Initiation au rayonnement thermique Master...

Heat and Mass Transfer 2Radiation in FirePas al BOULET and Anthony COLLINLEMTA - CNRS - Université de LorraineE ole thématique du CNRS sur la S ien e des In endies et ses Appli ations Porti io, 30/05/15 04/06/15

ContentsIntrodu tionFrom simple laws to radiative transfer equationRadiative properties of surfa es and amesEmission / Absorption, gas modelsNumeri al methods for radiative transferE ole thématique du CNRS sur la S ien e des In endies et ses Appli ations Porti io, 30/05/15 04/06/15

Introdu tion 1 / 60Obje tives of this short ourseNot an overview on radiative transfer!Rather a sele tion of topi s related to radiative transfer in the frame ofre appli ations, illustrated with real re appli ations.Built to . . . make you aware of the inuen e of radiative transfer show you the traps, some hidden assumptions and their onsequen es present you some tools dedi ated to radiative transfer make a link with numeri al methods and sensitivity analysis oursesWarning: we assumed that you already heard about radiative transfer. . . all the fundamentals will not be re alled here !E ole thématique du CNRS sur la S ien e des In endies et ses Appli ations Porti io, 30/05/15 04/06/15

Introdu tion 2 / 60Illustration of radiative transfer through three familiarappli ations - Some questions...Flux from ame? Absorptivity of vegetation? Flux measurement?Radiation from ame? Smoke?Radiation to/from surfa es? Radiation from oil? ame? Absorp-tivity? Emissivity?Relative ontribution? What is the required model omplexity?E ole thématique du CNRS sur la S ien e des In endies et ses Appli ations Porti io, 30/05/15 04/06/15

ContentsIntrodu tionFrom simple laws to radiative transfer equationSome denitionsTransfer through non-parti ipating mediaParti ipating mediaRadiative Transfer EquationRadiative properties of surfa es and amesEmission / Absorption, gas modelsNumeri al methods for radiative transferE ole thématique du CNRS sur la S ien e des In endies et ses Appli ations Porti io, 30/05/15 04/06/15

From simple laws to radiative transfer equation 3 / 60 Some denitionsSpe i ities of radiative transferSpatial, time + spe tral and dire -tional dependen e... but there are some available ap-proximations like diuse and greyassumptions.In the followings ν (or η) is the wavenumber (in m−1) = 10000λ(µm)The fundamental radiative variable is the spe tral intensity ("laluminan e spe trale")= part of ux per solid angle, surfa e andwavelength unit

dS

θ

φ

PSfrag repla ements ~Ω~n dΩ Lλ =d3ΦdS os θdΩdλ( os θ = ~Ω · ~n)E ole thématique du CNRS sur la S ien e des In endies et ses Appli ations Porti io, 30/05/15 04/06/15

From simple laws to radiative transfer equation 4 / 60 Some denitionsDenition of uxes from the spe tral intensityDo you remember the solid angles?o

dΩ = dΣR2 dΩ = os θdSR2dΩ

θ

φ

rdφ

Rdθr

R

PSfrag repla ements

Ω

dΩ = dΣR2 = RdθrdϕR2 = dθR sin θdϕR = sin θdθdϕE ole thématique du CNRS sur la S ien e des In endies et ses Appli ations Porti io, 30/05/15 04/06/15

From simple laws to radiative transfer equation 5 / 60 Some denitionsFlux, emissive power, integrated intensityBy denition the ux is obtained integrating the intensity :Φ =

∫

λ

∫S ∫Ω Lλ os θdΩdSdλHen e, the spe tral emissive power (émittan e) for a surfa e isMλ =

∫2π Lλ (~Ω) ~Ω · ~n dΩ =

∫2π Lλ os θdΩFor a Lambertian surfa e, this yields Mλ = πLλ!Not to be onfused with the ux sent by the surfa e whi h in ludesemission and reexion and orresponds to the radiosity :Jλ = Mλ + ρλEλOther useful integrated variables for any point in the domain of interest:integrated intensity Gλ =∫4π LλdΩand divergen e of radiative ux ~∇ · ~φ = κ(4πL0λ(T (s)) − Gλ)E ole thématique du CNRS sur la S ien e des In endies et ses Appli ations Porti io, 30/05/15 04/06/15

From simple laws to radiative transfer equation 6 / 60 Some denitionsBasi laws... that you probably know...Plan k's law : spe tral emissive power (Emittan e) of a bla k bodyM0λ (T ) =

C1λ5 (exp ( C2

λT )− 1) = πL0λ (T )with C1 = 3, 74 · 10−16 kg.m4/s3 and C2 = 1, 4388 · 10−2 m.KWien's law : the maximum of emissionis at λmaxλmaxT = 2898µm.K96% energy emitted in [0.5λmax - 5λmax Warning!! Spe i expressions for M0

ν and M0η!Stefan's law : integrated emissive power M0 (T ) = σT 4 with

σ = 5, 66897 · 10−8 W/m2/K4E ole thématique du CNRS sur la S ien e des In endies et ses Appli ations Porti io, 30/05/15 04/06/15

From simple laws to radiative transfer equation 7 / 60 Transfer through non-parti ipating mediaRadiative transfer through non parti ipating mediaRadiative ex hange between surfa es des ribed through view fa tors(fa teurs de forme)Φ1→2 = S1F1→2M01with the view fa tor from tables, relationships or denitionA usefull link : http://www.thermalradiation.net/indexCat.html

F1→2 = 1S1 ∫S1 ∫S2 os θ1 os θ2dS1dS2πd2Possible extensions introdu ing the radiosity (non bla k surfa e), atransmittan e (parti ipating medium), surfa e-to-volume view fa tors,...E ole thématique du CNRS sur la S ien e des In endies et ses Appli ations Porti io, 30/05/15 04/06/15

