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

Simulation of radiative heat transfer in participating media with

simplified spherical harmonics

Ralf Rettig, University of Erlangen

Ferienakademie Sarntal

18/09 – 30/09/2005

18/09 – 30/09/2005

Ralf Rettig – Ferienakademie Sarntal 2005 2

RADIATIVE HEAT TRANSFER WITH SIMPLIFIED SPHERICAL HARMONICS

Contents

1. Introduction

2. Physics of radiative heat transfer

3. 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

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Introduction

From: Larsen et al. (J Comp Phys 2002)

3D-simulation of thecooling a glass cube

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Introduction

• Radiative heat transfer in participating media:– Glass industry– Crystal growth of semiconductors– Engines– Chemical engineering

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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)

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Contents

1. Introduction



harmonics (PN)




7. Conclusion

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Physics of radiative heat transfer

12

)(

S

mm ddIBTkt

Tc

1

0

2

1

2 )),(),(()(

dTBTBn

nTThTkn bb

Energy balance equation

Boundary condition:

18/09 – 30/09/2005




)(1:, 2

1 IBIt

I

cS

),())(1(),()(),(:0 bTBnInIn

Equation of transfer

Boundary condition:

Initial condition:

)()0,(: 0 xTxTVx

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1

2),(

2

321

Tk

h

p

B

p

ec

hnTB

)(sin

)(sin

)(tan

)(tan

2

1)(

212

212

212

212

Planck‘s Law:

Reflectivity:

1

0

1 )(12 dnHemispheric emissivity:

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reference

lengthpathfree

refref x

x

x

1

12

)(22

S

ddIBTkt

T

)(:, 21 IBIS

Dimensionless equations:

dTBTBn

nTThTkn bb

1

0

2

1

2 ),(),()(

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Contents

1. Introduction



harmonics (PN)




7. Conclusion

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Mathematics of spherical harmonics

0sin

11sin

sin

1112

2

2222

2

rr

rrr

),()(),,( YrFr

022

2

m

Orthogonal solutions of Laplace equation in spherical coordinates

Separation of variables:

01

)1(2)1(2

2

2

22

P

s

mllP

ssP

ss

m>0: differential equation of associated Legendre polynomials

)()(cos),( PY(Spherical harmonics)

cosswith

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Mathematics of spherical harmonics

)(cos)!(

)!()1()ˆ( 2/)( m

limmmm

l Peml

mlsY

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

0

)ˆ()()ˆ,(l

l

lm

ml

ml sYrIsrI

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Contents

1. Introduction



harmonics (PN)




7. Conclusion

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PN 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

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)( IBkI

N

lll PII

0

)()(),(

RTE:

1. Spherical harmonics:

N

lllll BkIPkIP

0

)()()()(arccos

1

1 0

)()()()()(arccosN

lwllll dPIPkIP

Nw ,...,1,0

1

1 12

2)()( lwwl wdPP 5. Orthogonality:

2. Substitution:

3.+4. Multiplication with spherical harmonics and integration

withAdPBk w

1

1

)(

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Aw

Ikdww

I ll

12

2)(

12

2

1

1

12

2arccos)(

1

12

1

1

AIw

k

wI ll

)(12

2

12

2arcsin)(

1

1

AIw

kI

w ll

)(12

2)(

12

4 Nw ,...,1,0

Simplification:

6. System of differential linear equations independent of direction

(PN)

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Contents

1. Introduction



harmonics (PN)




7. Conclusion

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Simplified spherical harmonics for RTE

)( IBI

),(),,,(1 TBtxI

Less complicated equations especially in higher dimensions!

Neumann‘s series:

0

1)1(j

jaa

B

BI

...1

1

4

4

43

3

32

2

2

1

(RTE)

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...

945

44

45

4

314: 6

6

64

4

42

2

2

1 B

...753753753

1

75314

3

66

64

4

42

2

22

66

64

4

42

2

26

6

64

4

42

2

2

1

66

64

4

42

2

2

B

BIdS

...

75314 6

6

64

4

42

2

2

2

n

S

nn

nd

1

211

2

Flux:

with

(SPN)

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22

2

34 B

)4(3

1: 2

1 B

1

12

1

3

1

)4()(

2

d

dBddIBS

SP1

13

1

dTkt

T

Simplified SPN equation:

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13

1

dTkt

T

)4(5

32 B

SP2

)4(9

4

9

5B

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)4(11

21

2

B

)4(22

22

2

B

1

)(1

2211

daaTkt

T

SP3

5

6

7

2

7

321

5

6

7

2

7

322 with

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1

0

1 )( dr 1

0

22 )( dr

1

0

33 )( dr

SPN Boundary conditions, derivation from a variational principle

1

0

34 )()( dPr

1

0

35 )()( dPr 1

0

326 )()()( dPPr

1

0

337 )()()( dPPr

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)(4)(3

2

21

31)(

1

2 xBxnr

rx b

SP1 – boundary conditions

SP2 – boundary conditions

)(4)(45

6

41

21)(4)(

5

4

41

31)(

3

1

3

2 xBxBr

rxBxn

r

rx b

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bBxxnx 122111 )()()(

bBxxnx 211222 )()()(

3

61134

96

51

S3 – boundary conditions

5

61134

96

52

5

62

96

51

5

62

96

52

5

63

2

51

5

63

2

52

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Contents

1. Introduction



harmonics (PN)




7. Conclusion

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Comparison of computational cost and precision


1-dimensional slab geometry

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1-dimensional slab geometry

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Computational cost for 1-dimensional simulation

Rosseland SP1 SP2 SP3 RHT

Flops (x106)

8.2 14.3 14.3 26.9 490.0

Time (s)

21.0 30.0 30.3 42.2 812.8

From: Larsen et al. (J Comp Phys 2002) (AMD-K6 200, MATLAB 5)

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Jump in opacity

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3D-simulation

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Contents

1. Introduction



harmonics (PN)



6. Comparsion of computational cost and precision

7. Conclusion

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Conclusion

• In multidimensional geometries SPN equations are less complicated

• The simulations are derived for <1, i.e. short free pathes => higher temperatures

• Systems of second-order differential equations are easy to solve

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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

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

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