Structure and Dynamics of Combustion Waves in Premixed Gases

1

Structure and Dynamics of

Combustion Waves in Premixed Gases

Paul Clavin

Aix-Marseille Université ECM & CNRS (IRPHE)

Lecture VI

Thermal quenching of flames and flammability limits

Princeton Summer School on Combustion June 20 - June 26, 2021

Copyright 2021 by Paul Clavin

This material is not to be sold, reproduced or distributed

without permission of the owner, Paul Clavin

P.Clavin VI

2

Lecture 6: Thermal quenching and flammability limits

6-1. Extinction through thermal loss

6-2. Basic concepts in chemical kinetics

6-3. Flame speed near flammability limits

Combustion of hydrogen

Two-step model. Crossover temperature

One-step model with temperature cutoff

P.Clavin VI

3

VI-1) Extinction through thermal loss

Zeldovich 1941 Davy 1830

d /d⇥| _⇤=0 = e ^{(1 ⇥ f )/2} d

d + 1 Le

d d

0+

0

jumps across the thin reaction zone : = 0

⇥ < 0 : (⇥) = _f e [µ+h/( µ)]⇥ ,

⇤ (⇥) = 1 e ^{Le µ⇥} , ⇥ > 0 : ⁺ (⇥) = _f e ^{[h/( µ)]⇥} ,

⇤ ₊ (⇥) = 0, external solutions : w = 0

up to first order O(1/ )

a small heat loss can quench the flame

Zoom

slope

⇥ = : = 0, ⇤ = 1, ⇥ = + : = 0, ⇤ = 0 µ d

d⇥

d ²

d⇥ ² = w ⇤ _L

⇤ _cool , µ d⌅

d⇥

1 Le

d ² ⌅

d⇥ ² = w x/d _L µ = U _L /U _Ladia 1/ _cool D _T /R ²

Formulation (volumetric heat loss in a planar flame)

L cool

D _T R U _L

2

L D _T /U _L ² R = tube radius

(Joulin Clavin 1976)

w(⇥, ⇤) = ( ² /2)⇤ exp[ (1 ⇥)]

⇥ _L /⇥ _cool = h/ h = O(1) (1 ⇥ _f ) = O(1)

unknown flame temperature < adiabatic flame temperature : _f < 1

Asymptotic analysis for small heat release and a one step reaction

4 P.Clavin VI

d

d + 1 Le

d d

0+

0

= 0

⇥ < 0 : (⇥) = _f e [µ+h/( µ)]⇥ ,

⇤ (⇥) = 1 e ^{Le µ⇥} , ⇥ > 0 : ⁺ (⇥) = _f e ^{[h/( µ)]⇥} ,

⇤ ₊ (⇥) = 0,

Zoom

slope

(h/ µ)⇥ _f (µ + h/ µ)⇥ _f + µ = 0

(1 ⇥ _f ) = 2h/µ ²

⇥ _f 1 = O(1/ )

d /d⇥| _⇤=0 = e ^{(1 ⇥ f )/2}

µ ² ln µ ² = 2h µ = exp( h/µ ² )

Non dimensional flame speed

Non dimensional heat loss 0.0

1.0

µ

2h

C-shaped curve: no solution for 2 h > 1/e

quenching at finite flame velocity U _L /U _Ladia = 1/ e

5 P.Clavin VI

VI-2) Basic concepts in chemical kinetics

˜k _j = ˜ B _j T e ^{T aj /T}

units: moles/cm ³ , s ¹ and Kelvin

j = ˜k _j c _1j c _2j or _j = ˜k _j c _1j c _2j c _3j dc _ij /dt = j

Label Reaction ˜k j B ˜ _{j j} T _aj

˜k 1f 3.52 10 ¹⁶ -0.7 8590 1 O 2 + H OH + O ˜k 1b 7.04 10 ¹³ -0.264 72

˜k 2f 1.17 10 ⁹ 1.3 1825 2 H 2 + OH H 2 O + H ˜k 2b 1.29 10 ¹⁰ 1.196 9412

˜k 3f 5.06 10 ⁴ 2.67 3165 3 H 2 + O OH + H ˜k 3b 3.03 10 ⁴ 2.63 2433 4f O 2 + H + M HO 2 + M ˜k 4f 5.79 10 ¹⁹ -1.4 0 5f H + H + M H 2 + M ˜k 5f 1.30 10 ¹⁸ -1 0 6f H + OH + M H 2 O + M ˜k 6f 4.00 10 ²² -2 0 7f HO 2 + H OH + OH ˜k 7f 7.08 10 ¹³ 0 148 8f HO 2 + H H 2 + O 2 ˜k 8f 1.66 10 ¹³ 0 414 9f HO 2 + OH H 2 O + O 2 ˜k 9f 2.89 10 ¹³ 0 -250

(4f) chain breaking

M + H + O ₂ M + HO ₂ , rate : _4f = c _H c _{O 2} nk _4f

k _4f = B _4f

Combustion of hydrogen

Law 2006 Sanchez Williams 2014

H ₂ + O ₂ HO ₂ + H (8b) initiation

k _1f (T ) B _1f e ^{E 1 /k B T} = nB _4f hydrogen combustion:

(1f), (2f), (3f) shuffle reactions

chain branching

O ₂ + 3H ₂ 2H ₂ O + 2H rate: _1f c H c O 2 k 1f

Flammability limit

T _b = T q _R Y _u c _p (T T _u )

Simplified two-step model: crossover temperature

(B) R + X 2X, B = c R c _X B _B e ^{E/k B T} , E k _B T (R) M + X P + Q, _R = c _X nB _R , E _R = 0

c R B B e ^{E/k B T}

nB R

T T

no reaction Arrhenius law OK

c _R B _B e ^{E 1 /k B T} = nB _R T [900K 1400K]

Zeldovich 1961, Li˜nan 1971 , Peters Williams 1987 , Peters 1997

P.Clavin VI

6

Two-step model for rich hydrogen flames near the flammability limit

(consumption of hydroperoxide included) O ₂ + 3H ₂ 2H ₂ O + 2H,

1f = c _H c _{O 2} k _1f (T ), k _1f (T ) = B _1f e ^{E 1 /k B T} H + H H ₂ + Q,

4f + _5f = nc _H c _{O 2} B _4f + nc ² _H B _5f

nB _4f = B _1f e ^{E 1 /k B T} m d

dx ⇥D _T d ² dx ²

⇥

⇤ ⌅ ² J(T ) m d⇤

dx D _{O 2} d ² ⇤

dx ² ⇥ ⇤ ² e ^{kB E ( T 1 T 1 )} J(T )