From simple laws to radiative transfer equation 8 / 60 Parti ipating mediaAbsorption, emission and s attering phenomenaConsider a ray travelling through a parti ipating medium in dire tion ~Ω.The radiation intensity an be attenuated by absorption, proportional to the travelled distan e and the absorption oe ient κ (m−1) s attering (diusion) = deviation, proportional to the travelleddistan e and the s attering oe ient σ (m−1)while the radiation intensity an be re-infor ed by emission, involving the absorption oe ient, the bla kbodyintensity at the lo al temperature and the path length s attering, be ause a radiation oming from dire tion ~Ω′ an bes attered through the dire tion of interest ~Ω, involving a probabilitywhi h is alled the phase fun tion Pλ(~Ω

′ → ~Ω)First determine the radiative properties, then solve the RTE...E ole thématique du CNRS sur la S ien e des In endies et ses Appli ations Porti io, 30/05/15 04/06/15

From simple laws to radiative transfer equation 9 / 60 Radiative Transfer EquationThe RTE Radiative Transfer EquationEnergy balan e on this ray travelling through our parti ipating mediumdLλ(s, ~Ω)ds = −κλLλ(s, ~Ω)︸︷︷︸absorption − σλLλ(s, ~Ω)

︸︷︷︸out-s attering+ n2λκλL0λ(T (s))︸︷︷︸emission + . . .

· · ·+14π ∫Ω′=4π σλPλ(~Ω

′ → ~Ω)Lλ(s, ~Ω′)dΩ′

︸︷︷︸in-s attering+ boundary onditionsLλ(sw , ~Ω) = ǫwL0λ(T (sw )) + ∫2π ρ′′wLλ(sw , ~Ω′)|~n · ~Ω′|dΩ′Flux at the surfa e omputed from φ =∫ Lλ(s, ~Ω)~n · ~ΩdΩand ux divergen e from ~∇ · ~φ = κ(4πL0λ(T (s)) − ∫4π Lλ(s, ~Ω)dΩ)E ole thématique du CNRS sur la S ien e des In endies et ses Appli ations Porti io, 30/05/15 04/06/15

ContentsIntrodu tionFrom simple laws to radiative transfer equationRadiative properties of surfa es and amesDenition of main radiative propertiesOpaque surfa esParti ipating mediaRadiation from amesEmission / Absorption, gas modelsNumeri al methods for radiative transferE ole thématique du CNRS sur la S ien e des In endies et ses Appli ations Porti io, 30/05/15 04/06/15

Radiative properties of surfa es and ames 10 / 60 Denition of main radiative propertiesDenition of main radiative propertiesφ0 ρφ0 ρτ2φ0e−2κδτφ0 ρτφ0e−2κδτφ0e−κδ ρ2τφ0e−3κδτ2φ0e−κδx Aλ Rλ

Tλ

Interfa e between two media transmissivity τλ(~Ω, ~Ω

′)

ree tivity ρλ(~Ω, ~Ω′)Opaque surfa e

absorptivity αλ(~Ω)

emissivity ǫλ(~Ω)

Parti ipating medium transmittan e Tλ(~Ω, ~Ω

′)

ree tan e Rλ(~Ω, ~Ω′)

absorban e Aλ(~Ω)

emittan e Eλ(~Ω)Usefull tools: spe tros opi measurements, Fresnel's lawsE ole thématique du CNRS sur la S ien e des In endies et ses Appli ations Porti io, 30/05/15 04/06/15

Radiative properties of surfa es and ames 11 / 60 Opaque surfa esRadiative properties of opaque surfa esRemember : αλ + ρλ = 1 (sin e τλ = 0)Measuring ρλ provides αλ

Wavenumber [cm-1]

ρ,α

[-]

Inte

nsity

[W/(

m2 .s

r.cm

-1)]

1000 2000 3000 4000 5000 60000

0.2

0.4

0.6

0.8

1

0

2

4

6

8

10

12

14

16

18

20

αρBlackbody at 1500 KBlackbody at 1000 KBlackbody at 400 K

Plywood has a non-grey surfa e Wavenumber [cm-1]ρ,

α[-

]

1000 2000 3000 4000 5000 60000

0.2

0.4

0.6

0.8

1

Composite -α99 % confident intervalComposite -ρ

...while arbon omposite is neargreyAverage absorptivity should be omputed from < α >=∫αλL0λ(T )dλ∫ L0

λ(T )dλConsequen e... αλ(~Ω) = ǫλ(~Ω), but ... < α > 6=< ǫ >!!E ole thématique du CNRS sur la S ien e des In endies et ses Appli ations Porti io, 30/05/15 04/06/15

Radiative properties of surfa es and ames 12 / 60 Parti ipating mediaA parti ipating medium: the famous PMMA!!... amazing non-grey medium, whi h should be avoided for a ademi studies...Measuring transmittan e and ree tan e allows identifying κλ

Wavenumber [cm-1]

Tra

nsm

issi

vity

[-]

0 5000 10000 15000 200000

0.2

0.4

0.6

0.8

1

0.37 mm14.79 mm

Wavenumber [cm-1]

Ref

lect

ivity

[-]

0 5000 10000 15000 200000

0.02

0.04

0.06

0.08

0.1

1.7 mm96.8 mm

Wavenumber [cm-1]

Abs

orp

tion

coef

ficie

nt[m

-1]

0 5000 10000 15000

100

101

102

103

104

105

106

Transmittan e Ree tan e Absorption oe ientNi e exer ise: imagine what happens if you study PMMA with a one( lose to a bla kbody near 1000 K) or a FPA (tungsten lamp radiating inthe visible and the near IR)...E ole thématique du CNRS sur la S ien e des In endies et ses Appli ations Porti io, 30/05/15 04/06/15

Radiative properties of surfa es and ames 13 / 60 Parti ipating mediaCC vs FPA, the absorbed ux annot be the same!

