theory of combustion
TRANSCRIPT
1
Non-Premixed (Diffusion) Flames
IIT Kanpur1g µµµµg
• Candle flameyellow
luminous
region
blue
region
• Wax melts due to heat radiated from the flame
• Liquid wax is drawn up the wick by capillary action
• Liquid wax on the wick vaporizes by the heat transported (mostly radiation) from flame
• Wax vapor diffuses outward to make contact with oxygen diffusing in from surrounding air
• Chemical reaction occurs when fuel and oxygen mixes and temperature is high enough
• At critical temperature: wax molecules
(~C31H64) breaks � frees carbon atoms
� Incandescence (radiation) �
yellowish
• Most burning in blue reaction zone and
on surface of flame
• Chemical reaction time scale <<
diffusion time scale (diffusion is rate
controlling)
• Due to natural convection
flame has elongated shape
2
Common Diffusion Flame Configurations
IIT Kanpur
Fuel
Oxidizer
• Jet diffusion flame • Counter flow diffusion flame
Oxidizer
Fuel
drop
Fuel (vapor)
Oxidizer
• Spherical diffusion flame
• Safer � fuel and air are not premixed
• Wider range of operation � not restricted by flammability limits
• We will focus on laminar flames only
3
Non-Reacting Jet
IIT Kanpur
Fuel
Oxidizer
2R
potential
core
jet
edge
x
r
• Non-reacting laminar fuel jet flowing into infinite reservoir of
quiescent oxidizer
• Understand basic flow and diffusional process
• No effect of chemical reaction
• Circular fuel port, assume uniform (top hat) velocity profile at
tube exit
• Initial jet momentum is conserved (jet velocity decreases,
mass flow increases due to entrainment)
• � � 2���� �� ��
= � �� �� ��
• Initial jet fuel mass is conserved
• � � 2���� �� ��
= � �� �� ��,�
area
mass flow rateaxial (x)
momentum
jet fuel mass
4
Assumptions
IIT Kanpur
• 2D-axisymmetric, steady flow
• Constant (uniform) pressure (P)
• Constant (uniform) temperature (T)
• Same molecular weights for fuel and oxidizer (MWfuel = MWoxidizer) �
uniform density (ρ)
• Simple binary diffusion of species (diffusivity: DAB or D)
• Momentum and species diffusivities are equal or Schmidt number,
Sc = ν/D = 1
• Only radial diffusion (species and momentum) is important, axial
diffusion is neglected
• � solution valid some distance downstream of nozzle exit
5
Non-Reacting Jet (Mass Conservation)
IIT Kanpur
• Mass conservation:
�
�
�
������ +
�
����� = 0
(�� �)��∆�−(�� �)�+(�� �)��∆�−(�� �)�= 0
���∆�2� � + ∆� ∆� �� ��∆� − ��2��∆� �� �+ ���∆�2��∆� �� ��∆� − ��2��∆� �� � = 0
(�� �)� (�� �)��∆�
∆�
r
r+∆r
(�� �)�
(�� �)��∆�
2π(r+∆r)∆x
∆�
2πr∆x
2πr∆r
Area (A):
�
� + ∆�
area
divide by (∆r∆x), arrange
• uniform density
� !
��+
�
�
� "�
��= 0 mass conservation equation
6
IIT Kanpur
• Species conservation (fuel � F)
(�� �,�)� (�� �,�)��∆�
∆�
r
r+∆r
(�� �,�)�
(�� �,�)��∆�
2π(r+∆r)∆x
∆�
2πr∆x
2πr∆r
Area (A):
�
� + ∆�
Volume (V):
�� �### 2��∆�∆�
(�� �,�)��∆�−(�� �,�)�+(�� �,�)��∆�−(�� �,�)�
= �� �### 2��∆�∆�
(�� �,�## )��∆�2� � + ∆� ∆� − (�� �,�
## )�2��∆�+ (�� �,�
## )��∆�2��∆� − (�� �,�## )�2��∆� = �� �
###2��∆�∆�
area
�� �,�## = �� �
##�� − �$%&'
%�axial:
radial: �� �,�## = �� �
##�� − �$%&'
%�
0, axial diffusion
neglected
�� �##�� − �$
%&'
%� ��∆�� + ∆� ∆� − �� �
##�� − �$%&'
%� ��∆� + (�� �
##��)��∆��∆� −
(�� �##��)��∆� = �� �
###�∆�∆�
axial:
radial:
�� �## = ���
�� �## = ���
Non-Reacting Jet (Species (fuel) Conservation)
�&'
��≫
�&'
��
7
Non-Reacting Jet (Species (fuel) Conservation)
IIT Kanpur����� − �$
%&'
%� ��∆�� + ∆� ∆� − ����� − �$
%&'
%� ��∆� +
����� ��∆��∆� − ����� ��∆� = �� �###�∆�∆�
divide by (∆r∆x), arrange
�
������� +
�
������� − $
�
���
�&'
��= 0
• uniform density and diffusivity, no reaction
� � "
��+ �
� !
��= 0
• Mass conservation
����&'
��+ ��
� � "
��+ �
� !
