispheric emissivity:

10. RADIATIVE HEAT TRANSFER WITH SIMPLIFIED SPHERICAL HARMONICS 18/09 – 30/09/2005Ralf Rettig – Ferienakademie Sarntal 2005 10 Physics of radiative heat transfer Dimensionless equations:

11. RADIATIVE HEAT TRANSFER WITH SIMPLIFIED SPHERICAL HARMONICS 18/09 – 30/09/2005Ralf Rettig – Ferienakademie Sarntal 2005 11 Contents 1.Introduction 2.Physics of radiative heat transfer 3.Mathematics of spherical harmonics (P N ) 4. P N in radiative heat transfer 5. Simplified spherical harmonics for RTE 6. Comparison of computational cost and precision 7. Conclusion

12. RADIATIVE HEAT TRANSFER WITH SIMPLIFIED SPHERICAL HARMONICS 18/09 – 30/09/2005Ralf Rettig – Ferienakademie Sarntal 2005 12 Mathematics of spherical harmonics Orthogonal solutions of Laplace equation in spherical coordinates Separation of variables: m>0: differential equation of associated Legendre polynomials (Spherical harmonics) with

13. RADIATIVE HEAT TRANSFER WITH SIMPLIFIED SPHERICAL HARMONICS 18/09 – 30/09/2005Ralf Rettig – Ferienakademie Sarntal 2005 13 Mathematics of spherical harmonics Spherical harmonics: Properties of spherical harmonics: -Spherical harmonics are orthogonal -Spherical harmonics form a complete function system on unity sphere  Any function can be expressed by a series of spherical harmonics

14. RADIATIVE HEAT TRANSFER WITH SIMPLIFIED SPHERICAL HARMONICS 18/09 – 30/09/2005Ralf Rettig – Ferienakademie Sarntal 2005 14 Contents 1.Introduction 2.Physics of radiative heat transfer 3.Mathematics of spherical harmonics (P N ) 4. P N in radiative heat transfer 5. Simplified spherical harmonics for RTE 6. Comparison of computational cost and precision 7. Conclusion

15. RADIATIVE HEAT TRANSFER WITH SIMPLIFIED SPHERICAL HARMONICS 18/09 – 30/09/2005Ralf Rettig – Ferienakademie Sarntal 2005 15 P N in radiative heat transfer Aim: - Less variables - easier systems of differential equations 1.Expanding radiative intensity I into a series of spherical harmonics 2. Substituting radiative transfer equation (RTE) with the series 3. Multiplying the RTE with a spherical harmonic 4. Integrating the equation 5. Application of orthogonality => simplification 6. Set of coupled first order equations without directional variables

16. RADIATIVE HEAT TRANSFER WITH SIMPLIFIED SPHERICAL HARMONICS 18/09 – 30/09/2005Ralf Rettig – Ferienakademie Sarntal 2005 16 P N in radiative heat transfer RTE: 1. Spherical harmonics: 5. Orthogonality: 2. Substitution: 3.+4. Multiplication with spherical harmonics and integration with

17. RADIATIVE HEAT TRANSFER WITH SIMPLIFIED SPHERICAL HARMONICS 18/09 – 30/09/2005Ralf Rettig – Ferienakademie Sarntal 2005 17 P N in radiative heat transfer Simplification: 6. System of differential linear equations independent of direction (P N )

18. RADIATIVE HEAT TRANSFER WITH SIMPLIFIED SPHERICAL HARMONICS 18/09 – 30/09/2005Ralf Rettig – Ferienakademie Sarntal 2005 18 Contents 1.Introduction 2.Physics of radiative heat transfer 3.Mathematics of spherical harmonics (P N ) 4. P N in radiative heat transfer 5. Simplified spherical harmonics for RTE 6. Comparison of computational cost and precision 7. Conclusion

19. RADIATIVE HEAT TRANSFER WITH SIMPLIFIED SPHERICAL HARMONICS 18/09 – 30/09/2005Ralf Rettig – Ferienakademie Sarntal 2005 19 Simplified spherical harmonics for RTE Less complicated equations especially in higher dimensions! Neumann‘s series: (RTE)

20. RADIATIVE HEAT TRANSFER WITH SIMPLIFIED SPHERICAL HARMONICS 18/09 – 30/09/2005Ralf Rettig – Ferienakademie Sarntal 2005 20 Simplified spherical harmonics for RTE Flux: with (SP N )

21. RADIATIVE HEAT TRANSFER WITH SIMPLIFIED SPHERICAL HARMONICS 18/09 – 30/09/2005Ralf Rettig – Ferienakademie Sarntal 2005 21 Simplified spherical harmonics for RTE SP 1 Simplified SP N equation:

22. RADIATIVE HEAT TRANSFER WITH SIMPLIFIED SPHERICAL HARMONICS 18/09 – 30/09/2005Ralf Rettig – Ferienakademie Sarntal 2005 22 Simplified spherical harmonics for RTE SP 2