(from Bal et al., Int. J. Heat Mass Transfer, 2013)Degradation depends on the experimental onditions!E ole thématique du CNRS sur la S ien e des In endies et ses Appli ations Porti io, 30/05/15 04/06/15

Radiative properties of surfa es and ames 14 / 60 Radiation from amesRadiation from amesWhat is really a ame? A mixing of soot and gases... neither a solid wall,nor an ideal emitter (bla kbody), nor a grey medium!W av e num be r [c m

-1]

I n

[W/(

m2.s

r.cm

-1)]

1 0 0 0 2 0 0 0 3 0 0 0 4 0 0 0 5 0 0 0 6 0 0 00

2

4

6

8

1 0

1 2

1 4

1 6

1 8

2 0

2 2

2 4

W S _ 5 0 cm _ ac q_ 6

W S _ 1 m _ ac q_ 2

W S _ 2 m _ ac q_ 4

W S _ 3 m _ ac q_ 2

W S _ 4 m _ ac q_ 2

31 0 4 25 (µm )

B B at 1 0 0 0 K

B B at 1 2 0 0 K

B B at 1 4 0 0 K

B B at 1 6 0 0 K

Typi al intensity emitted by avegetation re during a re test -In reasing ontribution of soot withopti al thi kness. Intensity measured from a PMMAsample during degradation test -The role of gases and ba kgroundemission.E ole thématique du CNRS sur la S ien e des In endies et ses Appli ations Porti io, 30/05/15 04/06/15

Radiative properties of surfa es and ames 15 / 60 Radiation from amesHow to model the ux emitted by the ame? (1)(1) A usual but rough des ription: the solid ame approa hφ = ǫf Sf σT 4fSf ame surfa e, but ame temperature Tf ??? emissivity ǫf ???(2) Flame modelled as a volume of gas Vf with absorption oef κ (m−1)φ = 4κf Vf σT 4fTf ??? Absorption oe ient κf ???The well-known radiative fra tion χr in some numeri al odes ontrolsthe terms ǫf σT 4f or κf σT 4f(3) The truth should ome from the solution of the RTE RadiativeTransfer Equation ( oming next)E ole thématique du CNRS sur la S ien e des In endies et ses Appli ations Porti io, 30/05/15 04/06/15

Radiative properties of surfa es and ames 16 / 60 Radiation from amesHow to model the ux emitted by the ame? (2)Knowledge of temperature and on entrations for CO, CO2 H2O and sootin the ells of the ame area?... then omputation of κλ (next part).Find the solution for the RTE for non s attering medium :dLλ(s,~Ω)ds = −κλLλ(s, ~Ω) + κλL0λ(T (s)) with BC Lλ(0, ~Ω) = 01D solution for homogeneous medium with thi kness X?Lλ(s, ~Ω) = (1− e−κλs)L0λ(T ) from whi h, the emissive power at theboundary M = πL(X , ~Ω) = ǫσT 4 ... but L(X , ~Ω) 6= st! whi h shows thelimitations of ǫ = 1− e−κX1D solution without self-absorption? Lλ(s, ~Ω) = κλL0λ(T )s from whi hthe ux after integration over 4π sr and the volume: Φ = 4SXκσT 4These are rough approximations!Numeri al methods an lead to a more a urate solution of the RTE,in luding the spe tral dependen e and models for the gases.E ole thématique du CNRS sur la S ien e des In endies et ses Appli ations Porti io, 30/05/15 04/06/15

ContentsIntrodu tionFrom simple laws to radiative transfer equationRadiative properties of surfa es and amesEmission / Absorption, gas modelsRadiative emission / absorption by gasesLBL modelSNB modelWSGG approa hCon lusions on the models for gas absorptionNumeri al methods for radiative transferE ole thématique du CNRS sur la S ien e des In endies et ses Appli ations Porti io, 30/05/15 04/06/15

Emission / Absorption, gas models 17 / 60 Radiative emission / absorption by gasesRadiative emission / absorption by gases• Energy emission or absorption is due to a hange of a mole ule state toa lower (for emission) or higher (for absorption) energy state,• To hange its energy state, it must be a hange in the translational,rotational, vibrational or ele troni energy levels of the mole ule (or a ombination),• Quantum me hani s demonstrates that these energy levels are dis rete(not ontinuous) and proves that only ertain transitions between energylevels are possible:=⇒ homonu lear diatomi mole ules, N2 and O2 annot emit or absorbradiation,=⇒ In ombustion, the primary radiating gases are CO2, H2O and to alesser extent CO.E ole thématique du CNRS sur la S ien e des In endies et ses Appli ations Porti io, 30/05/15 04/06/15

Emission / Absorption, gas models 18 / 60 Radiative emission / absorption by gasesRadiative emission / absorption by gases• Between 300 K up to 3000 K, the main part of radiant emission andabsorption orresponds to a hange in rotational and vibrational energylevels of the mole ules.Rotation 1 Rotation 2 Bending Stret hing 1 Stret hing 2

E ole thématique du CNRS sur la S ien e des In endies et ses Appli ations Porti io, 30/05/15 04/06/15

Emission / Absorption, gas models 19 / 60 Radiative emission / absorption by gasesRadiative emission / absorption by gases• For the rotational and vibrational transitions, ea h allowable transitionprovides a line in the κν spe trum at ν0 = ∆Eh where ∆E is thedieren e in energy levels before and after the transition.

