Che materia stai cercando?

Anteprima

ESTRATTO DOCUMENTO

Fundamentals of Flame Stabilities

Hydrodynamic Instability

• Boundary Conditions

ρ = = = = −∞

1

, 1

, 1 :

T Y x

( )

ρ = + = + = = ∞

1 / 1 q , T 1 q , Y 0 : x

• Surface Kinematics ∇

∂ F

dF F d

x =

= + ∇ ⋅ = n

F 0 ∇

∂ F

dt t dt

F

x

d

⇒ ⋅ = − t

n ∇

dt F

F

⇒ = ⋅ + =

t 1 (Darrieus & Landau)

S v n ∇

L F

Nondimensional flame speed equation

Fundamentals of Flame Stabilities

Linear Analysis

• Steady Solution (1-D)

( )

= − <

v ,

0

,

0

U , 0

gx x

s ⎪

=

< ⎨

⎧ p gx

1

, x 0 − − >

= s

⎨ , x 0

q

U +

+ > ⎩ 1 q

1 q , x 0

=

= 0

x f s ( )

• f y , z , t

Linear Stability Analysis

′ ′ ′

= + = + = +

p p p f f f

v v v , ,

s s s

( )

′ ′

<< << <<

v v , p p , f f

s s s ( )

∇ ⋅ =

⎧ =

v 0 x f y , z , t

⎪ s

⇒ ′

⎨ ⎛ ⎞

v

ρ ′ ′ ′

+ ⋅ ∇ + ⋅ ∇ = −∇

⎜ ⎟

v v v v p

⎪ ∂ s s

⎝ ⎠

⎩ t

Fundamentals of Flame Stabilities

Method of Normal Modes

• Method of Normal Modes

( )

′ ω

= + +

v v

exp t ik y ik z

1 2

( )

ω

′ = + +

p exp t ik y ik z p

1 2

( )

ω

= + +

f exp t ik y ik z A

1 2 ω

and we seek a solution for which is the eigenvalue

of the system:

( )

ω >

⎧ Re 0 : unstable

⎨ ( )

ω <

⎩ Re 0 : stable ( )

ω = ℑ

• k

Dispersion Relation gravity effect

[ ]

( ) ( ) ( )

ω ω

+ + + + − + =

2

2 q 2 1 q k qk g 1 q k 0

Fundamentals of Flame Stabilities

The Dispersion Relation

[ ]

( ) ( ) ( )

ω ω

+ + + + − + =

2

2 q 2 1 q k qk g 1 q k 0

1. Without gravity: ω

( ) ( )

⎡ ⎤

+ +

q q 2 q

1

ω = − ± +

⎢⎣ ⎥

k 1 1

( )

+ +

2 q 1 q ⎦

ω > 0 k

Unconditionally unstable

1 ( )

<

2. Upward propagation g 0

( ) ( ) ( )

⎡ ⎤

⎛ +

+ +

1 2 2

q q q q q

ω ⎟

⎜ − >

= − + +

⎢ ⎥

2

1 0

k k gk

( ) +

+ +

1 ⎢ ⎥

2 1 1

q q q

⎣ ⎦

Unconditionally unstable

Hydrodynamic + Rayleigh-Taylor instability

Fundamentals of Flame Stabilities

The Dispersion Relation

[ ]

( ) ( ) ( )

ω ω

+ + + + − + =

2

2 q 2 1 q k qk g 1 q k 0

( )

>

3. Downward propagation 0

g

( ) ( ) ( )

⎡ ⎤

⎛ ⎞

+ + +

q q q q

1 q 2 2

ω ⎜ ⎟

= − + + −

⎢ ⎥

2

k 1 k gk

⎜ ⎟

( )

+ + +

1 ⎢ ⎥

⎝ ⎠

2 q 1 q 1 q

⎣ ⎦

( ) g

ω > > =

Re 0 for k k +

c 1 q ω

( )

ω < <

Re 0 for k k c

Long waves are stabilized. k k

c

Fundamentals of Flame Stabilities

Physical Mechanism of D-L

Streamline deflection due to heat release

S ω Higher heat release

u S b

u < S u < S

u b S

S b

u u > S u > S k

u b ω ~ k

Unconditionally unstable

Fundamentals of Flame Stabilities

Markstein’s Analysis

Darrieus and Landau analysis contradicts experimental

observation that stable flame can be established.