1/ (nB _4f ² c _{O 2 u} )/B _5f

T > T : J(T ) T _u

T [e ^{kB E ( T 1 T 1 )} 1]

T < T : J(T ) = 0

x = : = 0, x = + : = 1

reaction of order 2 with a temperature cutoff

m d

dx ⇥D _T d ² dx ²

⇥

⇤ ⌅ ² J(T ) m d⇤

dx D _{O 2} d ² ⇤

dx ² ⇥ ⇤ ² J(T )

T > T : J(T ) T _u T

E k _B T

T T

T T < T : J(T ) = 0

T _b T T

k _B T

E [e ^{kB E ( T 1 T 1 )} 1] E

k _B ( 1 T

1

T ) 1 very close to the flammability limit

dc _H

dt = B _1f e ^{E 1 /k B T} nB _4f c _{O 2} c _H nB _5f c ² _H

H in quasi-steady state

T > T : T < T : c _H = 0

(B _1f e ^{E 1 /k B T} nB _4f )/nB _4f 1 tri molecular recombination reaction (5f) c _H c _{O 2} [B _1f e ^{E 1 /k B T} nB _4f ]

nB _5f

One-step model (near the flammability limit) ^{c H B 5 c O 2 B 4f 5f 4f}

P.Clavin VI VI-3) Flame speed near flammability limits

> : j( ) b ( )

< : j( ) = 0 m d⌅

dx ⇥ _b D _{O 2} d ² ⌅ dx ²

⇥ _b

⇤ ⌅ ² j( ) m d

dx ⇥ _b D _T d ² dx ²

⇥ _b

⇤ ⌅ ² j( )

b T _u T

E k _B T

T _b T _u T (T T _u )

(T _b T _u ) T _b > T < 1 but close to 1

reaction zone: ⇤ = Le(1 ), D _T d ²

dx ² = Le ² b

⇥ (1 ) ² [( 1) ( 1)] Le D _T /D _{O 2}

7

T _b U _L

dU _L

dT _b = 2T _b T _b T

the least heat loss quenches the flame at a non zero velocity

Divergence of the thermal sensitivity: Thermal quenching

+ matching _{D T} ^d

dx Le b

6 (1 ) ² D _T

⇥

⇥ _u

⇥ _b

U _L

D _T /⇤ Le b

6 (1 ) ² d

dx + ¹ d

(T T _u )

(T _b T _u ) [ , 1]

the flame velocity decreases smoothly to zero when approaching the flammability limit T _b T the flame thickness d _L diverges, T _b T : d _L

d _L

1

2

T T _u T _b T

2

u U _L

D _T /⇥ Le b 6

T _b T T T _u

2

0 < T _b T

T _b T _u 1 Peters 1997

Peters Williams 1987 Peters 1997

H ₂ O ₂ flames Methane flames

Sanchez Williams 2014

1

Structure and Dynamics of

Combustion Waves in Premixed Gases

Paul Clavin

Aix-Marseille Université ECM & CNRS (IRPHE)

Lecture VII

Flame kernels and quasi-isobaric ignition

Princeton Summer School on Combustion June 20 - June 26, 2021

Copyright 2021 by Paul Clavin

This material is not to be sold, reproduced or distributed

without permission of the owner, Paul Clavin

2 P.Clavin VII

Lecture 7: Flame kernels and quasi-isobaric ignition

7-1. Introduction

7-2. Zeldovich critical radius

7-3. Critical radius near the flammability limits 7-4. Dynamics of slowly expanding flames

7-5. Quasi-steady dynamic of thin flames

Semi-phenomenological model

Opened-tip Bunsen flames

P.Clavin VII

VII-I) Introduction

Flammability limits Critical conditions of ignition

Limits for upstream propagation = downstream propagation

Turbulence facilitates ignition of hydrocarbon lean mixtures

Turbulence may suppress ignition of hydrocarbon rich mixtures

Some hydrocarbon lean mixtures that are flammable cannot be ignited quasi-isobarically Some hydrocarbon rich mixtures that are non-flammable can sustain curved flames

Ignition in turbulent flows

Princeton experiments (2014) Wu & al.

3

4

P.Clavin VII VII-2) Zeldovich critical radius

Flame kernel for a flame far from the flammability limits

Zeldovich

concentration temperature m as s

flux he at

burnt flux gas

reaction rate

R R _f : = _f , ⇥ = 0; R : = 0, ⇥ = 1

(T T _u )/(T _b T _u ) Y /Y _{u rb r} (T _b ) c _p (T _b T _u ) = q _R Y _u

Unstable steady spherical solution for the one-step model of adiabatic flames

= 1 R ²

d

dR R ² d

dR = 1

R ² d

dR R ² d dR

D _T 1 R ²

d

dR R ² d

dR = D 1 R ²

d

dR R ² d⇤

dR = ⇤

⇥ _rb e ^{(1 ⇥)}

No flow

Le = 1 T _f = T _b

Flame temperature

D _T d

dR R ² d

dR + D d

dR R ² d

dR = 0 (conservation energy)

Le > 1 T _f < T _b Le < 1 T _f > T _b

Le D _T /D

D _T = D(1 ⇥) _f = 1/Le

double integration from R = 0 to R =

5 P.Clavin VII

concentration temperature m as s

flux he at

burnt flux gas

reaction rate

D _T 1 R ²

d

dR R ² d

dR = D 1 R ²

d

dR R ² d⇤

dR = ⇤

⇥ _rb e ^{(1 ⇥)}

R R f : = f , ⇥ = 0; R : = 0, ⇥ = 1

(T T _u )/(T _b T _u ) Y /Y _{u rb r} (T _b ) c _p (T _b T _u ) = q _R Y _u

Unstable steady spherical solution for the one-step model of adiabatic flames

⇥ _b ² ⇥ _rb /2, d _L D _T ⇥ _b (Le=1)