00,51,01,52,0

3300 3310 3320 3330 3340 3350 3360 3370 3380Absorption oe ient[ m−1

Wavenumber [ m−1∆ν = 0,01 m−1ν0 = 3340 m−1x 10−10 P = 1 atmfvH2O = 0,1fvN2 = 0,9T = 1200 K


Emission / Absorption, gas models 20 / 60 LBL modelLBL model• The ontributions of all transitions are added: κν =

Nlines∑i=1 κiν

00,020,040,060,080,10

3300 3310 3320 3330 3340 3350 3360 3370 3380Absorption oe ient[ m−1

Wavenumber [ m−1∆ν = 0,1 m−1 P = 1 atmfvH2O = 0,1fvN2 = 0,9T = 1200 K


Emission / Absorption, gas models 21 / 60 LBL modelLBL model• LBL model takes into a ount for a given wavenumber all thetransitions of all parti ipating gases,• The LBL model parameters are available on,- HITRAN database, at low temperature, until 47 parti ipating gases,- HITEMP, at high temperature, onsidered as the referen e database forH2O and CO2,- CDSD, at high temperature for CO2.• HITEMP database gathers 114 millions of transitions for H2O and 11millions for CO2.E ole thématique du CNRS sur la S ien e des In endies et ses Appli ations Porti io, 30/05/15 04/06/15

Emission / Absorption, gas models 22 / 60 LBL modelLBL model• Main advantages of LBL model:- the most a urate model to represent the gas ontributions (absorption/ emission),- the spe tral dis retization ∆ν is set / dened by the user withoutlimitation,- no assumption is done on the studied media.• Main drawba k of LBL model:- omputational ost:for a CFD ode, 1 absorption oe ient must be estimated ell by ellfor ea h dis rete wavenumber,for a re safety appli ation, the spe tral range is approximately between1000 and 7000 m−1,for a LBL model, the spe tral dis retization is about 0.1 or 0.01 m−1.

=⇒ Not appli able for CFD odesE ole thématique du CNRS sur la S ien e des In endies et ses Appli ations Porti io, 30/05/15 04/06/15

Emission / Absorption, gas models 23 / 60 SNB modelSNB model• Statisti al Narrow Band model introdu ed in its useful form byMalkmus in 1969, well-known database available from Souani andTaine, 1997- the spe tral dis retization is about 25 m−1,- a mean spe tral transmissivity (and not a mean spe tral absorption oe ient) is provided for ea h spe tral band,

=⇒ Some assumptions are introdu edHomogenous and isothermal media- The non-homogeneity of the medium an be a ounted for using aCurtis Godson approa h,- The SNB model provides an absorption oe ient for ea h dis retewavenumber band.E ole thématique du CNRS sur la S ien e des In endies et ses Appli ations Porti io, 30/05/15 04/06/15

Emission / Absorption, gas models 24 / 60 SNB modelSNB model

00,040,080,120,163300 3350 3400 3450 3500 3550Absorption oe

ient[ m−1

Wavenumber [ m−1

P = 1 atmfvH2O = 0,1fvN2 = 0,9T = 1200 K ∆ν = 0,1 m−1


Emission / Absorption, gas models 25 / 60 SNB modelSNB model

00,0020,0040,0060,0083300 3350 3400 3450 3500 3550Absorption oe

ient[ m−1

Wavenumber [ m−1E ole thématique du CNRS sur la S ien e des In endies et ses Appli ations Porti io, 30/05/15 04/06/15

Emission / Absorption, gas models 26 / 60 SNB modelSNB model• Main advantages of SNB model:- Computational ost saved:for a SNB model, the spe tral dis retization is about 25 m−1,for a re safety appli ation, 367 wavenumber bands are ne essary.

=⇒ Appli able for CFD odes• Main drawba k of SNB model:- Assumption done on the studied media (homogeneous and isothermalmedia),- Computational ost to treat the radiative transfer in many reappli ations.E ole thématique du CNRS sur la S ien e des In endies et ses Appli ations Porti io, 30/05/15 04/06/15

Emission / Absorption, gas models 27 / 60 SNB modelOther spe tral methods• Other spe tral methods- SNB-Ck approa h:Spe tral dis retization is about 25 m−1,Formalism based on absorption oe ient,Same assumptions than SNB model,Use of a quadrature (7 points in its lassi form) → Huge omputational ost.- Large band models su h as RADCAL used by FDS:6 spe tral bands whi h depends on the ombustible sour e.- Gray gas models:A unique value for the absorption oe ient is used.E ole thématique du CNRS sur la S ien e des In endies et ses Appli ations Porti io, 30/05/15 04/06/15

Emission / Absorption, gas models 28 / 60 SNB modelUse of spe tral approa h for re appli ations ?• Spe tral approa h required for CFD odes in re appli ations ?Obje tive: a urate al ulation of radiative energy (not its spe tralrepresentation)div ~φr = ~∇ ·

∫

λ

∫

Ω=4π Lλ(s, ~Ω)~ΩdΩdλ=

∫

λ

κλ

[4πn2λL0λ (T ) −

∫

Ω=4π Lλ(s, ~Ω)dΩ]dλ=⇒ Use of global modelsE ole thématique du CNRS sur la S ien e des In endies et ses Appli ations Porti io, 30/05/15 04/06/15

Emission / Absorption, gas models 29 / 60 WSGG approa hWSGG approa h• WSGG approa h: Weighted Sum of Gray Gases from Hottel (1954)- κ is divided into N dis rete values(N gray gases and κ0 = 0),- At ea h κj , a oe ient aj is as-signed ,aj = 1M0 ∑i M0