Markstein: Consideration of Flame Structure

(Phenomenological model)

μ μ

= − ∇ = −

2

S 1 f 1 / R

L curvature Markstein length

Dispersion Relation μ

⎡ ⎤

( ) ( ) ( ) ( )

2

ω ω

+ + + − + − + =

2 2 ⎢ ⎥

2 q 2 1 q k q 1 q k 1 k 1 q 0

⎣ ⎦

q

ω

q

= *

* k

k ( )

μ +

2 1 q k

μ > 0

Structure can stabilize if

Fundamentals of Flame Stabilities

Diffusive-Thermal Instability

• Turing (1950), Sivashinsky (1979)

- Recognizing the importance of flame structure when

the wavelength of perturbation becomes comparable

( ( )

)

δ

k ~ O 1

to the flame thickness

- Activation energy asymptotics

• Assumptions ( )

ε = 2

- T / T

Large activation energy ad a

( )

ε

= +

- Le 1 l / q

Near-unity Lewis number ( )

( )

ε

= +

- T T O

Small temperature gradient behind flame ad

( )

=

- Y 0

No reactant behind the flame

( )

ε

- O

Enthalpy remains within of supply value

Fundamentals of Flame Stabilities

Schematic of D-T Analysis

• Asymptotic Structure of D-T Analysis

( ) ( )

δ =

O O 1

Flame zone

(Conv + Diff)

Unburned Burned

gases gases

( )

ε

O

Reaction zone

(Diff + React)

=

x f

Fundamentals of Flame Stabilities

Formulation for D-T Analysis

ρ δ

= =

constant & 1

( )

v x , y , z , t given

⎛ ⎞ ( )

Y 1 ρ

= Λ −

ρ + ⋅∇ − ∇ = −

2 w Y exp T / T

⎜ ⎟

v Y Y w

∂ a

⎝ ⎠

t Le

⎛ ⎞

T

ρ + ⋅∇ − ∇ =

2

⎜ ⎟

v T T qw

⎝ ⎠

t = +

Define the enthalpy function (S-Z variable): H T qY

ε

= +

and Le 1 l / q

⎛ ⎞

H l ( )

ρ ε

+ ⋅∇ − ∇ = − ∇ −

2 2

⎜ ⎟

H H H T

v

⎝ ⎠

t q

which replaces the species equation.

Fundamentals of Flame Stabilities

Summary of Asymptotic Analysis (1)

( )

=

Outer Zone w 0 :

ε ρ ρ ερ ε

= + + = + + = + + +

" " "

T T T , , H 1 q h

0 1 0 1

⎛ ⎞

h l

ρ + ⋅∇ − ∇ = − ∇ ≠

2 2

⎜ ⎟

h h T , x f

v

0 0

⎝ ⎠

t q

⎛ ⎞

T

ρ + ⋅∇ − ∇ = <

2

0

⎜ ⎟

T T 0, x f

v

0 0 0

⎝ ⎠

t = + >

T 1 q , x f

0

∂ ∂

⎛ ⎞ ⎛ ⎞

T T

ρ ρ

+ ⋅∇ + + ⋅∇ − ∇ = ≠

2

0 1

⎜ ⎟

⎜ ⎟

T T T 0, x f

v v

∂ ∂

1 0 0 1 1

⎝ ⎠

⎝ ⎠

t t

Fundamentals of Flame Stabilities

Summary of Asymptotic Analysis (2)

( ) ( )

ζ ξ ε ε

= = −

Inner Zone reactive : / x f /

εθ εψ

= + + = + +

" "

T T , H T

ad ad

θ

∂ 2

( ) ( )

ψ θ θ

+ + = − −

2 2

1 f f D e

ζ

y z 2

ψ

∂ 2 = 0

ζ

∂ 2

The solutions to these equations provide the matching conditions

for the outer solutions. ⎡ ∂ ⎤

( ) T ( )

1/ 2

+ + = − h 0 / 2

2 2 0

⎢ ⎥

1 f f qe

ξ

y z ⎣ ⎦ ξ = 0

⎡ ⎤ ⎡ ∂ ⎤

∂ T

h l

+ =

0

⎢ ⎥ ⎢ ⎥ 0

ξ ξ

∂ ∂

⎣ ⎦ ⎣ ⎦

q

ξ ξ

= =

0 0

Fundamentals of Flame Stabilities

Linear Stability Analysis of D-T Model (1)

To further simplify the analysis, an additional assumption is made:

q 1 (constant density approximation)

thereby eliminating the D-L effect.