Radius of the kernel ₁

Le D _T

R _f = e ² ( Le ¹ 1 ) LeD ^T

b

R _f

d _L = Le ^3/2 e ² ( ¹ Le ¹ ),

matching

Thin reaction zone thickness flame radius R _f

Asymptotic analysis 1

x R R _f , |x| R _f : D _T d ²

dx ² = D d ²

dx ² =

rb

e ^{(1 )}

Le( _f ) _e ^{(1 )} _{= e} ^{( f 1)} _{(Le/ ) e} d ²

d ² = R ² _f D _T

e ^{( f 1)}

2 rb

Le e

Le > 1 : R _f d _L Le < 1 : R _f d _L

(⇥ _f ⇥) = O(1)

(x/R _f ) = O(1), [ , + ] [0, ] Inner variables

: lim D _T d⇥

dR = e ^{(⇥ f 1)/2} 2 Le D _T

2 ⇤ _rb d

d +

0

d

External zones

d

dR R ² d

dR = 0 R ² d

dR = cst. R = R f : = f = 1/Le

R R _f : = 1 Le

R _f

R , R = R _f : D _T d

dR = 1 Le

D _T

R _f R < R _f : = _f

D _{C n H m} < D _{O 2} D _T , Le D _{O 2} /D _{C n H m} > 1

Lean hydrocarbon mixtures Le > 1 are difficult to ignite (R _f > d _L )

Extension of the Zeldovich analysis to a constant energy source Flame balls in micro-gravity experiments (Ronney 1985-2004)

lean hydrogen mixtures, diameter = 2 15 mm (Deshaies Joulin 1984)

R _f convective flux dR _f /dt < 0

R _f R _f < R _f : |diffusion fluxes| 1/R R _f R _f > R _f : |diffusion fluxes| 1/R

heat flux towards the preheated zone

convective flux dR _f /dt > 0 R _f

convection should be added to restore equilibrium

f = 1/Le = cst.

= _f R _f /R preheated zone at rest

d /dR| ^{R=R f} = _f /R _f Instability ? (adiabatic condition)

Stabilization in the presence of heat loss for Le < 1 (Buckmaster, Joulin, Ronney 1990)

Stability analyses

positive feed back: amplification

Quasi-isobaric ignition as a nucleation problem

Le < 1 : R _f d _L Le > 1 : R _f d _L

6

= Le( _f )

P.Clavin VII

7

T > T : ˙w = ( ³ /4)Le ² e ^{(⇥ f 1)} (⇥ _f ⇥) ² e ^{(⇥ ⇥ f )} e ^{⇤ (⇥ f 1)} T < T : ˙w = 0

< _f

⇥ T T _u

T _b T _u = 1 ⇤

E k B T b

1 T u

T b

⇥ (T _b T ) 1

(T _b T _u ) > 0 = O(1) : near to flammability limits (⇥ = 0 : quenching ˙ W = 0) 1, e 0 : far from flammability limits

d d

2

= Le ²

2 e ^{(⇥ f 1)}

⇥ _f

⇥ =1 ⇤/

3 ( _f ) ² e ^{(⇥ ⇥ f )} e ^{⇤ (⇥ f 1)} d d

d

at the exit of the reaction layer (entrance of the preheated zone)

˙w = 0

VII -3) Critical radius near the flammability limits

(He Clavin 1993-94)

R R _f : = _f , ⇥ = 0; R : = 0, ⇥ = 1

D _T 1 R ²

d

dR R ² d

dR = D 1 R ²

d

dR R ² d⇥

dR = ˙ W

1

rb

1 e ^{kB E}

“ 1 T

1 Tb

”

= B 4f

B 5f

B _1f e ^{kB Tb E} c _{O 2 u}

T > T : W = ˙ ^b

u

⇤ ²

⇥ _rb [e ^{(1 ⇥)} e ^⇤ ] T < T : W = 0 ˙

model of lecture VI p. 6, 7

= _f : = 0 d ² ⇥/d ² = ˙w(⇥, ⇤), (1/Le)d ² ⇤/d ² = ˙w(⇥, ⇤)

Thin reaction zone (non-dimensional form = x/d _L , d _L for ⇥ 1, Le = 1, 2 ^nd reaction order)

D T

d ² _L = 4 ^b

u

1

3 rb

8 P.Clavin VII

d d

2

= Le ²

2 e ^{(⇥ f 1)}

⇥ _f

⇥ =1 ⇤/

3 ( _f ) ² e ^{(⇥ ⇥ f )} e ^{⇤ (⇥ f 1)} d at the exit of the reaction layer (entrance of the preheated zone)

(d⇥/d ) ² = Le ² e ^{(⇥ f 1)} J( _f ) J( _f ) 1 2

f

0

2 (e e ^f )d [0, 1]

J( _f ) = 1 e ^f 1 + _f +

2 f

2! +

3 f

3!

f 1 : J 1

0 _f 1 : J ⁴ _f /(4!)

: d _L

R _f = Le ² e ^{2 ( Le 1 1)} J( _f )

d _L for 1, Le = 1, 2 ^nd reaction order

Preheated zone and matching

R R _f : d

dR = _f R _f

R ² , _f = 1 Le

flame temperature of the spherical flame

T _f T R _f /d _L

f 0

f

T _f T

T _b T _u = (⇥ _f ⇥ ) = ⇤ + (⇥ _f 1) 0

= (⇥ _f ⇥) [0, _f ]

measure of the distance from the flammability limit: f [0, ]

d = d

9

Le < 1 T _f > T _b

Flammability limit

non-flammable mixtures that can be ignited

(flame balls) Flammability limit

flammable mixtures that cannot be ignited (infinite critical radius)

T _f = T

Flammable mixtures

Flame kernels (a)

d _L /d L

T _b = T

Le > 1 T _f < T _b

f

T _f T _u

T _b T _u = 1 Le

temperature of the spherical flame kernel

f = 1/Le

energetic mixtures

T _b

energetic mixtures

T _b

T T _u T _b T _u depends on the composition

depends on the composition

T T _u T _b T _u

temperature of the planar flame

T _b is determined by the composition of the mixture Y _Ru (mass fraction of the limiting component)

T is determined by the chemical kinetics Y _Ru

T _b = T T _f = T

Flammable mixtures Flame kernels

(b)

d _L /d _L T _f = T

Le > 1 : Heavy hydrocarbon lean mixtures Hydrogen rich mixtures

Le < 1 : Heavy hydrocarbon rich mixtures

Hydrogen lean mixtures

P.Clavin VII

10

Flammability limits Critical conditions of ignition

Some hydrocarbon lean mixtures that are flammable cannot be ignited quasi-isobarically Some hydrocarbon rich mixtures that are non-flammable can sustain curved flames

Turbulence facilitates ignition of hydrocarbon lean mixtures

Turbulence may suppress ignition of hydrocarbon rich mixtures

Ignition in turbulent flows

Princeton experiments (2014) Wu & al.

