Fundaments of Combustion & Flame

typical burner flame

fuel

airair

premixedflow

Inner flame

outerflame

air

Stream lines

temperatur

e

distance along streamline

reaction range

oxygenfuel

diffusion layer

diffusion diffusion

reactionlayer

burnt gas

unburnt gas

flamepropagati

on

pre-heatedlayer

premixed flame (inner flame)

diffusion flame (outer flame)

火炎解析へのアプローチ　–予混合火炎モデル -

予混合火炎例：ブンゼンバーナ、ガスコンロ•混合比の定まった予混合ガスの未燃、既燃ガス界面における燃焼•火炎は伝播性を持つ•反応は温度律速•火炎特性（火炎面など）を表現する関数反応進行度、火炎伝播速度

反応帯既燃ガス温度

未燃ガス温度

伝播

予熱帯

air fuel

uin

ubsininu uu

unburnt burnt

Tu

Streamline

u

uu

Tb

b

ub

bbuu uu

( 燃焼速度 )

uuu

予混合火炎の flamelet モデル

GSx

Gu

t

GL

jj

● 実用燃焼機器における予混合燃焼流れでは実用燃焼機器における予混合燃焼流れでは　　　　 KolmogrovKolmogrov スケール スケール ≫≫ 火炎面厚さスケール 火炎面厚さスケール

流れ変動時間スケール 流れ変動時間スケール ≫≫ 化学反応素過程の時間スケール 化学反応素過程の時間スケール

● ● GG 方程式方程式 (Kerstein, 1988) ; 火炎面の輸送を表す .

SL;(層流 )火炎速度SL

T(t1) T(t0)

uSLCpT ～ h　 　 　 　 　 ～ Cp∂T∂x)δ

T ＝ Tb － TuG

GnnSvv Lp

,

GG

GSGvGnSv LL

δ

Weak points of G-equation modeling

1. A pure convection equation tends unstable in numerical solution without diffusion term.

2. An initial profile is conserved in time evolution ， even if inappropriate.

a. Use upwind scheme or add numerical diffusion.

b. Reset the profile adjusted to physical or mathematical meaning; ex. distance function.

　　　（ level-set method ）

？？

X

G

未燃 既燃

Analysis of local profile near the flame surface 1D plane flame

• Considering a finite thickness – Add a diffusion term explicitly, – Give a spatial profile of S 　 by Taylor’s expansion around G= G0

0)(

x

GSu

t

Guu

x

uuSu

Unburnt(G<0)

Burnt(0<G)

G=G0

......)()( 010 GGSSGS

0

1GGdG

dSS

uuSS 0 ，・・・，

2

2

010 )}({x

G

x

GGGSSu

t

G

Burger’s eq.

2

2

xxt

010 GGSSu 2

2

010 )}({

x

G

x

GGGSSu

t

G

011 0 GG1010 SSuSSu

tSuxS

S

Su

01

1

0

2tanh

tSuxS

G 01

2tanh

tvx

G ftanh

2

11

0

S

x

GGG0Suv f ,

hyperbolic tangent profile

Analysis of profile in the flame

• Dependency on variation of 　　　(analogy of u = const.)

