X.S. Bai Laminar Non-premixed Flames
Content
• Structures of laminar non-premixed flames • Mixture fraction • Burke-Schumann flame-sheet model • Jet diffusion flames • Laminar diffusion flame at high mixing rate
X.S. Bai Laminar Non-premixed Flames
Where is the reaction zone?
Zst
Burnable mixture
A F
Diffusion velocity: Fick’s Law
YiVi = Di!Yi, Vi = Di / Yi!Yi "Di / !
!Mixing length
X.S. Bai Laminar Non-premixed Flames
Where is the reaction zone?
Zst
Burnable mixture
A F
O2
O2
F
F
Preheat zone Post flame zone
X.S. Bai Laminar Non-premixed Flames
Where is the reaction zone?
Zst
Burnable mixture
A F
O2
O2
F
F
Preheat zone Post flame zone
X.S. Bai Laminar Non-premixed Flames
Where is the reaction zone?
Zst
Burnable mixture
A F
O2
O2
F
F
Preheat zone Post flame zone
X.S. Bai Laminar Non-premixed Flames
Where is the reaction zone?
Zst
Burnable mixture
A F
O2
O2
F
F
Preheat zone Post flame zone
X.S. Bai Laminar Non-premixed Flames
Where is the reaction zone?
Zst
Burnable mixture
A F
O2
O2
F
F
Preheat zone Post flame zone
X.S. Bai Laminar Non-premixed Flames
Where is the reaction zone?
Zst
Burnable mixture
A F
O2
O2
F
F
Preheat zone Post flame zone
X.S. Bai Laminar Non-premixed Flames
Where is the reaction zone?
Zst
Burnable mixture
A F
O2
O2
F
F
Fuel-rich mixing zone Reaction zone Oxidizer-rich mixing zone
X.S. Bai Laminar Non-premixed Flames
Where is the reaction zone?
Zst
Burnable mixture
Fuel-rich Fuel-lean
T
Z Zst
Tst
T1 T2
0 1
Tcr
A F
X.S. Bai Laminar Non-premixed Flames
Burke-Schumann flame-sheet structure
A P P F
Zst
1
Yi
Z
Reaction zone
Fuel-rich Fuel-lean
Reactions occur in a zero thickness sheet at Z=Zst
Soot problem
X.S. Bai Laminar Non-premixed Flames
Burke-Schumann flame-sheet structure
Reaction zone
Reactions occur in a zero thickness sheet at Z=Zst
Fuel-rich Fuel-lean
T
Z Zst
Tst
T1
T2
0 1
NOx problem
X.S. Bai Laminar Non-premixed Flames
Content
• Structures of laminar non-premixed flames • Mixture fraction • Burke-Schumann flame-sheet model • Jet diffusion flames • Laminar diffusion flame at high mixing rate
X.S. Bai Laminar Non-premixed Flames
Laminar jet diffusion flames
Zst
Mixing layer
A non-reactive fuel jet
Fuel
Air
Zst
Burnable mixture
X.S. Bai Laminar Non-premixed Flames
Laminar jet diffusion flames
Air
A+P
Air F+P
A jet diffusion flame Fuel
Fuel rich
Air rich
Reaction zone
a-b
c-d c-d
O2 F P
a-b
O2 F
1, 0, 1 1
stF A P
st st
Z Z ZY Y YZ Z− −
= = =− −
0, 1 , F A Pst st
Z ZY Y YZ Z
= = − =
soot NOx
Fuel rich zone
Fuel lean zone
X.S. Bai Laminar Non-premixed Flames
Jet diffusion flame: mathematical description
Zst
Mixing layer
A non-reactive fuel jet
Fuel
Air
11F P F P st
A
Z Y Y Y Y Zγ
= + = ++
FZ Y=
Combustion
No combustion (pure mixing)
!!Z!t
+"# (!!vZ) ="# (!D"Z)
!!Yi!t
+"#!Yiv!="#!Di"Yi +"i
!!YF!t
+"#!YFv!="#!DF"YF +"F
!!YP!t
+"#!YPv!="#!DP"YP +"P
DF = DP = D, !F +!PZst = 0
X.S. Bai Laminar Non-premixed Flames
Height of laminar jet flame
R
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F + P
A + P air
Lf
R
vf
x
Lf !vFR
2
D
!!Z!t
+"# (!!vZ) ="# (!D"Z)
!vF / Lf
!D / R20 Mixing layer analysis
X.S. Bai Laminar Non-premixed Flames
Height of laminar jet flame
R
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F + P
A + P
air Lf
R
vf
x
Lf !vFR"D!vFR
2
D
X = vFt,
Y = R !VO2t = R !1!DO2t
Mixing layer analysis
YiVi = Di!Yi, Vi = Di / Yi!Yi "Di / !