Limits for upstream propagation = downstream propagation

Buoyancy promoted curved flames

non-flammable hydrocarbon rich flames

simplest explanation:

Turbulent diffusion coe⇥cients are all equal ^{Le > 1} Le = 1 Le < 1

laminar turbulent

P.Clavin VII

11

VII-4) Dynamics of slowly expanding flame kernels

t _relax R ²

D _T t _evol R _f

˙R _f

quasi-steady state ?

˙Q = cst.

d

T T _u = 1 4 D _T

˙Q

⇥c _p 1 R

2

R/ 4D T t

X e ^{X 2}

X R/ 4D _T ^{dX = 2D} T R d

(4D _T ) ^3/2 T (R, t) T _u = ^t

0

˙Q(t ⇤)

⇥c _p

exp( R ² /4D _T ⇤ ) (4 D _T ⇤ ) ^3/2 d⇤

T / t = D _T T

exact solution of the heat equation with a point energy source point source, R = 0, t > 0 : ˙Q(t)

⇥

⇥t ˙R _f ⇥

⇥R D _T 1 R ²

⇥

⇥R R ² ⇥

⇥R = 0

preheated zone in the reference frame attached to R _f (t) _{˙R f} ^{dR f} dt

Quasi-steady preheated zone of flame kernel ?

The evolution of spherical flame kernel cannot be quasi-steady at large distance R D _T t _evol , not valid at large distance

relax time toward (T T _u ) 1/R increases with R like R ² /D _T

R ² /(4D _T t) 0

(Joulin 1985)

t/t _{ref t ref} ^{(1 Le}

1/2 ) Le

R _{f Z}

(4⇥D _T ) ^1/2 r _f R _f /R _{f Z}

R f Z

R

t

Joulin’s equation

1

r _f = exp

0

d

1/2 ˙r _f ( )

2 ^f

1

Le =

0

d

1/2 ˙r _f ( )

Le < 1

0 1 2

0 10 20 30 40 50 60 70

Reduced time

Reduced radius

Extension to a short pulse of an energy source (Joulin 1985)

Extension to the proximity of flammability limits + heat loss (Clavin 2015)

The structure and the dynamics of flame kernels = planar flames even for R _f d _L ( _f /R)

(Ronney 1985-1990)

Self-extinguished flames in micro-gravity experiments of lean methane-air mixtures Dynamical quenching of flame kernels in nonflammable mixturesfor Le < 1

(Clavin 2016)

1

r _f + H _b r ² _f = 1 I( ) where I( )

0

d

1/2 ˙r _f ( )

For Le < 1 and near to the Zeldovich radius the slow evolution of flame kernels is governed by the diffusion at large distance

12

R _f

d _L 1, Quasi-steady state approx valid only when ˙R _f U _L

U _L 1, d d _L ,

VII-5) Quasi-steady dynamics of thin flames ?

P.Clavin VII

13

thin flames: flame thickness ^{( D T / ˙ R f ) R f}

⇥

⇥t ˙R _f + 2 D _T R

d

dx D _T d ²

dx ² = 0 x (R R _f )

moving frame

(R > R _f )

quasi-steady preheated zone

˙R _f R _f

D _T

(D _T / ˙ R _f ) ² R _f ˙R _f D _T

diffusion time on the flame thickness evolution time

also negligible !

Almarcha Quinard (2015) rich propane-air flame

U _f

U _L ln U _f

U _L = 1

Le 1 d _L R _f

no solution below a minimum radius

U _f

U _L ln U _f U _L

1/e Le < 1

U _f /U _L

(Le ¹ 1) d _L /R _f 1/e

U _f = ˙R _f > 0 d _L / ˙ R _f = O(1/ )

Steady converging flames. Opened-tip Bunsen flames

Frankel Sivashinsky (1984) Le < 1

heavy hydrocarbon rich flame numerical results of He (2000) extended to R _f ˙R f D T !

F ( ˙ R _f , R _f ) = 0 numerical results for a constant heat source

extension of the Deshaies Joulin analysis (1983) by He (2000)

+ asymptotic analysis thin reaction zone

qualitative agreements with the experiments by Kelley et al. (2009)

Semi-phenomenological model

˙R _f + 2 D _T R

d

dx D _T d ²

dx ² = 0

˙R _f + 2 D R

d

dx D d ²

dx ² = 0

Critical conditions for spherical flame initiation 167

Figure 3. Variation of the flame velocity obtained from equation (12) with an ignition power Qs = 2 and an infinitely long duration as a function of the flame radius. R Z, − and R Z,+ are two solutions for the super-adiabatic stationary flame ball.

Figure 4. Variation of the flame velocity obtained from equation (12) with different values of the heat power as a function of the flame radius.

Downloaded by [Aix-Marseille Université] at 03:16 27 May 2014

He (2000)

R f /d L

˙R f /U L

1

Structure and Dynamics of

Combustion Waves in Premixed Gases

Paul Clavin

Aix-Marseille Université ECM & CNRS (IRPHE)

Lecture VIII

Thermo-acoustic instabilities. Vibratory flames

Princeton Summer School on Combustion June 20 - June 26, 2021

Copyright 2021 by Paul Clavin

This material is not to be sold, reproduced or distributed

without permission of the owner, Paul Clavin

2 P.Clavin VIII

Lecture 8: Thermo-acoustic instabilities

Lecture 8-1. Rayleigh criterion

Lecture 8-2. Admittance & transfer function

Lecture 8-3. Vibratory instability of flames

Acoustic waves in a reactive medium

Sound emission by a localized heat source Linear growth rate

Flame propagating in a tube Pressure coupling

Velocity and acceleration coupling

Acoustic re-stabilisation and parametric instability (Mathieu’s equation) Flame propagating downward (sensitivity to the Markestein number)

Bunsen flame in an acoustic field

P.Clavin VIII

VIII-1) Rayleigh criterion

Lord Rayleigh 1878

a ² = (c _p /c _v )(c _p c _v )T p = (c _p c _v ) T

Ideal gas

Acoustic waves in a reactive medium

D/Dt / t + u.