• Density weighted flame speed

202

*01

*0

** )( GGSGGSSGSS

00

00***~

T

TGGTS

T

TGTS

SS i

uu

ui

i

i

T

TT

TT

TTG u

ub

u

uuTT an

dif

20

*2

*1

000

*1

0

20

00

00*20

00

00*1

*0

202010

* ~~~

GGSST

TGGSS

GGT

GGTTSGG

T

GGTTSS

GGSGGSSS

uu

2

2

01*

0 x

G

x

GGGSSu

t

Guu

Analysis of profile in the flame (cont.’d)

.~

0 constSuSu uu In laminar plane flame

fx x v t 00

~Suv f ( ) ;

2

2

00

1*

x

G

x

GGG

S

t

G

x

SGG

0

1*

0 2tanh

0

1*

2

10

S

x

GGG

tvx

G ftanh

Dependency on Variation of 　

2

2

0

21

x

G

x

GGG

t

G

2

0

*1S( )

１ D example

0

0.2

0.4

0.6

0.8

1

0 200 400 600 800 1000

10step20step30step40step50step

100stepinitial

Fig.1 Time marching solution of new eq. from linear initial profile for CH4:O2:N2=1:2:3 premixed flame.

X [m]

old+diff.100 steps

new25 steps old

100 steps

0

0.5

1

1.5

2

0 100 200 300 400 500 600 700 800 900 1000

G[-

]

Fig.2 Time marching solutions of new and old eqs. from the same linear initial profile.

1D Flame propagation by 　

new G-eq.

old G-eq.

X [m]

2

2

010 )}({x

G

x

GGGSS

Dt

DG

2

2

x

G

x

GS

Dt

DGL

２ D example

0.1102[s]

0.0102[s]

0.0002[s]

Burnt gas flow out

Fig.4 G-profile and steam lines at 0.1802[s]

•Fluid dynamic flame instability by converging/diverging stream lines.

0.0302[s]

0.0702[s]

•Curvature effect on local flame speed rounds the flame shape.

Analysis of solution around a spherical flame

R

nSuu 0

r

GGG

S

r

GS

r

G

r

n

r

G

r

Gu

t

G

00

1*

0*

2

2

GGGS

Sx

G

x

Gu

t

Guu

jjj

00

1*

2

2

3D formulation

• Cylindrical (r-) coordinate 　 (n: dimension)

　　　　　

　　　　　 on flame surface

r

G

r

nS

t

G

)(0

*

R

nSS uu 00

0*

:shrink(extinct)

:expandr

Su

i ｆ u=0, G=G0, 02

2

r

G

Analysis of solution with a streching flow • Flame in a stretching flow

• Flame speed dependency

LDL 00 SSS

nvnn D , L: Markstain No. ( ～ 1)

2

2

1 x

G

x

GGS

G

x

D

(Calvin 1985)

2

'

'

1 1

0

DS

x

GGG

DDD

1

2

1 211

'Sif

S

2

01

000 11'

'

DD

SS

SSS

0

0

S

x

GS

dt

dGif

Constant flame speed

burnt

unburntx

x

u

D

u

Methane-air lifted non-premixed jet flame Muniz and Mungal,Combustion and flame 111, 1997fuel tube inner diameter D=4.8 [mm] SL

0max of methane-air flame=0.37[m]

(for ex. Re=4900,avaraged Lift-off height is about 30D=150 [mm])

Velocity(Co-flow )

Fuel

CoflowInlet diameter

Reynolds No.

AIR

D=4.8 　(mm)

4900

Velocity (Fuel )

CH4 (99% Vol.)

Ujet=15.0 [m/s]

Uco=0.74 [m/s]

D

60D

Fuel

air 20D

Domain ( X,R,θ)= (60D, 20D, 2π)No. of Grid cells(X,R, θ)=(200,82, 32)

(www.efluids.com)

Modeling for Partially Premixed Flame

Edge flame extinction Triple flame

propagation Frame-front propagation

Blow-out criterion

Air

Fuel

Classification of the flame position

FLAME C FLAME A

Classification ofChen et al.(2000)

Flamelet equation

Schematic Figure of Triple flame

2 scalar flamelet model of partially premixed flame

u L

GG S G

t

u～

FlameletG-equation

Premixed flamepropagation

jj j j

ut x x x

Mixture fraction equation

Diffusion flame,Mixing of Fuel

