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Lecture 6. Laminar Non-premixed Flames
part 2
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?
Mixing length !
Zst
A
F
Burnable mixture
Diffusion velocity: Fick’s Law
Yi Vi = Di !Yi ,
X.S. Bai
Vi = Di / Yi !Yi " Di / !
Laminar Non-premixed Flames
Where is the reaction zone?
O2
F
F
O2
Zst
A
Post flame zone
Preheat zone
F
Burnable mixture
X.S. Bai
Laminar Non-premixed Flames
Where is the reaction zone?
O2
F
F
O2
Zst
A
Post flame zone
Preheat zone
F
Burnable mixture
X.S. Bai
Laminar Non-premixed Flames
Where is the reaction zone?
O2
F
F
O2
Zst
A
Post flame zone
Preheat zone
F
Burnable mixture
X.S. Bai
Laminar Non-premixed Flames
Where is the reaction zone?
O2
F
F
O2
Zst
A
Post flame zone
Preheat zone
F
Burnable mixture
X.S. Bai
Laminar Non-premixed Flames
Where is the reaction zone?
O2
F
F
O2
Zst
A
Post flame zone
Preheat zone
F
Burnable mixture
X.S. Bai
Laminar Non-premixed Flames
Where is the reaction zone?
O2
F
F
O2
Zst
A
Post flame zone
Preheat zone
F
Burnable mixture
X.S. Bai
Laminar Non-premixed Flames
Where is the reaction zone?
O2
F
F
O2
Oxidizer-rich mixing zone
Zst
A
Reaction zone
Fuel-rich mixing zone
F
Burnable mixture
X.S. Bai
Laminar Non-premixed Flames
Where is the reaction zone?
T
Tst
Tcr
T1
T2
0
Zst
Fuel-lean
Z
1
Fuel-rich
Zst
A
F
Burnable mixture
X.S. Bai
Laminar Non-premixed Flames
Burke-Schumann flame-sheet structure
Soot problem
Yi
1
A
P
P
F
Zst
Fuel-rich
Fuel-lean
Reaction zone
X.S. Bai
Z
Reactions occur in
a zero thickness sheet
at Z=Zst
Laminar Non-premixed Flames
Burke-Schumann flame-sheet structure
T
NOx problem
Tst
T1
T2
Z
0
Zst
Fuel-rich
Fuel-lean
Reaction zone
X.S. Bai
1
Reactions occur in
a zero thickness sheet
at Z=Zst
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
Air
Fuel
A non-reactive fuel jet
Zst
Burnable mixture
X.S. Bai
Laminar Non-premixed Flames
Laminar jet diffusion flames
soot NOx
F
Reaction zone
A+P
Air rich
Air
P
O2
c-d
F+P
Fuel
rich
c-d
Air
a-b
F
O2
Fuel
A jet diffusion flame
a-b
YF =
Z − Z st
1− Z
, YA = 0, YP =
1 − Z st
1 − Z st
YF = 0, YA = 1 −
X.S. Bai
Z
Z
, YP =
Z st
Z st
Fuel rich zone
Fuel lean zone
Laminar Non-premixed Flames
Jet diffusion flame: mathematical description
Zst
Mixing
layer
Air
D F = D P = D,
Fuel
A non-reactive fuel jet
Z = YF +
! F + ! P Zst = 0
!
!! Z
+ " # ( ! vZ) = " # ( ! D"Z)
!t
1
YP = YF + YP Z st
1+ γ A
Z = YF
X.S. Bai
!
!! Yi
+ " # ! Yi v = " # ! Di "Yi + " i
!t
!
!! YF
+ " # ! YF v = " # ! D F "YF + " F
!t
!
!! YP
+ " # ! YP v = " # ! D P "YP + " P
!t
Combustion
No combustion (pure mixing)
Laminar Non-premixed Flames
Height of laminar jet flame
Mixing layer analysis
0
x
!
!! Z
+ " # ( ! vZ) = " # ( ! D"Z)
!t
R
A+P
Lf
F+P
!D / R 2
air
!v F / Lf
vf
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R
X.S. Bai
vFR 2
Lf !
D
Laminar Non-premixed Flames
Height of laminar jet flame
Mixing layer analysis
X = v F t,
x
1
Y = R ! VO2 t = R ! DO2 t
!
R
A+P
X = Lf = v F t f ,
Lf
F+P
air
1
Y = 0 = R ! DO2 t f
!
vf
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memory to open the image, or the image may have been corrupted.
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appears, you may have to delete the image and then insert it again.
R
Yi Vi = Di !Yi ,
X.S. Bai
v F R" v F R 2
Lf !
!
D
D
Vi = Di / Yi !Yi " Di / !
Laminar Non-premixed Flames
Jet diffusion flame: experimental obervation
vFR 2 ! vFR $
Lf !
=#
& R = Pe ' R
D
" D %
O
x
U
F
O
Pe 50
X.S. Bai
Pe 50
Pe 200
Pe 1
Laminar Non-premixed Flames
Propagation of laminar diffusion flame
!
!!
+ " # !v = 0
!t
!
!! Yi
+ " # ! Yi v = " # ! Di "Yi + " i
!t
!
!!
!! v
+ "! vv = " # pI + !
!t
(
N+5 PDE equations
)
N $
$
'
Dh Dp
1 '
! + # : "v"
!
!
= " # && !""h ! !" *&&1!
)) h i "Yi )) + Q
r
Dt Dt
Le
%
(
%
(
i=1
i
X.S. Bai
Laminar Non-premixed Flames
Burke-Schumann flame-sheet model
!
!!
+ " # !v = 0
!t
6 PDE equations
!
!! Z
+ " # ( ! vZ) = " # ( !""Z)
!t
!
!!
!! v
+ "! vv = " # pI + !
!t
(
)
N $
$
'
Dh Dp
1 '
! + # : "v"
!
!
= " # && !""h ! !" *&&1!
)) h i "Yi )) + Q
r
Dt Dt
Le
%
(
%
(
i=1
i
Yi = a + bZ
T = c + dZ
X.S. Bai
Burke-Schumann flame sheet relation
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
Flamelet assumption:
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))
!
!! Yi
+ " # ! Yi v = " # ! Di "Yi + " i
!t
!
!! Z
+ " # ( ! vZ) = " # ( ! D"Z)
!t
X.S. Bai
Species transport
Mixture fraction
Laminar Non-premixed Flames
Flamelet equation
2
d
Yi
1
! !"
= #i
2
2
dZ
d 2 Yi
2! i
=!
2
"#
dZ
! = 2D(!Z " !Z)
Scalar dissipation rate: mixing rate
Damköhler number Da = (mixing time)/(chemical reaction time)
X.S. Bai
Da !
2!i
"#
Laminar Non-premixed Flames
Z st
Finite rate chemistry effect
Yi
Yi
d 2Yi
→∞
2
dZ
d 2Yi
= finite
2
dZ
F
F
O2
O2
Z st
Damköhler number infinity
X.S. Bai
Z
Z st
Z
Damköhler not very high
Laminar Non-premixed Flames
χ↑
Effect of mixing rate
Yi
Yi
O2
χ↑
F
χ↑
Z
Z st
Z
Z st
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
Quenched
Flamelet
TP
Flame sheet
With preheating
Adiabatic flame
Tq
With radiation heat loss
χ P −1 [ s ]
χ q −1
X.S. Bai
Laminar Non-premixed Flames
Laminar jet diffusion flame
Diffusion
flame
F+P
A+P
Lean premixed flame
Rich premixed
flame
Lift-off
hight
Fuel
inlet
Air
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.
X.S. Bai
Laminar Non-premixed Flames