fundaments of combustion & flame typical burner flame fuel air premixed flow inner flame outer...
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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.