Lifted diffusion flame RANS:Muller et al(1994), Herrmann et al.(2000)LES:Duchamp et al (2001),LES:Hirohata et al(2001)

Air

Fuel

Un-burnt gas:Mixing zone

Burnt gas:Diffusion flame zone

Partial Premixed flame front

st

G=G0

The G-equation is used to distinguish between the unburnt and burnt regions,the iso-surface of G is used to express flame surface.

Mass fraction model using 2scalar flamelet.

Unburnt mixing zone (G=0) flame surface(G=0.5)Diffusion flame zone (G=1)

～ ～ ,uY Y

～ ～ 1 '2,0

, ,bY Y P d

TYG ,,

10 1 YYY

YY

～ ～～

2

0 1 , 1i i

T L pg q q qL st q

u u aS S C f f C

S a

～

Quenching effect of turbulent burning velocity

Burnt gas without quenchingmodel fq=1

Burnt gas with quenching model

Flame tips can not quench where the strong shear existsLift-off height and flame shape cannot be predicted without quenching model.

LDL 00 SSS L: Markstain No. ( ～ 1)

nvnn D ,

18

Premixed flame with the mixture rate gradientPremixed flame with the mixture rate gradient

• Flame speed dependency on defined position

based on the flamelet approach

G=0.25 G=0.5 G=0.75

19

Premixed flame with the mixture rate gradientPremixed flame with the mixture rate gradient

– Fuel ratio (ξ) gradient normal to flame surface (G)– Flame speed SL is basically depend on ξ

Is flame speed same as simple plane flame?

iso-surface of G=0.5

20

Premixed flame with the mixture rate gradientPremixed flame with the mixture rate gradient

• mixture rate gradient normal to flame face

⇒ gradient of flame speed

⇒ thinner flame

Flame speed gradienton the flame surface

: turbulent flame thickness

: turbulent flame speed gradient

Level-set to phase field in flame• Distance function (scale in space)

　

• Progress variable (scale in time)

Gxxx 21tanh2 10

*

’ ～　 ： observed thickness (if　　s’/’　 ～　 s/

x (~x’) real distance along streamlimes

speedflameobserved:'

flameintimeprogress'/'

''

0

*

ss

sxt

GfxftYCO

corrugated or wrinkling turbulent flame

x’: observed distance in averaged flame

speedflame:

flamein timeprogress/

reaction)(

0

0

*

s

sxt

GfxftY

plane laminar flame

x : distance from flame surface

00 GxG

G=G(x)

x X

G

unburnt burnt

gas flow

Level-set to phase field in flameSteady propagating flame solution:

x

GG

S

xx

GSu

t

G 210 2

=const

2111010010 2

)()( GS

CGSGSSx

GGGSS

=0

if S1→0 　 (S1/=const)

x

gxxx

gDt

D 2

2

'

)eq-G clasical(' 1

01

cg

ccg

Level-set form Phase field form

2

' 2gd

ggF

(F ~ quadratic)

thin flame assumption

=const

211 2

GS

Cx

G

If steady solution exists,

Level-set to phase field in flame

ExampleCHEMKIN 　（ GRI-Mech 3.0 ）CH4- O2 (φ=1) ＋ N2 50% 　 300K

0

500

1000

1500

2000

2500

3000

0 0.05 0.1 0.15x ［ mm ］

tem

pera

ture

1400+1100tanh{(x-x0)/}

CHEMKIN

Inage’s Hyperbolic Tangent Approximation (Inage et.al. 1989)

22 18

S

Dt

D: progress variable

Assumed solution for laminar plane flame:

Suv

tvx

f

f

2tanh1

2

1

PH for non-equilibrium flame interfaceAllen-Cahn equation

0'22

FMt

(Allen & Cahn 1979, Chen 1992)

01214

22

t

x2

tanh12

1

Model of steady liquid-solid phase interface (vf=0)