c _p c _p c _v

Dp

Dt = c _p T D

Dt + c _p DT Dt c _p D

Dt T = c _p c _p c _v

D

Dt p c _v

c _p c _v a ² D Dt

⇥c _p DT

Dt = Dp

Dt + .( T ) +

j

Q ^(j) W ˙ ^(j)

Energy conservation

isentropic acoustic p = a ² ⇥ Dp/Dt a ² D /Dt = ˙q

˙q ( 1) .(⇥ T ) +

j

Q ^(j) W ˙ ^(j) c _p /c _v heat transfer + heat release

˙q (r, t) =

(rate of energy transfert per unit volume) c _v

c _p c _v Dp

Dt

c _v

c _p c _v a ² D

Dt = .( T ) +

j

Q ^(j) W ˙ ^(j)

3

4

Green’s retarded propagator (1/a ² )⇤ ² G/⇤t ² G = (r) (t), G(r, t) = a (r at)/4⇥r spherical geometry _{p (r, t) =} ¹

4 a ²

1 r

⇥

⇥t ˙q (r , t r/a)d ³ r = ¨(t r/a)

4 a ² r , r = |r|

⇥p /⇥t a ² ⇥ /⇥t = ˙q

2 / t ² = p

acoustic wavelength size of the combustion zone

Sound emission by a localized heat source in free space

⇥ ˙q (r, t)/⇥t = (r) ¨ (t) ¨(t) ˙ (t)/ t, ˙ (t) = ˙q (r , t)d ³ r classical problem

of acoustics

Approximations / t a, c _p , c _v ; constant

Dp/Dt a ² D /Dt = ˙q

˙q ( 1) .(⇥ T ) +

j

Q ^(j) W ˙ ^(j) c _p /c _v

Linearization around a uniform state

= + , p = p + p , u = u + u , ˙q = ˙q + ˙q

a 0,

1 D

Dt = .u, Du

Dt = p ⇥ /⇥t = .u , ⇥u /⇥t = p ,

u. 0

Mean flow velocity neglected in front of the sound speed

2 p / t ² a ² p = ˙q / t

elimination of

5 P.Clavin VIII

Liner growth rate

Unsteady

heat release Acoustic

waves Sound generation

Retroaction

retro-action loop:

hot wire acoustic

wave

Rijike 1859 burner

Bunsen Singing flame

Higgings 1802

Rocket engines, gas turbines...

2 p / t ² a ² p = ˙q / t d ² ˜p _k dt ²

b

ins

d˜p _k

dt + a ² k ² ˜p _k = 0

b > 0 fluctuations of heat release and pressure in phase: instability b < 0 fluctuations of heat release and pressure out of phase: stability Im( ) = ⇤ _k + .., Re( ) = b/(2⇥ _inst ) + ..

/2 < ⇤ _k ⇥ _d (⇤ _k ) < + /2 : Instability Nonlinear study: limit cycles in the unstable case

˜p _k = e ^{t 2 ins} = b ± b ² 4a ² k ^{2 2} _ins 1

ins

⇥ _k = ak

+ 1-D geometry

p(x, t) =

k=

˜p _k (t)e ^ikx k = 2 n/L

Simplest retro-action mechanism: pressure coupling

˙q = b p/⇥ _ins

˙q (x, t) = 1

⇥ _ins

t

b(t t ) p (x, t )t b( ) =

+

r(⇥)e ^{i d ( )} e ⁱ d⇥ + c.c. r( ) > 0 _d ( ) is the phase lag

More general retro-action mechanism

P.Clavin VIII

6

VIII-2) Admittance & transfer function

p = ⇥a u p/ a ² = O(1)

acoustic pressure

jump of the pressure is negligible

( p _b p _u )/p = O(U _L /a) p _f : fluctuation of the pressure at the flame gas expansion jump of the fluctuations of the flow velocity (acoustics)

( u _b u _u )/U _L = O(1)

˙E t = ( u _b u _u ) p _f averaged energy flux (/period) combustion acoustic

˙E ^t = 1

4 _b a _b (Z ˆp ^f ˆp _f + Z ˆp ^f ˆp _f ) = 1

2 [Re Z(⇥)]| ˆp _f | ²

b a _b instability : Re(Z) > 0 .u = 1

T

DT

Dt = ˙q /( 1)

⇥c _p T = ˙q

⇥a ²

mass conservation (quasi-isobaric combustion) ( u _b u _u ) = ˙q

⇥a ² dx

flame brush

Analytical study of a planar flame submitted to a fluctuation of pressure ( ) T _f /T _f p _f /p _f

x

Solid propellant

Reaction Zone

Clavin Lazimi 1992

Zeldovich 1942

Clavin Lazimi 1992

u(t) = Re ˆ u(⇥)e ^{i t} p(t) = Re ˆ p(⇥)e ^{i t}

Definition of the admittance function Z( )

(ˆu _b ˆu _u ) = Z(⇥) ˆp ^f / _b a _b (Rayleigh: ˙q v.s. p)

Pressure coupling

burned gas

fresh mixture u _b

u _u laser tomography (Boyer 1980)

thickness of the flame brush acoustic wavelength

Flame propagating in a tube

⇥ _a

⇥ _ins ( 1)M b

E k _B T _b coeff depends on the position in the tube

as p _f does

gaseous flame

Clavin et al. 1990 Clavin Searby 2008

|Z| = O(M ^b )

P.Clavin VIII

7

Velocity and acceleration coupling

fluctuating velocity modification to flame geometry

fluctuation of heat release through the flame surface

acceleration of a curved flame modulation of the flame surface ^{S =} dy 1 + _y ² Weakly cellular flame propagating downward in an acoustic wave

ˆuˆp _f = ˆu ˆp _f ˙E ^t = Im T ^r ( )(iˆu _u ˆp _f )/2

Real number

phase quadrature (acoustic mode of a tube) (sign depends on position)

fluctuation of heat release rate/ cross-section aera

˙q dx = ⇥ _u U _L c _p (T _b T _u ) S/S _o u _b u _u = ˙q u _b u _u = (T _b /T _u 1)U _L S/S _o

⇥a ² dx

u(t) = Re ˆ u(⇥)e ^{i t} p(t) = Re ˆ p(⇥)e ^{i t}

˜ ₁ vs ˆu _u ?

