theory of combustion

19
1 Non-Premixed (Diffusion) Flames IIT Kanpur 1g μ μ μg Candle flame yellow 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 (~C 31 H 64 )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

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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%�

�� = −�

�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%�

= −@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

%�

= @E νB

�Fξ�

ξB

%�

%ξ−

ξ%�

−�

ξ%B�

%ξB +

%�

%ξ%B�

%ξB

��� !

��+ ��

� !

��= @E νB

�Fξ�

ξB

%�

%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%�

= @D νB

�B

A

ξ�

��

A

%B�

%ξB −

ξ%�

%ξ= @E νB

�Fξ�

A

��

%B�

%ξB −

ξ%�

= @E νB

�Fξ�

A

�ξ�r

%

%ξ%B�

%ξB −

ξ%�

%ξ= @E νB

�Fξ%

%ξ%B�

%ξB −

ξ%�

15

Solution

IIT Kanpur�

ξB

%�

%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=

%B�

%ξB −

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

18

Species Field

IIT Kanpur

Jet decay

19

Jet

IIT Kanpur