∆νij∆νij- RTE be omes,∂Lj∂s = −κjLj + ajκjLbwith L =

N∑j=0 LjE ole thématique du CNRS sur la S ien e des In endies et ses Appli ations Porti io, 30/05/15 04/06/15

Emission / Absorption, gas models 30 / 60 WSGG approa hOther global approa hes• SLW (Spe tral Line Weighted sum of gray gases) from Denisonand Webb (1994)An approa h similar to WSGG,Approa h extended from LBL database (whereas WSGG usesexperimental data on spe tral emissivities),• ADF (Absorption Distribution Fun tion) and FSK (Full-Spe trumCorrelated-k distribution)Approa hes similar to SLW,Other assumptions are done by the use of distribution fun tion.Databases available in the literature.E ole thématique du CNRS sur la S ien e des In endies et ses Appli ations Porti io, 30/05/15 04/06/15

Emission / Absorption, gas models 31 / 60 Con lusions on the models for gas absorptionCon lusions on absorption gas modelsModels Easy implementation Fast omputation A ura ySpe tral modelsLBL model + o +++SNB model ++ +(+) ++SNB-Ck model ++ +(+) +(+)Large band model ++ +++ +Global modelsWSGG model ++(+) +++ +SLW/ADF/FSK ++ +++ +from o (the worst) to +++ (the best)E ole thématique du CNRS sur la S ien e des In endies et ses Appli ations Porti io, 30/05/15 04/06/15

Emission / Absorption, gas models 32 / 60 Con lusions on the models for gas absorptionA few words about soot parti les...Absorption oe ient an be approximated as κλ = C0fvλ

a ording to Solovjov C0 ≈ 5.5 a ording to the SFPE handbook C0 = 6πnk

(n2−k2+2)2+4n2k2 introdu ingthe omplex index of refra tion for soot m = n− ik(problem : n and k are varying with wavelength and sour es...)from whi h the average oe ient an be omputed asκ = 3.72C0C2 fvT (with C2 = 1.4388× 10−2m.K )S attering an be negle ted for energy ex hange evaluation.E ole thématique du CNRS sur la S ien e des In endies et ses Appli ations Porti io, 30/05/15 04/06/15

ContentsIntrodu tionFrom simple laws to radiative transfer equationRadiative properties of surfa es and amesEmission / Absorption, gas modelsNumeri al methods for radiative transferZonal methodRosseland approximationDis rete Ordinate MethodFinite Volume MethodForward Monte Carlo approa hCon lusions on numeri al methodsE ole thématique du CNRS sur la S ien e des In endies et ses Appli ations Porti io, 30/05/15 04/06/15

Numeri al methods for radiative transfer 33 / 60 Zonal methodZonal method• Determination of radiative heat uxes for an absorbing, emitting andisotropi ally s attering medium,• Extension of the net radiation method developed for surfa e ex hanges,• Prin iple:- Volumes and surfa es are divided into a nite number of isothermalelements ( alled zones),- An energy balan e is arried out for the radiative ex hange between anytwo zones,- Radiative uxes, as fun tions of temperature, are obtained by a matrixsystem.E ole thématique du CNRS sur la S ien e des In endies et ses Appli ations Porti io, 30/05/15 04/06/15

Numeri al methods for radiative transfer 34 / 60 Zonal methodZonal method• Bla k surfa e ex hange - no parti ipating medium,- Net ex hange of radiative energy between any two surfa es,

Φi↔j = −Φj↔i = si sj (M0i −M0j )with si sj = sj si = AiFi→j = AjFj→i = ∫Ai ∫Aj os θi os θjπd2ij dAjdAiand M0k is the bla kbody emission (emissive power) oming from Aksurfa e.E ole thématique du CNRS sur la S ien e des In endies et ses Appli ations Porti io, 30/05/15 04/06/15

Numeri al methods for radiative transfer 35 / 60 Zonal methodZonal methodsi sj = sj si = AiFi→j = AjFj→i = ∫Ai ∫Aj os θi os θjπd2ij dAjdAix y

z b AjθjAiθi dij


Numeri al methods for radiative transfer 36 / 60 Zonal methodZonal method- In summing all the radiative ontributions oming from the N elements,the net heat ux at zone i isΦi = N∑j=1 Φi↔j = N∑j=1 si sj (M0i −M0j ) = AiM0i −

N∑j=1 si sjM0jsin e, N∑j=1 si sj = Ai- Matrix system,Φ = MM0with Mij = −sisj for i 6= j else Mii = Ai − si siE ole thématique du CNRS sur la S ien e des In endies et ses Appli ations Porti io, 30/05/15 04/06/15

Numeri al methods for radiative transfer 37 / 60 Zonal methodZonal method• Gray diuse surfa e - no parti ipating medium,- Net ex hange of radiative energy between any two surfa es (emissivepower M0 is substituted by radiosity J),

Φi↔j = −Φj↔i = si sj (Ji − Jj )Radiosity is dened as J = M0 − (1− ǫ

ǫ

) qrThe net heat ux at zone i is estimated by,Φi = Aiqri = N∑j=1 Φi↔j = N∑j=1 si sj (Ji − Jj ) = AiJi − N∑j=1 si sjJjE ole thématique du CNRS sur la S ien e des In endies et ses Appli ations Porti io, 30/05/15 04/06/15

Numeri al methods for radiative transfer 38 / 60 Zonal methodZonal methodAiqri = N∑j=1 si sj (Ji − Jj ) = AiJi − N∑j=1 si sjJj and Ji = M0i −

(1− ǫiǫi ) qriIt implies,N∑j=1 (Aiδij

ǫi − si sj (1− ǫjǫj )) qrj = N∑j=1 (Aiδij − si sj )M0iAll the net heat ux densities are known, knowing all the temperature ofany zone.


