Simulation of radiative heat transfer in participating media with simplified spherical harmonics

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. Contents • Introduction • Physics of radiative heat transfer • Mathematics of spherical harmonics (PN) 4. PN in radiative heat transfer 5. Simplified spherical harmonics for RTE 6. Comparison of computational cost and precision 7. Conclusion Ralf Rettig – Ferienakademie Sarntal 2005 2

3. Introduction 3D-simulation of the cooling a glass cube From: Larsen et al. (J Comp Phys 2002) Ralf Rettig – Ferienakademie Sarntal 2005 3

4. Introduction • Radiative heat transfer in participating media: • Glass industry • Crystal growth of semiconductors • Engines • Chemical engineering Ralf Rettig – Ferienakademie Sarntal 2005 4

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) Ralf Rettig – Ferienakademie Sarntal 2005 5

6. Contents • Introduction • Physics of radiative heat transfer • Mathematics of spherical harmonics (PN) 4. PN in radiative heat transfer 5. Simplified spherical harmonics for RTE 6. Comparison of computational cost and precision 7. Conclusion Ralf Rettig – Ferienakademie Sarntal 2005 6

7. Physics of radiative heat transfer Energy balance equation Boundary condition: Ralf Rettig – Ferienakademie Sarntal 2005 7

8. Physics of radiative heat transfer Equation of transfer Boundary condition: Initial condition: Ralf Rettig – Ferienakademie Sarntal 2005 8

9. Physics of radiative heat transfer Planck‘s Law: Reflectivity: Hemispheric emissivity: Ralf Rettig – Ferienakademie Sarntal 2005 9

10. Physics of radiative heat transfer Dimensionless equations: Ralf Rettig – Ferienakademie Sarntal 2005 10

11. Contents • Introduction • Physics of radiative heat transfer • Mathematics of spherical harmonics (PN) 4. PN in radiative heat transfer 5. Simplified spherical harmonics for RTE 6. Comparison of computational cost and precision 7. Conclusion Ralf Rettig – Ferienakademie Sarntal 2005 11

12. Mathematics of spherical harmonics Orthogonal solutions of Laplace equation in spherical coordinates Separation of variables: (Spherical harmonics) with m>0: differential equation of associated Legendre polynomials Ralf Rettig – Ferienakademie Sarntal 2005 12

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 Ralf Rettig – Ferienakademie Sarntal 2005 13

14. Contents • Introduction • Physics of radiative heat transfer • Mathematics of spherical harmonics (PN) 4. PN in radiative heat transfer 5. Simplified spherical harmonics for RTE 6. Comparison of computational cost and precision 7. Conclusion Ralf Rettig – Ferienakademie Sarntal 2005 14

15. PN in radiative heat transfer Aim: - Less variables - easier systems of differential equations • 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 Ralf Rettig – Ferienakademie Sarntal 2005 15

16. PN in radiative heat transfer RTE: 1. Spherical harmonics: 2. Substitution: 3.+4. Multiplication with spherical harmonics and integration with 5. Orthogonality: Ralf Rettig – Ferienakademie Sarntal 2005 16

17. PN in radiative heat transfer Simplification: 6. System of differential linear equations independent of direction (PN) Ralf Rettig – Ferienakademie Sarntal 2005 17

18. Contents • Introduction • Physics of radiative heat transfer • Mathematics of spherical harmonics (PN) 4. PN in radiative heat transfer 5. Simplified spherical harmonics for RTE 6. Comparison of computational cost and precision 7. Conclusion Ralf Rettig – Ferienakademie Sarntal 2005 18

19. Simplified spherical harmonics for RTE Less complicated equations especially in higher dimensions! (RTE) Neumann‘s series: Ralf Rettig – Ferienakademie Sarntal 2005 19

20. Simplified spherical harmonics for RTE with Flux: (SPN) Ralf Rettig – Ferienakademie Sarntal 2005 20

21. SP1 Simplified spherical harmonics for RTE Simplified SPN equation: Ralf Rettig – Ferienakademie Sarntal 2005 21

22. SP2 Simplified spherical harmonics for RTE Ralf Rettig – Ferienakademie Sarntal 2005 22

23. SP3 Simplified spherical harmonics for RTE with Ralf Rettig – Ferienakademie Sarntal 2005 23

24. Simplified spherical harmonics for RTE SPN Boundary conditions, derivation from a variational principle Ralf Rettig – Ferienakademie Sarntal 2005 24

25. Simplified spherical harmonics for RTE SP1 – boundary conditions SP2 – boundary conditions Ralf Rettig – Ferienakademie Sarntal 2005 25

26. Simplified spherical harmonics for RTE S3 – boundary conditions Ralf Rettig – Ferienakademie Sarntal 2005 26

27. Contents • Introduction • Physics of radiative heat transfer • Mathematics of spherical harmonics (PN) 4. PN in radiative heat transfer 5. Simplified spherical harmonics for RTE 6. Comparison of computational cost and precision 7. Conclusion Ralf Rettig – Ferienakademie Sarntal 2005 27

28. Comparison of computational cost and precision 1-dimensional slab geometry From: Larsen et al. (J Comp Phys 2002) Ralf Rettig – Ferienakademie Sarntal 2005 28

29. Comparison of computational cost and precision 1-dimensional slab geometry From: Larsen et al. (J Comp Phys 2002) Ralf Rettig – Ferienakademie Sarntal 2005 29

30. Comparison of computational cost and precision Computational cost for 1-dimensional simulation (AMD-K6 200, MATLAB 5) From: Larsen et al. (J Comp Phys 2002) Ralf Rettig – Ferienakademie Sarntal 2005 30

31. Comparison of computational cost and precision Jump in opacity From: Larsen et al. (J Comp Phys 2002) Ralf Rettig – Ferienakademie Sarntal 2005 31

32. Comparison of computational cost and precision 3D-simulation From: Larsen et al. (J Comp Phys 2002) Ralf Rettig – Ferienakademie Sarntal 2005 32

33. Contents • Introduction • Physics of radiative heat transfer • Mathematics of spherical harmonics (PN) 4. PN in radiative heat transfer 5. Simplified spherical harmonics for RTE 6. Comparsion of computational cost and precision 7. Conclusion Ralf Rettig – Ferienakademie Sarntal 2005 33

34. Conclusion • In multidimensional geometries SPN equations are less complicated • The simulations are derived for e<1, i.e. short free pathes => higher temperatures • Systems of second-order differential equations are easy to solve Ralf Rettig – Ferienakademie Sarntal 2005 34

35. Literature • Larsen, E.W. et. al: Simplified PN 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 Ralf Rettig – Ferienakademie Sarntal 2005 35