simulation of radiative heat transfer in participating media with simplified spherical harmonics
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Simulation of radiative heat transfer in participating media with simplified spherical harmonics. Ralf Rettig, University of Erlangen Ferienakademie Sarntal 18/09 – 30/09/2005. Contents. Introduction Physics of radiative heat transfer Mathematics of spherical harmonics (P N ) - PowerPoint PPT PresentationTRANSCRIPT
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
18/09 – 30/09/2005
Ralf Rettig – Ferienakademie Sarntal 2005 3
RADIATIVE HEAT TRANSFER WITH SIMPLIFIED SPHERICAL HARMONICS
Introduction
From: Larsen et al. (J Comp Phys 2002)
3D-simulation of thecooling a glass cube
18/09 – 30/09/2005
Ralf Rettig – Ferienakademie Sarntal 2005 4
RADIATIVE HEAT TRANSFER WITH SIMPLIFIED SPHERICAL HARMONICS
Introduction
• Radiative heat transfer in participating media:– Glass industry– Crystal growth of semiconductors– Engines– Chemical engineering
18/09 – 30/09/2005
Ralf Rettig – Ferienakademie Sarntal 2005 5
RADIATIVE HEAT TRANSFER WITH SIMPLIFIED SPHERICAL HARMONICS
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)
18/09 – 30/09/2005
Ralf Rettig – Ferienakademie Sarntal 2005 6
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
18/09 – 30/09/2005
Ralf Rettig – Ferienakademie Sarntal 2005 7
RADIATIVE HEAT TRANSFER WITH SIMPLIFIED SPHERICAL HARMONICS
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
Ralf Rettig – Ferienakademie Sarntal 2005 8
RADIATIVE HEAT TRANSFER WITH SIMPLIFIED SPHERICAL HARMONICS
Physics of radiative heat transfer
)(1:, 2
1 IBIt
I
cS
),())(1(),()(),(:0 bTBnInIn
Equation of transfer
Boundary condition:
Initial condition:
)()0,(: 0 xTxTVx
18/09 – 30/09/2005
Ralf Rettig – Ferienakademie Sarntal 2005 9
RADIATIVE HEAT TRANSFER WITH SIMPLIFIED SPHERICAL HARMONICS
Physics of radiative heat transfer
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:
18/09 – 30/09/2005
Ralf Rettig – Ferienakademie Sarntal 2005 10
RADIATIVE HEAT TRANSFER WITH SIMPLIFIED SPHERICAL HARMONICS
Physics of radiative heat transfer
reference
lengthpathfree
refref x
x
x
1
12
)(22
S
ddIBTkt
T
)(:, 21 IBIS
Dimensionless equations:
dTBTBn
nTThTkn bb
1
0
2
1
2 ),(),()(
18/09 – 30/09/2005
Ralf Rettig – Ferienakademie Sarntal 2005 11
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
18/09 – 30/09/2005
Ralf Rettig – Ferienakademie Sarntal 2005 12
RADIATIVE HEAT TRANSFER WITH SIMPLIFIED SPHERICAL HARMONICS
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
18/09 – 30/09/2005
Ralf Rettig – Ferienakademie Sarntal 2005 13
RADIATIVE HEAT TRANSFER WITH SIMPLIFIED SPHERICAL HARMONICS
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
18/09 – 30/09/2005
Ralf Rettig – Ferienakademie Sarntal 2005 14
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
18/09 – 30/09/2005
Ralf Rettig – Ferienakademie Sarntal 2005 15
RADIATIVE HEAT TRANSFER WITH SIMPLIFIED SPHERICAL HARMONICS
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
18/09 – 30/09/2005
Ralf Rettig – Ferienakademie Sarntal 2005 16
RADIATIVE HEAT TRANSFER WITH SIMPLIFIED SPHERICAL HARMONICS
PN in radiative heat transfer
)( 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
)(
18/09 – 30/09/2005
Ralf Rettig – Ferienakademie Sarntal 2005 17
RADIATIVE HEAT TRANSFER WITH SIMPLIFIED SPHERICAL HARMONICS
PN in radiative heat transfer
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)
18/09 – 30/09/2005
Ralf Rettig – Ferienakademie Sarntal 2005 18
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
18/09 – 30/09/2005
Ralf Rettig – Ferienakademie Sarntal 2005 19
RADIATIVE HEAT TRANSFER WITH SIMPLIFIED SPHERICAL HARMONICS
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)
18/09 – 30/09/2005
Ralf Rettig – Ferienakademie Sarntal 2005 20
RADIATIVE HEAT TRANSFER WITH SIMPLIFIED SPHERICAL HARMONICS
Simplified spherical harmonics for RTE
...