��+ ���
�&'
��− $
�
���
�&'
��= 0
0, mass conservation
�) = 1 − ��
oxidizer fuel
���&'
��+ ��
�&'
��= $
�
�
�
���
�&'
��Species (fuel) conservation equation
� !
��+
�
�
� "�
��= 0
�
�
�
�������� +
�
�
�
�������� −
�
�
�
����$
�&'
��= �� �
###
net production of F by
chemical reaction
mass flow of F by
radial diffusion
mass flow of F by
radial convection
mass flow of F by
axial convection
8
Non-Reacting Jet (Axial-Momentum Conservation)
IIT Kanpur
• Momentum conservation(�� ���)��∆�− �� ��� � + (�� ���)��∆�− �� ��� �= +, � − +, ��∆� + -��, ��∆� − -��, � −�./
���∆�2��∆� �� ��∆� �� ��∆� − ��2��∆� �� � �� �+ ���∆�2� � + ∆� ∆� �� ��∆� �� ��∆� − ��2��∆� �� � �� �= + �2��∆� − + ��∆�2��∆�+ -�� ��∆�2� � + ∆� ∆� −(-��)� 2��∆� − �2��∆�∆�/
divide by (∆r∆x), arrange
�
�������� +
�
�������� = −
�
���+ +
�
���-�� − �/�
∆�
r
r+∆r
(�� ���)�
(�� ���)��∆�
2π(r+∆r)∆x
∆�
2πr∆x
2πr∆r
Area (A):
(+,)��∆�
(+,)�
(-��,)��∆�(-��,)�
(�� ���)� (�� ���)��∆�g
Volume (V):2��∆�∆�
-�� = 0� !
��+
� "
��
dynamic
viscosity� !
��≫
� "
��
0, neglected
-�� = 20� !
��
� �1"!
��≫
� �1!!
��
9
Non-Reacting Jet (Axial-Momentum Conservation)
IIT Kanpur• Order of magnitude analysis:
�
��≫
�
��, �� ≫ ��
�2
��→ �4�5 6�788
�2
��≈
�2:
��= −�/
�
�
�
�������� +
�
�
�
�������� =
�
�
�
���0
� !
��+ � − � /
buoyant forceviscous forceaxial-momentum
flow by axial
convection
axial-momentum
flow by radial
convection
• uniform density and viscosity (no buoyancy)
�
������� +
�
������� = ν
�
���
� !
��
kinematic viscosity
�� !
��+
� � "
��= 0
• Mass conservation
���� !
��+ �� �
� !
��+
� � "
��+ ���
� !
��= ν
�
���
� !
��
0, mass conservation
radial-momentum equation
��� !
��+ ��
� !
��= ν
�
�
�
���
� !
�� axial-momentum conservation equation
� !
��+
�
�
� "�
��= 0
10
Summary
IIT Kanpur
��� !
��+ ��
� !
��= ν
�
�
�
���
� !
��axial-momentum conservation equation
���&'
��+ ��
�&'
��= $
�
�
�
���
�&'
��species (fuel) conservation equation
� !
��+
�
�
� "�
��= 0 mass conservation equation
Unknowns: �� �, � , �� �, � 7;� �� �, �
Along jet centerline (r = 0):
�� 0, � = 0 no source or sink along jet centerline
� !
��0, � = 0
�&'
��0, � = 0
symmetry
Large radius (r ���� ∞∞∞∞):
�� ∞, � = 0 stagnant fluid
�� ∞, � = 0 no fuel
Jet exit (x = 0):
�� � ≤ �, 0 = ��
�� � > �, 0 = 0
�� � ≤ �, 0 = ��,� = 1
�� � > �, 0 = 0
• If we divide by �� (replace �� by ��/��)
then species and momentum equations
are identical (Sc = ν/D = 1)
top hat profile
11
Solution
IIT Kanpur
��� !
��+ ��
� !
��= ν
�
�
�
���
� !
��axial-momentum conservation equation
� � 2���� �� ��
= � �� �� �� axial-momentum is constant
• Take similarity variable: ξ = @�
�
•
��
=A
�
•
��
= −@�
�B
• Stream function: ψ = ν�C ξ
constant
�� =�
�
�ψ��
=ν�
�
��
��=
ν�
�
%�
%ξ�ξ��
=ν�
�
%�
%ξA
�= @ ν
�ξ%�
%ξ
�� = −�
�
�ψ��
= −ν�
���
��+ C = −@
νξ�
�%�
%ξ�ξ��
+ C = @ν�
�
ξ%�
%ξ@
�
�B −�
ξ= @
ν�
%�
%ξ−
�
ξ
12
Solution
IIT Kanpur
� !
��=
�
��@ ν
�ξ%�
%ξ= @ ν
�
��
�
�ξ%�
%ξ= @ ν
�
�ξ�
�ξ%�
%ξ�ξ��
= @ ν�
�ξ�
�ξ%�
%ξ−@
�
�B
= −@D ν�
�B
�
�ξ�
�ξ%�
%ξ= −@D ν�
�B
�
�ξ%B�
%ξB +
�
�
%�
%ξ−
�
ξB +
�
ξ%�
%ξ−
�
�B
��
�ξ
= −@D ν�
�B
�
�ξ%B�
%ξB +
�
�
%�
%ξ−
�
ξB +
�
ξ%�
%ξ−
�
�B −�B
A�
� !