X = Lf = vFtf ,
Y = 0 = R ! 1!DO2tf
X.S. Bai Laminar Non-premixed Flames
Jet diffusion flame: experimental obervation
O
F
O
U x
Pe 50 Pe 50 Pe 200 Pe 1
Lf !vFR
2
D=vFRD
!
"#
$
%&R = Pe 'R
X.S. Bai Laminar Non-premixed Flames
Propagation of laminar diffusion flame
N+5 PDE equations
!!!t+"#!v
!= 0
!!!v
!t+"!
!v!v ="# pI+!( )
!DhDt
!DpDt
="# !""h ! !" 1! 1Lei
$
%&&
'
())hi"Yi
i=1
N
*$
%&&
'
())+!Qr +# :"
"v
!!Yi!t
+"#!Yiv!="#!Di"Yi +"i
X.S. Bai Laminar Non-premixed Flames
Burke-Schumann flame-sheet model
!!!t+"#!v
!= 0
!!Z!t
+"# (!!vZ) ="# (!""Z)
Yi = a + bZT = c+dZ
6 PDE equations
!!!v
!t+"!
!v!v ="# pI+!( )
!DhDt
!DpDt
="# !""h ! !" 1! 1Lei
$
%&&
'
())hi"Yi
i=1
N
*$
%&&
'
())+!Qr +# :"
"v
Burke-Schumann flame sheet relation
X.S. Bai Laminar Non-premixed Flames
Content
• Structures of laminar non-premixed flames • Mixture fraction • Burke-Schumann flame-sheet model • Jet diffusion flames • Laminar diffusion flame at high mixing rate
X.S. Bai Laminar Non-premixed Flames
Finite-rate chemistry effect
• Burke-Schumann flame-sheet model is not always true; it is true only when the chemical reactions are much faster than the mixing of fuel and the oxidizer
• If the mixing is very fast, what is going to happen?
Zst
Burnable mixture
X.S. Bai Laminar Non-premixed Flames
Flamelet equation
!!Yi!t
+"#!Yiv!="#!Di"Yi +"i
Yi(x,y,z, t) =Yi(Z(x,y,z, t)),
!(x,y,z, t) =!(Z(x,y,z, t)),T(x,y,z, t) = T(Z(x,y,z, t))
Flamelet assumption:
!!Z!t
+"# (!!vZ) ="# (!D"Z)
Species transport
Mixture fraction
X.S. Bai Laminar Non-premixed Flames
Flamelet equation
!12!"d2YidZ2
=#i
d2YidZ2
= !2!i
"#
! = 2D(!Z "!Z) Scalar dissipation rate: mixing rate
Damköhler number Da = (mixing time)/(chemical reaction time) Da ! 2!i"#
X.S. Bai Laminar Non-premixed Flames
Finite rate chemistry effect
F O2
F O2
2
2id Y
dZ→∞
2
2 finiteid YdZ
=
Damköhler number infinity Damköhler not very high
iY iY
Z ZstZ
stZ
stZ
X.S. Bai Laminar Non-premixed Flames
Effect of mixing rate
F O2
iYiY
Z Z
χ ↑
χ ↑
χ ↑
stZ stZ
Increase of scalar dissipation rate leads to • the leakage of fuel to the oxygen rich side increases • the leakage of oxygen to the fuel rich side increases • the flame temperature decreases, as a result of insufficient oxidation of
the fuel
X.S. Bai Laminar Non-premixed Flames
Flame quenching at high scalar dissipation rate
Adiabatic flame
With preheating
With radiation heat loss
Flame sheet Flamelet Quenched
PT
[ ]1P sχ −
qT
1qχ−
X.S. Bai Laminar Non-premixed Flames
Laminar jet diffusion flame
Air
A+P F+P
Lift-off hight
Diffusion flame
Lean premixed flame Rich premixed
flame
Fuel inlet
c
Zst
Burnable mixture
X.S. Bai Laminar Non-premixed Flames
Effect of mixng rate
• For very small scalar dissipation rate the flame temperature is high; changing the scalar dissipation a little would not affect the flame temperature. This regime can be described by the simple Burke-Schumann flame sheet model.
• For intermediate scalar dissipation rate, the flame temperature is lower than the Burke-Schumann flame temperature. Increasing the scalar dissipation rate leads to a decrease of the flame temperature until to a critical condition, at which the flame temperature changes rapidly. This regime may be called flamelet
• For scalar dissipation rate higher than the quenching scalar dissipation rate, no flame exists. The flame is quenched.