Numeri al methods for radiative transfer 39 / 60 Zonal methodZonal method• Radiative ex hange in gray absorbing / emitting media,- Surfa e-surfa e radiative ex hange between zone i and zone j is denedby

Φi→j = si sjJiin onsidering now si sj assi sj = ∫Ai ∫Aj e−κdij os θi os θjπd2ij dAjdAiwhere κ is the absorption oe ient of the medium between zone i andzone j .E ole thématique du CNRS sur la S ien e des In endies et ses Appli ations Porti io, 30/05/15 04/06/15

Numeri al methods for radiative transfer 40 / 60 Zonal methodZonal method- Volume-surfa e radiative ex hange between zone i and zone j is denedbyΦi→j = gisjM0iin onsidering gisj by,gisj = ∫Vi ∫Aj e−κdijκ os θj

πd2ij dAjdVi- Volume-volume radiative ex hange between zone i and zone j is denedbyΦi→j = gigjM0iin onsidering gigj by,gigj = ∫Vi ∫Vj e−κdij κ2

πd2ij dVjdViE ole thématique du CNRS sur la S ien e des In endies et ses Appli ations Porti io, 30/05/15 04/06/15

Numeri al methods for radiative transfer 41 / 60 Zonal methodZonal method- Radiative heat balan e at a surfa e (zone i) be omes,Φsi = Aiqri = Ns∑j=1 sjsi (Ji − Jj) + Nv∑k=1 gksi (Ji −M0k)in onsidering Ns∑j=1 sj si + Nv∑k=1 gksi = Ai for isothermal losure.- Radiative heat balan e at a volume (zone i) be omes,

Φvi = κVi (4M0i − Gi) = Ns∑j=1 sjgi (M0i − Jj)+ Nv∑k=1 gkgi (M0i −M0k )in onsidering Ns∑j=1 sjgi + Nv∑k=1 gkgi = 4κVi for isothermal losure.E ole thématique du CNRS sur la S ien e des In endies et ses Appli ations Porti io, 30/05/15 04/06/15

Numeri al methods for radiative transfer 42 / 60 Rosseland approximationRosseland approximation• Mandatory assumption: opti ally thi k media (τ = βL 1),• Examples of situations: heat transfer in hot glass or semi-transparentmaterials, sometimes used for PMMA,• This method is so- alled diusion approximation

~φr = −kr −−→grad Twith kr = 16n2σT 33β• Main drawba k: the diusion approximation is not valid near aboundary (that implies huge dis repan ies in radiative heat ux andtemperature).This approximation an be improved by Deissler's jump boundary onditions.E ole thématique du CNRS sur la S ien e des In endies et ses Appli ations Porti io, 30/05/15 04/06/15

Numeri al methods for radiative transfer 43 / 60 Dis rete Ordinate MethodDis rete Ordinate Method• Approa h proposed by Chandreskhar in 1960.• Aim:- Divide the medium into nite volumes,- Dis retize the angular spa e into dis rete dire tions,- Consider the RTE a ording to these dis rete dire tions, in using aTaylor expansion to substitute the derivative term.dLλds = −κλLλ(s, ~Ω)

︸︷︷︸Absorption − σλLλ(s, ~Ω)︸︷︷︸out-s attering+ n2λκλL0λ(T (s))

︸︷︷︸Emission + . . .

· · ·+14π ∫Ω′=4π σλPλ(~Ω

′ → ~Ω)Lλ(s, ~Ω′)dΩ′

︸︷︷︸in-s atteringE ole thématique du CNRS sur la S ien e des In endies et ses Appli ations Porti io, 30/05/15 04/06/15

Numeri al methods for radiative transfer 44 / 60 Dis rete Ordinate MethodDis rete Ordinate Method• Dire tional dis retization (−→Ω i , ωi) has to verify several onditions(moments):- Ndire tions∑i=1 ωi ≈ ∫

Ω=4π dΩ = 4π,- 0th moment: ∫Ω=4π ~ΩdΩ =

Ndire tions∑i=1 ωi ~Ωi = ~0,- 1st moment: ∫~n·~Ω<0 ~n · ~ΩdΩ =

∫

~n·~Ω>0 ~n · ~ΩdΩ =

Ndire tions∑i=1 ωi~n · ~Ωi = 0- 2nd moment: ∫Ω=4π ~Ω · ~ΩdΩ =

Ndire tions∑i=1 ωi ~Ωi · ~Ωi = 4π3E ole thématique du CNRS sur la S ien e des In endies et ses Appli ations Porti io, 30/05/15 04/06/15

Numeri al methods for radiative transfer 45 / 60 Dis rete Ordinate MethodDis rete Ordinate Method• In 1968, Carlson and Lathrop propose the SN quadrature based onN (N + 2) dire tions:m Ωmx Ωmy Ωmz ωm01 0.9709 0.1691 0.1691 0.146102 0.7987 0.5773 0.1691 0.159803 0.7987 0.1691 0.5773 0.159804 0.5773 0.7987 0.1691 0.159805 0.5773 0.5773 0.5773 0.173306 0.5773 0.1691 0.7987 0.159807 0.1691 0.9709 0.1691 0.146108 0.1691 0.7987 0.5773 0.159809 0.1691 0.5773 0.7987 0.159810 0.1691 0.1691 0.9709 0.1461 x yz

b b

b

b

b

b

b

bb

b

S8 LSH quadrature (80 dire tions)E ole thématique du CNRS sur la S ien e des In endies et ses Appli ations Porti io, 30/05/15 04/06/15

Numeri al methods for radiative transfer 46 / 60 Dis rete Ordinate MethodDis rete Ordinate Method• For a given dire tion (~Ωm), RTE be omes,dLmλds (s) = −κλLmλ (s)− σλLmλ (s) + n2λκλL0λ (T ) + . . .