��=
�
��@ ν
�ξ%�
%ξ= @ ν
�
�
��
�
ξ%�
%ξ= @ ν
�
%
%ξ�
ξ%�
%ξ�ξ��
= @ ν�
%
%ξ�
ξ%�
%ξA
�
@D ν�B
%
%ξ�
ξ%�
%ξ= @D ν
�B
�
ξ%B�
%ξB +
%�
%ξ−
�
ξB = @D ν
ξ�B
%B�
%ξB −
�
ξ%�
%ξ
= −@D ν�
�B
�
�ξ%B�
%ξB +
�
�
%�
%ξ−
�
ξB +
�
ξ%�
%ξ�
ξ�= −@ ν
�B
%B�
%ξB
13
Solution
IIT Kanpur
��� !
��= @ ν
�ξ%�
%ξ−@ ν
�B
%B�
%ξB = −@E νB
�Fξ%�
%ξ%B�
%ξB
��� !
��= @E νB
�Fξ%�
%ξ−
�
ξ%B�
%ξB −
�
ξ%�
%ξ= @E νB
�Fξ%�
%ξ%B�
%ξB −
�
ξ%�
%ξ%�
%ξ−
�
ξ%B�
%ξB +
�
ξB
%�
%ξ
= @E νB
�Fξ�
ξB
%�
%ξ−
�
ξ%�
%ξ
−�
ξ%B�
%ξB +
%�
%ξ%B�
%ξB
��� !
��+ ��
� !
��= @E νB
�Fξ�
ξB
%�
%ξ−
�
ξ%�
%ξ
−�
ξ%B�
%ξB
14
Solution
IIT Kanpur
ν�
�
�
���
� !
��= ν
�
�
�
��� @D ν
ξ�B
%B�
%ξB −
�
ξ%�
%ξ= @D νB
�B
�
�
�
��
�
ξ%B�
%ξB −
�
ξ%�
%ξ
= @D νB
�B
A
ξ�
�
��
�
A
%B�
%ξB −
�
ξ%�
%ξ= @E νB
�Fξ�
A
�
��
%B�
%ξB −
�
ξ%�
%ξ
= @E νB
�Fξ�
A
�ξ�r
%
%ξ%B�
%ξB −
�
ξ%�
%ξ= @E νB
�Fξ%
%ξ%B�
%ξB −
�
ξ%�
%ξ
15
Solution
IIT Kanpur�
ξB
%�
%ξ−
�
ξ%�
%ξ
−�
ξ%B�
%ξB =
%
%ξ%B�
%ξB −
�
ξ%�
%ξaxial-momentum conservation equation
integrate
C%�
%ξ=
%�
%ξ− ξ
%B�
%ξB
ξ = 0, C = 0,%�
%ξ= 0
solution:
C =ξ
B
��ξ
B
G
�� = @ ν�
�
��ξ
B
G
B�� = @
ν�
��ξ
F
G
��ξ
B
G
B
H� = � � 2���� �� ��
= � �� �� �� = @ �I
D��ν
@ =DJKL
�IM
�/ �
N
%
%ξ−
�
ξ%�
%ξ=
%
%ξ%B�
%ξB −
�
ξ%�
%ξ
−�
ξ%�
%ξ=
%B�
%ξB −
�
ξ%�
%ξ
CC# = C# − ξC##
ξ = 0, C = 0, C# = 0
16
Flow Field
IIT Kanpur �� =D
OM
KL
N�1 +
ξB
E
P
�� =DKL
�IMJ
�/ �
�
ξPξF
G
��ξB
G
B
H� = ��� ��
ξ =DJKL
�IM
�/ �
N
�
�
!(�Q)
L= 0.375�4V
�
W
P�Centerline velocity (vx(r=0)):
�4V =J LW
N jet Reynolds number
solution not valid near
nozzle (�
W< 0.375�4V)
• Decay is more rapid for low Re jets
�Y/B
�= 2.97 �4V
P�Jet half-width (r1/2): (radial location
where the jet velocity has decayed
to one-half of centerline value)
Spreading angle (αααα): [ = \7;P� �Y/B
�
• Low Re jets are wider
!
!(�Q)= ](ξ)
similarity
variable
17
Species Field
IIT Kanpur�� =
D
OM
^'
_�1 +
ξB
E
P
`� = ����
��(� = 0) = 0.375�4V�
W
P�Centerline fuel mass
(vx(r=0)):
solution not valid
near nozzle: (�
W<
0.375�4V)
• Decay is more rapid for low Re jets
�Y/B
�= 2.97 �4V
P�Jet half-width (r1/2): (radial location
where the fuel mass fraction has
decayed to one-half of centerline value)
Spreading angle (αααα): [ = \7;P� �Y/B
�
• Low Re jets are wider
As: Sc = 1
YF and vx/ve are same
volumetric flowrate of fuel&'
&'(�Q)= ](ξ)
similarity
variable