⇥S/S _o = (k ² /2)˜ _o ˆ ₁ e ^{i t} + c.c.

k ˜ ₀ 1 |˜ ¹ | ˜ ₀ (linear response ok)

(Pelc´e Rochewerger 1992)

Analytical expression (k = k _c )

T r (⇥) (k _{c o} ) ²

Transfer function for a flame in a tube T r ( ) (ˆu _b ˆu _u ) = T r ( )ˆu _u ˙E ^t = (1/4)(T ^r ˆu _u ˆp _f + T r ˆu _u ˆp _f )

1 + ⇥ _b

⇥ _u

d ² ˜

dt ² +2(U _L k) d˜

dt

⇥ _u

⇥ _b 1 k ⇥ _b

⇥ _u [g + g (t)] + U _L ² k 1 k

k _m ˜ = 0 lecture IV:

ok for the primary instability

g (t) = Re i ˆ u _u e ^{i t} g > 0 x = (y, t) (y, t) = ˜(t)cos(ky) _{˜(t) = ˜ 0 + ˆ 1 e} ^{i t} _{+ c.c}

Consider a curved front slightly perturbed unperturbed perturbation

burned gas

fresh mixture u b

u u

8 P.Clavin VIII

VIII-3) Vibratory instability of flames

-5000 0 5000 10000

-0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0

-0.5 0.0 0.5 1.0 1.5

Acoustic pressure (Pa) Relative flame position

Time (secs) Curved flame

Flat flame Cellular flame Case (c)

primary instability + re-stabilisation + parametric instability

N( ) ⇥ G ^o + ² / _m G ^o _b ¹ |g|d ^L /U _L ²

D _b ^b 1

b + 1

N

⇥ _{b u} / _b > 1 ^{B b}

b + 1 C ^b 1

b + 1

u _a

⇥ ,

u _a = 0

Acoustic re-stabilisation and parametric instability

Markstein 1964

d ² ˜

d ² + 2B d˜

d + D + ² C cos( ) ˜ = 0

t/ _h , _h 1/(U _L k), ⇤ ⇥ _h , = (⇥ _L )/(kd _L ) = kd _L g (t) = u _a U _L cos( t)

Mathieu’s equation. Kapitza pendulum

t Y (t) e ^B ˜ d ² Y

dt ² + { + h cos(t)} Y = 0 = (D + B ² )

2 h = C

Kapitza 1951

P.Clavin VIII

Mathieu’s equation. Kapitza pendulum d ² Y

dt ² + { + h cos(t)} Y = 0

> 0 : Oscillator whose frequency is modulated Prametric instability (Faraday 1831)

Faraday 1831 Kapitza 1951

< 0 : Re-stabilization of an unstable position of a pendulum by oscillations (Kapitza 1951)

9

I II

Stability limits of the solutions to Mathieus’s equation White regions: stable. Grey regions unstable

Parametric instability u _aI < u _a < u _aII

Reduced acoustic velocity

a b

c

d

= kd _L u _a /U _L

Re-stabilization tongue unstable wavelengths

u _aI ² 2 _b ( _b + 1)

( b 1) 1 U _Lc U L

, k _I k m

1 2

U _Lc U L

u _aII = 2 _b /( _b 1)

Bychkov 1999 Clavin 2015

ok with experiments Searby Rochewerger 1991

Parametric instability Re-stabilization tongue

Unstable situation

d ² ˜

d ² + 2B d˜

d + D + ² C cos( ) ˜ = 0

g (t) = u _a U _L cos( t) Flame propagating downward

u _a = 0

unstable wavelengths

Markstein 1960

Reduced acoustic velocity

a b

c

d

acoustic equipotential surface Flattening of Bunsen flames in an acoustic field

Rich Bunsen methane flame

(Hahnemann Ehret 1943, Durox et al. 1997, Baillot et al. 1999)

Sensitivity of the acoustic instability to the Markstein number an the acoustic frequency

140 Hz

+ intense axial acoustic field

10

1

Structure and Dynamics of

Combustion Waves in Premixed Gases

Paul Clavin

Aix-Marseille Université ECM & CNRS (IRPHE)

Lecture IX

Turbulent flames

Princeton Summer School on Combustion June 20 - June 26, 2021

Copyright 2021 by Paul Clavin

This material is not to be sold, reproduced or distributed

without permission of the owner, Paul Clavin

P.Clavin IX

2

Turbulent flames

Lecture 9 : 9-1. Introduction

Einstein-Taylor’s diffusion coefficient

9-2. Turbulent diffusion

Rough model of turbulent transport Well-stirred flame regime

Kolmogorov’s cascade Gibson’s scale

Elements of fractal geometry

Self similarity of strongly corrugated flames Co-variant laws

Monopolar sound emission

Sound generated by a turbulent flame Blow torch noise

9-3. Strongly corrugated flammelets regime

9-4. Turbulent combustion noise

3

Turbulent flow!

of unburnt gas

Average position!

of stabilised flame ^!

U _tur

x y

z

IX-1) Introduction

G/ t + v(r, t). G = U _n | G|

stochastic eikonal eq.

The problem of premixed flames in a turbulent flow is still widely open

v = (u, w _y , w _z ) x = (y, z, t)

G G ₀ = x (y, z, t) _{⇥ /⇥t u(r f , t) + w(r f , t). ⇥ yz = U tur U n 1 + |⇥ yz |} ²

U _tur S _o = U _L S S ⇥ = dxdy 1 + |⇤ ^yz | ² U _tur /U _L = 1 + | ^yz | ²

The very existence of S and d _tur is questionable

U _n = U _L ) (l _tur d _L , _{tur L}

G(r, t) = G ₀ G/ t + (dr/dt). G = 0 dr/dt = v(r, t) U _n n n = G/| G|

eq. flame surface

Same model in the wrinkled flame regime

U _turb ? d _turb ? d _turb

Experiments are difficult. Experimental data are very scattered

P.Clavin IX

|v| ⇥ U ^L : U _tur /U _L 1 + (|v|/U ^L ) ²

(Shchelkin 1943, Clavin Williams 1979)

|v| ⇥ U L : U _tur |v|

(Damk¨oler 1940)

Bending effect

modification to the laminar flame struture

(no gas expansion)

⌅ /⌅t + v(r, t). D _T = ⇤ ( )/⇥ _rb .