· · ·+14π Ndire tions∑m′=1 σλP∗

λ (m′ → m) Lm′

λ (s)ωm′P∗ is the renormalized phase fun tion.• The derivative term an be developed in 2 dimensions by,dLmλds (s) = Ωmx dLmλdx (s) + Ωmy dLmλdy (s)with a Taylor expansion and in using standard notation for the ellneighbouring,dLmλds (s) ≈ Ωmx Lmλ (sE )− Lmλ (sW )

∆x +Ωmy Lmλ (sN)− Lmλ (sS)∆yE ole thématique du CNRS sur la S ien e des In endies et ses Appli ations Porti io, 30/05/15 04/06/15

Numeri al methods for radiative transfer 47 / 60 Dis rete Ordinate MethodDis rete Ordinate Method• Closure problem: the radiative intensity at ea h ell has to be al ulated with respe t to the propagation dire tion.For instan e, when the propagation dire tion omes from West to East,the al ulation of Lmλ (s) needs to know Lmλ (sE ), whi h are unknowns.• The use of losure s hemes over omes this kind of problem,- Step law (upwind s heme),- Bilinear s heme: Lmλ (s) = Lmλ (sE ) + Lmλ (sW )2 ,- Exponential s heme: (Lmλ (s))2 = Lmλ (sE ) Lmλ (sW ),- Weighted s heme: Lmλ (s) = αLmλ (sE ) + (1− α) Lmλ (sW ),- . . .E ole thématique du CNRS sur la S ien e des In endies et ses Appli ations Porti io, 30/05/15 04/06/15

Numeri al methods for radiative transfer 48 / 60 Finite Volume MethodFinite Volume Method• Aim:- Divide the medium into nite volumes,- Dis retize the angular spa e into dis rete dire tions,- Consider the RTE in integrating both over a given solid angle and overa given nite volume.• Radiative transfer equation is re alled,

~Ω · ~∇Lλ(s, ~Ω) = −βλLλ(s, ~Ω) + Rλ(s, ~Ω)with Rλ(s, ~Ω) is the sour e (re-infor ement) term dened by,Rλ(s, ~Ω) = n2λκλL0λ(T (s)) + 14π ∫Ω′=4π σλPλ(~Ω′ → ~Ω)Lλ(s, ~Ω′)dΩ′E ole thématique du CNRS sur la S ien e des In endies et ses Appli ations Porti io, 30/05/15 04/06/15

Numeri al methods for radiative transfer 49 / 60 Finite Volume MethodFinite Volume Method• RTE is integrated over a given nite volume and over a givenpropagation dire tion,∫

Ωm ∫VP ~Ω·~∇Lλ(s, ~Ω)dVPdΩ =

∫

Ωm ∫VP (−βλLλ(s, ~Ω) + Rλ(s, ~Ω)) dVPdΩ• Ostrogradsky's theorem simplies RTE to,∫

Ωm ∫SP Lλ(s, ~Ω)(~Ω · ~n)dSPdΩ =

∫

Ωm ∫VP (−βλLλ(s, ~Ω) + Rλ(s, ~Ω)) dVPdΩDis retization of the nite volume boundaries (in 2 dimensions),e,w ,n,s∑i ∫

Ωm Lλ,i (~Ω)(~Ω · ~ni)SidΩ =

∫

Ωm ∫VP (−βλLλ,P(~Ω) + Rλ,P(~Ω)) dVPdΩE ole thématique du CNRS sur la S ien e des In endies et ses Appli ations Porti io, 30/05/15 04/06/15

Numeri al methods for radiative transfer 50 / 60 Finite Volume MethodFinite Volume Method• Assumption: ea h intensity is onstant over all elements (surfa es orvolumes),e,w ,n,s

∑i Lλ,i (Ωm)Si ∫Ωm (~Ω · ~ni) dΩ =

(−βλLmλ,P + Rm

λ,P)VP∆Ωm• Ea h dire tion oe ient is stri tly al ulated from the angulardis retization, Ci = ∫

Ωm (~Ω · ~ni)dΩ• Final form,e,w ,n,s

∑i Lλ,i (Ωm)SiCi = (−βλLmλ,P + Rmλ,P)VP∆ΩmE ole thématique du CNRS sur la S ien e des In endies et ses Appli ations Porti io, 30/05/15 04/06/15

Numeri al methods for radiative transfer 51 / 60 Finite Volume MethodFinite Volume Method• The al ulation of Lmλ,P depends on the knowledge of Lλ,i (Ωm) (at theboundaries !): it is a losure problem.• Chai et al. propose, in 1994, the modied exponential s heme:

∂LmP∂s = −βma LmP + RmaP

βma = βm −σm4π P (m,m)∆Ωmand Rma = κL0(T ) +

nd∑m′=1,m′ 6=m σ4πLm′P P (m′,m)∆Ωm′Lme = LmP e(−βma )Pdme +

(Rmaβma )P [1− e(−βma )Pdme ]E ole thématique du CNRS sur la S ien e des In endies et ses Appli ations Porti io, 30/05/15 04/06/15

Numeri al methods for radiative transfer 52 / 60 Finite Volume MethodDOM vs. FVM: what is the dieren e ?• DOM: the al ulated intensity is the dis rete intensity a ording to agiven propagation dire tion,• FVM: the al ulated intensity is a mean value obtained in a solid anglea ording to a propagation dire tion.