τ ρ

= + + = + + =

" "

T 1 q , 1 qR , h h

⎛ ⎞

h τ

+ ⋅∇ − ∇ = ∇ ≠

2 2

⎜ ⎟

v h h l , x f

⎝ ⎠

t

τ

⎛ ⎞ τ

τ τ = >

+ ⋅∇ − ∇ = <

2

⎜ ⎟

v 0, x f ; 1, x f

⎝ ⎠

t τ τ

∂ ∂ ∂

⎡ ⎤ ⎡ ⎤ ⎡ ⎤

[ ] [ ] h

τ = = + = = − / 2

h

h 0; 0;

l e

⎢ ⎥ ⎢ ⎥ ⎢ ⎥

∂ ∂ ∂

⎣ ⎦ ⎣ ⎦ ⎣ ⎦

n n n )

( ⎛ ⎞

∂ ∂ ∂ ∂

1

= ⋅∇ = + + − + + +

2 2 2 2 ⎜ ⎟

1 1

n f f f f f f

ξ

∂ ∂ ∂ ∂

y z y z y z

⎝ ⎠

n y z

Fundamentals of Flame Stabilities

Linear Stability Analysis of D-T Model (2)

( )

=

v

For a prescribed flow field 1, 0, 0 , a steady solution is derived as:

⎧ ⎧

< − <

x x

, 0, , 0,

e x lxe x

τ

= = =

⎨ ⎨

0; ;

f h

> >

s s s

⎩ ⎩

1, 0, 0, 0.

x x

Methods of normal modes:

τ τ

= + Φ = + Ψ = +

, h h , f f F

s s s

with ( )

φ ω

Φ = + +

( x ) exp ik y ik z t ,

1 2

( )

ψ ω

Ψ = + +

( x ) exp ik y ik z t ,

1 2

( )

ω

= + +

( ) exp ,

F A x ik y ik z t ( )

1 2 ω >

⎧ Re 0 : unstable

and derive the dispersion relation to determine ( )

ω <

⎩ Re 0 : stable

Fundamentals of Flame Stabilities

Results of the Linear Stability Analysis

The Dispersion Relation

( )

ω ω

+ + + −

3 2 2 2

64 192 k 32 8

l l

( )( ) ( )

ω 2

+ + + + + + + =

2 2 2 2

2 2 8

k l 1 12 k 2 8

k l k 0

ω ω − 2

~ ak bk ↓

Le k

Fundamentals of Flame Stabilities

Physical Description of D-T Instability

Diffusive-Thermal Imbalance

α

=

Le Lewis Number ω ω − 2

~ ak bk

D

Mass ↓

Le

Heat < (Unstable)

Le 1

Products k

Reactants > (Stable)

Le 1 Short wave stabilizing

Fundamentals of Flame Stabilities

Neutral Stability Curves

k ( )

( ) ω ≠

ω = Im 0

Im 0 Stable Unstable

Unstable Pulsating

Cellular ( )

− +

2 l

4 1 3

( )

( ) ε

= − − < < +

l Le 1 / 2 4 1 3

l ε

⇒ < < =

0

.

2 Le 2

.

1 ( 0

.

1

)

Fundamentals of Flame Stabilities

Experimental Observation

φ φ φ φ

= 0.56 = 3.00 = 0.80 = 1.30

Le < 1 (Unstable):

- Cellular instability

• Lean H -air

2

• Rich C H -air

3 8

Le > 1 (Stable):

- Hydrodynamic

instability

• Rich H2-air

• Lean C3H8-air Hydrogen-air flames (2atm) Propane-air flames (10atm)

Experiment by C. K. Law et al.


PAGINE

20

PESO

438.75 KB

AUTORE

Atreyu

PUBBLICATO

+1 anno fa


DESCRIZIONE DISPENSA

Materiale didattico per il corso di Combustione del Prof. Mauro Valorani, all'interno del quale sono affrontati i seguenti argomenti: instabilità intrinseca della fiamma; instabilità idrodinamica; metodo dei modi normali; analisi di Markstein; instabilità diffusiva termica; curve di stabilità neutrale; equazione di Kuramoto - Sivashinsky; instabilità pulsante.


DETTAGLI
Esame: Combustione
Corso di laurea: Corso di laurea magistrale in ingegneria aeronautica
SSD:
A.A.: 2007-2008

I contenuti di questa pagina costituiscono rielaborazioni personali del Publisher Atreyu di informazioni apprese con la frequenza delle lezioni di Combustione e studio autonomo di eventuali libri di riferimento in preparazione dell'esame finale o della tesi. Non devono intendersi come materiale ufficiale dell'università La Sapienza - Uniroma1 o del prof Valorani Mauro.

Acquista con carta o conto PayPal

Scarica il file tutte le volte che vuoi

Paga con un conto PayPal per usufruire della garanzia Soddisfatto o rimborsato

Recensioni
Ti è piaciuto questo appunto? Valutalo!

Altri appunti di Combustione

Stechiometria ed equilibrio chimico
Dispensa
Stabilità della fiamma
Dispensa
Combustione omogenea e combustione eterogenea
Dispensa
Cinetica chimica
Dispensa