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)
18/09 – 30/09/2005
Ralf Rettig – Ferienakademie Sarntal 2005 21
RADIATIVE HEAT TRANSFER WITH SIMPLIFIED SPHERICAL HARMONICS
Simplified spherical harmonics for RTE
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:
18/09 – 30/09/2005
Ralf Rettig – Ferienakademie Sarntal 2005 22
RADIATIVE HEAT TRANSFER WITH SIMPLIFIED SPHERICAL HARMONICS
Simplified spherical harmonics for RTE
13
1
dTkt
T
)4(5
32 B
SP2
)4(9
4
9
5B
18/09 – 30/09/2005
Ralf Rettig – Ferienakademie Sarntal 2005 23
RADIATIVE HEAT TRANSFER WITH SIMPLIFIED SPHERICAL HARMONICS
Simplified spherical harmonics for RTE
)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
18/09 – 30/09/2005
Ralf Rettig – Ferienakademie Sarntal 2005 24
RADIATIVE HEAT TRANSFER WITH SIMPLIFIED SPHERICAL HARMONICS
Simplified spherical harmonics for RTE
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
18/09 – 30/09/2005
Ralf Rettig – Ferienakademie Sarntal 2005 25
RADIATIVE HEAT TRANSFER WITH SIMPLIFIED SPHERICAL HARMONICS
Simplified spherical harmonics for RTE
)(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
18/09 – 30/09/2005
Ralf Rettig – Ferienakademie Sarntal 2005 26
RADIATIVE HEAT TRANSFER WITH SIMPLIFIED SPHERICAL HARMONICS
Simplified spherical harmonics for RTE
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
18/09 – 30/09/2005
Ralf Rettig – Ferienakademie Sarntal 2005 27
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
18/09 – 30/09/2005
Ralf Rettig – Ferienakademie Sarntal 2005 28
RADIATIVE HEAT TRANSFER WITH SIMPLIFIED SPHERICAL HARMONICS
Comparison of computational cost and precision
From: Larsen et al. (J Comp Phys 2002)
1-dimensional slab geometry
18/09 – 30/09/2005
Ralf Rettig – Ferienakademie Sarntal 2005 29
RADIATIVE HEAT TRANSFER WITH SIMPLIFIED SPHERICAL HARMONICS
Comparison of computational cost and precision
From: Larsen et al. (J Comp Phys 2002)
1-dimensional slab geometry
18/09 – 30/09/2005
Ralf Rettig – Ferienakademie Sarntal 2005 30
RADIATIVE HEAT TRANSFER WITH SIMPLIFIED SPHERICAL HARMONICS
Comparison of computational cost and precision
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)
18/09 – 30/09/2005
Ralf Rettig – Ferienakademie Sarntal 2005 31
RADIATIVE HEAT TRANSFER WITH SIMPLIFIED SPHERICAL HARMONICS
Comparison of computational cost and precision
From: Larsen et al. (J Comp Phys 2002)
Jump in opacity
18/09 – 30/09/2005
Ralf Rettig – Ferienakademie Sarntal 2005 32
RADIATIVE HEAT TRANSFER WITH SIMPLIFIED SPHERICAL HARMONICS
Comparison of computational cost and precision
From: Larsen et al. (J Comp Phys 2002)
3D-simulation
18/09 – 30/09/2005
Ralf Rettig – Ferienakademie Sarntal 2005 33
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. Comparsion of computational cost and precision
7. Conclusion
18/09 – 30/09/2005
Ralf Rettig – Ferienakademie Sarntal 2005 34
RADIATIVE HEAT TRANSFER WITH SIMPLIFIED SPHERICAL HARMONICS
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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Ralf Rettig – Ferienakademie Sarntal 2005 35
RADIATIVE HEAT TRANSFER WITH SIMPLIFIED SPHERICAL HARMONICS
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