This formulation insure the second law of thermal mechanics, so that F() (=Free energy) decreases in time

2222

,12

MF

Model of growing liquid-solid phase interface

1D plane surface:

01121 22

bat

236

12

2222

ba

F

bva

tvx

f

f

4,

2

2tanh1

2

1

F （）

PH for non-equilibrium flame interface

01

F

Source term fitting to Inage’s model

16

42 Sat

Sba

3

21

21

2 222 SSF

- 40000

- 30000

- 20000

- 10000

00 1000 2000 3000

　(K)

G (kJ/kg)

Reactionfast

Gibbs’ free energy: G=H-TS for gas reaction in constant pressure

Thermal equilibrium CAN’T be assumed in a flame with large temperature change.

Reactionslow

Reactionslow

Gibbs’ free energy progresses in CH4/air flameFunctional F() of Allen-Cahn eq.

Based on Phase Field MethodModified A-C eq. for temperature variation

Ct

Tt

T

TFMt

0,'22

(Fife 2000)

wqTvTF ,

F

Mt

Estimated by numerical solution

CHEMKIN solution of homogeneous conditionCH4-O2 (stoichimetric 600K)

Compare to Inage’s model 0

4000

8000

0.0 0.5 1.0

Mf Inage’s model

dt 23

62

b

22 12

a

PH for non-equilibrium flame interfaceModified A-C eq. with internal heat sink

0

0,22

TDt

DSTTc

t

Tc

S

TSGSM

Dt

DS

v

(Fife 2000)

→S ： EntropyH(=cT): Enthalpy is conserved F→G(S,T) ： Gibbs’free energy

Internal heat sink by convection holds a steady flame.

Tu

u

Tb

022

S

GSM

Dt

DS

Local homogeneous

Local equilibrium assumed in a steady flame(T locally balanced). ⇔ analogy to spinodal phase change

0

qv

S

GMTcT

t

cT

Approx.

S

GM

qvcT Dt

DST

x

T

S

Su

Sb

PH for non-equilibrium flame interface

k

T

dS

dT

T

TkS

ST

GT

S

G

constHifSTHG

dS

dT

T

G

S

G

S

TSG

ST

ST

,ln

,

.

,

0

Estimation in flame:

bb T

TT

T

TT

Dt

DSTT lnln10thenif

0

Modified A-C eq. for gas reaction with large temperature raise

bT

TTTM

S

TSGTM

Dt

DSln

,

0

0,22

cTDt

DSTcT

t

cT

S

TSGSM

Dt

DS

v Tb: burnt gas temperature(under H=const. & cTu ≪TS)

Reactions stop at burnt region ：

M(T) corresponds to the reaction speed which should increase as temperature raise ：

b

b

TTM(T)

T

TTM

/:lawlinear 2)

exp)( :law 'Arrheniuts analogy to 1)

0

1000

2000

3000

4000

0 500 1000 1500 2000 2500

T*(dS/ dt)M(T)*T*(dG/ dS)T*T*(dG/ dS)T*T 2̂*ln(T/ Tb)T*T*(Tb-T)

PH for non-equilibrium flame interfaceModified A-C eq. for gas reaction with large temperature raise

bT

TTTM

S

TSGTM

Dt

DSln

,

0

0,22

TDt

DSTTc

t

Tc

S

TSGSM

Dt

DS

v Tb: burnt gas temperature(under H=const. & cTu ≪TS)

[A] S

TSGTM

Dt

DS

,

by num. solutionin homogeneous

[B] )exp(T

TTM b

[C1] bTTTM /

dS

dT　　　 is estimated

by num. solutionin homogeneous

[C2] bTTTM /bT

TT

S

Gln

and

[D]

bb T

T

T

TS1

82

Inage’s modelT[K]

TS[KJ]

[A][B]

[C2]

[C1]

[D]

Ex. CH4/Air premixed flame

Conclusive remarks• Modified level-set function for premixed flamelet

(G-eq) is derived to consider the flame thickness.

• Inage’s flame model (progress variable) is considered by phase-field method based on Allen-Cahn eq.

• Modified level-set function is consistent to phase-field method based on Allen-Cahn eq., where the hyperbolic tangent profile is a common approximated solution.