→ The onvergen e of FVM is faster than the one of DOM be auseFVM ensures a good radiative balan e.Nevertheless, . . . for a stru tured Cartesian mesh, the two approa hes arequite similar, and the dieren e lies on the approximation,∫

Ωm (~Ω · ~nx)dΩ ≈ Ωmx ∆ΩmE ole thématique du CNRS sur la S ien e des In endies et ses Appli ations Porti io, 30/05/15 04/06/15

Numeri al methods for radiative transfer 53 / 60 Forward Monte Carlo approa hForward Monte Carlo approa h• Obje tives:- Follow several millions of quanta (bundles of photons with a givenenergy amount) from the radiative sour e (emission) until these quantadisappear or are totally absorbed,- Propagation of these quanta through a parti ipating medium withrespe t to radiative laws: s attering, absorbing and emitting phenomena.• Ea h quantum is hara terised by:- an energy amount (total or spe tral),- a propagation dire tion.E ole thématique du CNRS sur la S ien e des In endies et ses Appli ations Porti io, 30/05/15 04/06/15

Numeri al methods for radiative transfer 54 / 60 Forward Monte Carlo approa hForward Monte Carlo approa h• Quantum emission from a surfa e (Nquanta followed)- the quantum position is randomly dened on the surfa e,- the propagation dire tion (dened by θ and ϕ) issele ted a ording to these laws:

θ = 2πRθ and os2 ϕ = Rϕwhere Rθ and Rϕ are random variables between 0and 1. x yzθ

ϕ- the radiative energy transported by ea h quantum is dened by,Qr = SǫσbT 4Nquanta for grey surfa eQr ,λ =SǫλπL0λ (T )Nquanta for non-grey surfa eE ole thématique du CNRS sur la S ien e des In endies et ses Appli ations Porti io, 30/05/15 04/06/15

Numeri al methods for radiative transfer 55 / 60 Forward Monte Carlo approa hForward Monte Carlo approa h• Quantum emission from a volume (Nquanta followed)- the quantum position is randomly dened inside the nite volume,- the propagation dire tion (dened by θ and ϕ) issele ted a ording to these laws:

θ = 2πRθ and osϕ = 1− 2Rϕwhere Rθ and Rϕ are random variables between 0and 1. x yzθ

ϕ- the radiative energy transported by ea h quantum is dened by,Qr = 4κVσbT 4Nquanta for grey surfa eQr ,λ =4πκλVL0λ (T )Nquanta for non-grey surfa eE ole thématique du CNRS sur la S ien e des In endies et ses Appli ations Porti io, 30/05/15 04/06/15

Numeri al methods for radiative transfer 56 / 60 Forward Monte Carlo approa hForward Monte Carlo approa h• Quantum tra king inside the parti ipating media- The propagation of ea h quantum follows a straight line until anabsorption or s attering phenomenon,- Several approa hes exist to tra k the quanta.• MCM 1.1 - Ea h quantum propagates inside the parti ipating mediumuntil an intera tion distan e (Sβ) estimated by,Rβ = exp(− ∫ Sβ0 βds)- At this position, a new random number is sampled and ompared withalbedo (ω = σ/β) to sele t the quantum/medium intera tion,- If Rω < ω , the quantum is s attered into a new propagation dire tion,- If Rω > ω , the quantum energy is fully absorbedwhere Rβ and Rω are random variables between 0 and 1.E ole thématique du CNRS sur la S ien e des In endies et ses Appli ations Porti io, 30/05/15 04/06/15

Numeri al methods for radiative transfer 57 / 60 Forward Monte Carlo approa hForward Monte Carlo approa h• MCM 2 - Ea h quantum propagates inside the parti ipating mediumuntil an intera tion distan e (Sσ) estimated by,Rσ = exp(− ∫ Sσ0 σds)- At this position, an amount of the quantum energy dened by,Pabsorbed = 1− exp(− ∫ Sσ0 κds)is absorbed by the medium and another part,Ps attered = exp(− ∫ Sσ0 κds)is s attered in a new propagation dire tion. Rσ is a random variablebetween 0 and 1.E ole thématique du CNRS sur la S ien e des In endies et ses Appli ations Porti io, 30/05/15 04/06/15

Numeri al methods for radiative transfer 58 / 60 Forward Monte Carlo approa hForward Monte Carlo approa h• Other solutions exist: MCM 1.2, MCM 2.2, MCM 3, Re ipro al MC,. . .Propagationdire tion

Newdire tionΘ

Φ• S attering phenomenon:- A new propagation dire tion issele ted with respe t to the phasefun tion as follows,RΘ =

12 ∫ Θ0 P (Θ) sinΘdΘΦ = 2πRΦwhere RΘ and RΦ are random variables.E ole thématique du CNRS sur la S ien e des In endies et ses Appli ations Porti io, 30/05/15 04/06/15

Numeri al methods for radiative transfer 59 / 60 Con lusions on numeri al methodsCon lusions on numeri al methodsApproa h Easy implementation Fast omputation A ura yRosseland +++ +++ +Zonal method +(+) ++(+) +DOM + ++ +(+)FVM + ++ ++Monte Carlo ++(+) + +++from o (the worst) to +++ (the best)


Numeri al methods for radiative transfer 60 / 60 Con lusions on numeri al methodsReferen e booksM.F. Modest, Radiative Heat Transfer 3rd Ed., Elsevier, 2013.J.R. Howell, R. Siegel, P. Mengüç, Thermal Radiation Heat Transfer, 6thEd., Taylor & Fran is, 2015.


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