prescribed turbulent flow (stochastic field)

Reaction-diffusion wave in a turbulent flow

The simplest model has no known solution (Nonlinear stochastic equation)

4

P.Clavin IX

IX-2) Turbulent diffusion

Einstein 1905

G.I Taylor 1922

Taylor’s diffusion coe⇥cient

x ² (t) = 2 ^t

0

dt ^t

0

d v(t )v(t )

integration by parts x ² (t) = 2 v ^{2 t}

0

(t )g( )d where

0

g( )d = O( _I ² ) t _I : g = 0 v(t)v(t ) = v ² g( )

turbulence: homogeneous in time g(0) = 1, lim g = 0 I

0

g( )d

integral time scale

v ² _I t _I , 1-D : x ² (t) = 2D _tur t, 3-D : x ² (t) = 6D _tur t, where D _tur

v ² = (turbulence intensity) ² (analogy with Einstein random walk for molecular diffusion) 1-D for simplicity: dx/dt = v(t), x(t) = ^t

0

v(t )dt

stochastic Lagrangian velocity

x ² (t) = ^t

0

dt ^t

0

dt v(t )v(t )

ensemble average

U _tur D _tur / _b U _tur U _L

l _I d _L

v _I

U _L 1, d _tur D _tur /U _tur d _L

Rough model for the turbulent transport (analogy with molecular diffusion)

v D _tur , .(v ) D _tur

Well-stirred flame regime of Damk¨ohler ⁽¹⁹⁴⁰⁾ l _I d _L and ^D tur D _T little practical importance

)

limited to scalar mixing with small displacement / size (blobs, sheets ..)

integral length scale

l _I v _{I I}

turbulence intensity

l _I L ( v I v ^{2 1/2} ,

D _tur = l _I v _I

P.Clavin IX

5

IX-3) Strongly corrugated flammelets regime

Kolmogorov’s cascade (homogeneous, isotropic and fully-developed turbulence)

l _i , _i , v _i l _i / _i

turn-over velocity

Kolmogorov 1941

viscous diffusion coeff µ/⇥

local Reynolds nb Re _i l _i v _i /

Integral scale l _I , _I , v _I Re _I 1 l _I > l _i v _I > v _i i Kolmogorov scale l _K , _K , v _K Re _K = 1 l _i > l _K v _i > v _K i

schematic representation l i = l I /2 ⁱ

Decomposition into a sum of vorties of different length and time scales

Scaling laws (dimensional analysis) l _K l _i l _I

energy transfer in NS eqs : (v. )v ² /2 v _i ³ /l _i cst v _i ^1/3 l _i ^1/3 , v _i ^{2 2/3} l ^2/3 _i , ⇥ _i ^1/3 l _i ^2/3 , dissipation rate of energy : v. v ^{= ⇥ v} _K ^{2 /l} _K ^{2 = v} _I ^{3 /l} I

Re _K v _K l _K / = 1 _{l I /l K Re} ^3/4 _{I , v I /v K Re} ^1/4 _{I , I / K Re} ^1/2 _I Re _I 1 energy spectrum : v ² /2 =

0

dk E(k) E(k) ^2/3 k ^5/3

turn-over time = transit time across the vortex i l _i /U _L v _i U _L l _G U _L ³ / l _K l _G l _I

many scales of wrinkles l _G l _I fractal geometry of the flame front definition of strongly corrugated flames

v _K U _L v _I

definition of the Gibson scale: smallest size of the wrinkles on the flame front Gibson scale ^l G (Peters 1986) v _K /l _K , U _L D _T /d _L , , D _T

no modification to

the laminar flame structure

d _L l _K , _{L K}

P.Clavin IX

6

l _i

l ⁱ /2 l ⁱ /4

l ⁱ /8 l ⁱ /16

Elements of fractal geometry

Fractal dimension D _f > 2 : N _i,j (l _i /l _j ) ^{D f} , S _i,j /l _i ² (l _i /l _j ) ^{D f 2}

S i /l ² _i = lim

l _j 0 S i,j /l _i ² = finite cst.

Regular surface: D _f = 2 total area S _i in a box of size l _i

S _i N _i,G l _G ² . Total surface area in a cube of size l _i , l _G < l _i < l _I

nb of cubes of size l G that intersect the surface within the volume l ³ _i

S _i,j+k < S _i,j < S _i N _i,j+k < N _i,j < N _i Details of small scales are lost as the size of the box l _j increases

S _i,j N _i,j l _j ² Weaker resolution

nb of cubes of size l j that intersect the surface within the volume l _i ³

l _j , l _G < l _j < l _i

The fractal dimension D _f > 2 can concern only scales greater than the smallest wrinkles

d _L < l _G < l _j < l _i < l _I Fractal dimension of a turbulent flame can be meaningful only for

For a flame of thickness d _L its area is well defined for wrinkles whose scale is larger than d L , l _j > d _L

7

P.Clavin IX Self similarity of strongly corrugated flames ^v K U _L v _I d _L l _K l _G l _I

U _i /U _j = S _i,j /l _i ² A flame velocity U _i can be defined at each scale if U _i = (S _i,j /l ² _i )U _j

Fastest contamination: integral scale v _I v _i . U _tur = v _I ?

ok if the combustion time of the vortex is not longer than the turnover time

U _L = v _G At the Gibson scale the combustion time of the vortex = turnover time

v I

v _I v _I

l _I l _I

Kolmogorov cascade _{⇥ i} ^1/3 _{l i} ^2/3 l _{i v i} ^1/3 _{l i} ^1/3 l _i

Contamination time vs combustion time

Self similar law

An effective front of thickness l _i is defined at each scale

U _tur = v _I , l _tur = l _I

Self similarity: same law at all scales combustion time of the vortex = the turnover time l _i l _i /U _i = _i U _i = v _i

Assumption: the Kolmogorov cascade is not modified by gas expansion

Fractal dimension of the flame surface:

U _i /U _j = S _i,j /l _i ² u _i /u _j = S _i,j /l _i ² S _i,j /l _i ² = (l _i /l _j ) ^1/3 S _i,j /l ² _i = (l _i /l _j ) ^{D f 2}

D _f = 7/3

The result is the same for all mixtures...??