23. RADIATIVE HEAT TRANSFER WITH SIMPLIFIED SPHERICAL HARMONICS 18/09 – 30/09/2005Ralf Rettig – Ferienakademie Sarntal 2005 23 Simplified spherical harmonics for RTE SP 3 with

24. RADIATIVE HEAT TRANSFER WITH SIMPLIFIED SPHERICAL HARMONICS 18/09 – 30/09/2005Ralf Rettig – Ferienakademie Sarntal 2005 24 Simplified spherical harmonics for RTE SP N Boundary conditions, derivation from a variational principle

25. RADIATIVE HEAT TRANSFER WITH SIMPLIFIED SPHERICAL HARMONICS 18/09 – 30/09/2005Ralf Rettig – Ferienakademie Sarntal 2005 25 Simplified spherical harmonics for RTE SP 1 – boundary conditions SP 2 – boundary conditions

26. RADIATIVE HEAT TRANSFER WITH SIMPLIFIED SPHERICAL HARMONICS 18/09 – 30/09/2005Ralf Rettig – Ferienakademie Sarntal 2005 26 Simplified spherical harmonics for RTE S 3 – boundary conditions

27. RADIATIVE HEAT TRANSFER WITH SIMPLIFIED SPHERICAL HARMONICS 18/09 – 30/09/2005Ralf Rettig – Ferienakademie Sarntal 2005 27 Contents 1.Introduction 2.Physics of radiative heat transfer 3.Mathematics of spherical harmonics (P N ) 4. P N in radiative heat transfer 5. Simplified spherical harmonics for RTE 6. Comparison of computational cost and precision 7. Conclusion

28. RADIATIVE HEAT TRANSFER WITH SIMPLIFIED SPHERICAL HARMONICS 18/09 – 30/09/2005Ralf Rettig – Ferienakademie Sarntal 2005 28 Comparison of computational cost and precision From: Larsen et al. (J Comp Phys 2002) 1-dimensional slab geometry

29. RADIATIVE HEAT TRANSFER WITH SIMPLIFIED SPHERICAL HARMONICS 18/09 – 30/09/2005Ralf Rettig – Ferienakademie Sarntal 2005 29 Comparison of computational cost and precision From: Larsen et al. (J Comp Phys 2002) 1-dimensional slab geometry

30. RADIATIVE HEAT TRANSFER WITH SIMPLIFIED SPHERICAL HARMONICS 18/09 – 30/09/2005Ralf Rettig – Ferienakademie Sarntal 2005 30 Comparison of computational cost and precision Computational cost for 1-dimensional simulation RosselandSP 1 SP 2 SP 3 RHT Flops (x10 6 ) 8.214.3 26.9490.0 Time (s) 21.030.030.342.2812.8 From: Larsen et al. (J Comp Phys 2002) (AMD-K6 200, MATLAB 5)

31. RADIATIVE HEAT TRANSFER WITH SIMPLIFIED SPHERICAL HARMONICS 18/09 – 30/09/2005Ralf Rettig – Ferienakademie Sarntal 2005 31 Comparison of computational cost and precision From: Larsen et al. (J Comp Phys 2002) Jump in opacity

32. RADIATIVE HEAT TRANSFER WITH SIMPLIFIED SPHERICAL HARMONICS 18/09 – 30/09/2005Ralf Rettig – Ferienakademie Sarntal 2005 32 Comparison of computational cost and precision From: Larsen et al. (J Comp Phys 2002) 3D-simulation

33. RADIATIVE HEAT TRANSFER WITH SIMPLIFIED SPHERICAL HARMONICS 18/09 – 30/09/2005Ralf Rettig – Ferienakademie Sarntal 2005 33 Contents 1.Introduction 2.Physics of radiative heat transfer 3.Mathematics of spherical harmonics (P N ) 4. P N in radiative heat transfer 5. Simplified spherical harmonics for RTE 6. Comparsion of computational cost and precision 7. Conclusion

34. RADIATIVE HEAT TRANSFER WITH SIMPLIFIED SPHERICAL HARMONICS 18/09 – 30/09/2005Ralf Rettig – Ferienakademie Sarntal 2005 34 Conclusion In multidimensional geometries SP N equations are less complicated The simulations are derived for  higher temperatures Systems of second-order differential equations are easy to solve

35. RADIATIVE HEAT TRANSFER WITH SIMPLIFIED SPHERICAL HARMONICS 18/09 – 30/09/2005Ralf Rettig – Ferienakademie Sarntal 2005 35 Literature Larsen, E.W. et. al: Simplified P N approximations to the equations of radiative heat transfer and applications. J Comp Phys 183 (2002) 652-675 Seaid, M. et al.: Generalized numerical approximations for the radiative heat transfer problems in two space dimensions. In: Proceedings of the Eurotherm Seminar 73. Lybaert, P. et al., Mons, April 15-17, 2003 Modest, M.F.: Radiative heat transfer. San Diego, Academic Press, second edition 2003 Jung, M. et al: Methode der finiten Elemente für Ingenieure. Stuttgart, Teubner, 1.Auflage 2001