Kolmogorov cascade small vortices burn faster than larger ones

ok for l _i l _G

P.Clavin IX

8

Co-variant laws

More general law independent of the turbulent scaling and satisfying additivity Turbulent energy contained in the range [l _i , l _j ] : _{v i,j} ²

j 1

k=i

v _k ² v _k ² : energy in [k, k + 1]

Co-variant law = same for each couple of length scales l _i , l _j l _i > l _j

The only co-variant law for the flame velocity U _i at scale l _i satisfying additivity is U _i ² = U _j ² + c v _i,j ²

Pocheau 1994

Co-variance ? ^l i > l _k > l _j , v _i,j ² = v _i,k ² + v _k,j ² U _i ² = U _j ² + c v _i,k ² + c v _k,j ² = U _k ² + c v _i,k ²

U _i ² = U _k ² + c v _i,k ² co-variance ok

U _tur ² = U _L ² + c v ² v ²

n

v _n ² turbulence intensity Not limited to a strong turbulence

The case c = 1 covers the known results at low and large turbulence intensity

Pocheau 1996

Reasonably good agreement with experiments v/U _L = O(1), l _I /l _K 180

Pocheau 1994

P.Clavin IX

IX-4) Turbulent combustion noise

T urbulent flow

Flame front

Acoustic

wavelength a/ L size of the flame wave

Monopolar sound emission

acoustic potential ⇥(r, t) = ˙V (t a r)

u = (r, t) 4 r r |r|, ˙V (t) dV/dt

Deformable (small) body with fluctuating volume V (t)

r L : v = (4 ar) ¹ V (t ¨ r/a), V (t) ¨ d ² V (t)/dt ² Radiated flux of energy (intensity of sound) I a v ²

I = (⇥/4 a) d ² V /dt ^{2 2}

9

Sound generated by a turbulent flame ^{dV /dt = ˙ M b / b}

mass flow rate of burned gas in the lab frame

M ˙ _b = _b

S

(D ^f + U _b )d ² ⇥ M ˙ _u = _u

S

(D ^f + U _L )d ² ⇥

normal flame velocity in the lab frame

mass flow rate of fresh gas in the lab frame

u U _L = _b U _b

elimination of D ^{f dV /dt = ˙ M b / b = ˙ M u / u + (U b U L )S} dV /dt = (U _b U _L )S(t) constant

I = (⇥/4 a)(U _b U _L ) ² (dS/dt) ² intensity of sound

Strahle 1985

power spectrum of sound d˜ I(⇤)/d⇤ = ⇥

4 a (U _b U _L ) ²

0

dt e ^{i t} ˙S(t) ˙S(0)

P.Clavin IX

10

I = (⇥/4 a)(U _b U _L ) ² (dS/dt) ²

intensity of sound power spectrum of sound

d˜ I(⇤)/d⇤ = ⇥

4 a (U _b U _L ) ²

0

dt e ^{i t} ˙S(t) ˙S(0) Stongly corrugated regime with a Kolmogorov cascade:

D _f = 4/3 _I ¹

4

T _b

T _u 1

2

(⇥ V ) v _I ⁴

al _I d˜I( ) ^5/2 d

Clavin Siggia 1991

in agreement with experiments on very large burners

(Abugov Obrezkov 1978)

the noise is not resulting from the direct interaction of upstream turbulence on the flame front amplification by the intrinsic flame instability is essential (Searby 2001)

total volume of the flame brush

Combustion noise is two orders of magnitude higher

Blowtorch noise

1

Structure and Dynamics of

Combustion Waves in Premixed Gases

Paul Clavin

Aix-Marseille Université ECM & CNRS (IRPHE)

Lecture X

Supersonic waves

Princeton Summer School on Combustion June 20 - June 26, 2021

Copyright 2021 by Paul Clavin

This material is not to be sold, reproduced or distributed

without permission of the owner, Paul Clavin

P.Clavin X

2

Lecture 10 : Supersonic waves

10-1. Background

Model of hyperbolic equations for the formation of discontinuity Riemann invariants

Rankine-Hugoniot conditions for shock waves

Mikhelson (Chapman-Jouguet) conditions for detonations

10-2. Inner structure of a weak shock wave

Formulation

Dimensional analysis Analysis

10-3. ZND structure of detonations

10-4. Selection mechanism of the CJ wave

thickness: few mean free paths

distance (D v ^p )t

U

<latexit sha1_base64="+1iOEdzHDt8T9H4Z7qI17oQ+yFU=">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</latexit><latexit sha1_base64="xcoQRUeyB865ZtgYd2eVA8ty7Bw=">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</latexit><latexit sha1_base64="xcoQRUeyB865ZtgYd2eVA8ty7Bw=">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</latexit><latexit sha1_base64="rdkXvATxyR00bWNxMHYprzke0M0=">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</latexit>

_N

U

<latexit sha1_base64="+1iOEdzHDt8T9H4Z7qI17oQ+yFU=">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</latexit><latexit sha1_base64="xcoQRUeyB865ZtgYd2eVA8ty7Bw=">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</latexit><latexit sha1_base64="xcoQRUeyB865ZtgYd2eVA8ty7Bw=">AAACsnicjVHLSsNAFD2Nr9r6qHXpJliFrkoigq6k4MaVVLQPrKVM0mkNTTshmQil1C/QreJH+QH+gf6Fd6YpqEV0Apkz5557uA8n8L1IWtZbylhYXFpeSa9msmvrG5u5rXwtEnHo8qorfBE2HBZx3xvyqvSkzxtByNnA8Xnd6Z+qeP2Oh5EnhldyFPDWgPWGXtdzmSTqsto+b+cKVsnSx5wHdgIK5fzLw31xL1sRuVfcoAMBFzEG4BhCEvbBENHXhA0LAXEtjIkLCXk6zjFBhnJjUnFSMGL79O/Ra0xYkEaQfgIT+xQXpAwJK2dTx2PtotjffRjVpOoY0e0kXgNiJW6J/StvpvxvXpNYiS6OdbcezSLQjJqKm7jEegKqcvNLV5IcAuIU7lA8JOzqzNlMTZ0T6d7VHJmOv2ulYtXbTbQxPlSVtEz75+rmQe2gZFsl+4K2eoLpSWMHuyjS7o5QxhkqqJJ3D494wrNxaFwbzHCnUiOV5Gzj2zH8T6LPjdg=</latexit><latexit sha1_base64="rdkXvATxyR00bWNxMHYprzke0M0=">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</latexit>

_N