lecture 12 - diffusion flame strcuture...lecture 12 the structure of diffusion flames mixture...

$: Lecture 12 - Diffusion Flame Strcuture...Lecture 12 The Structure of diffusion flames Mixture fraction formulation 1 !c p (!T!t +v"#T)$#"(!#T)=Q! "(!Y F!t +v"#Y F)$#"(!D F #Y F)=$!$
6/26/2011

Copyright ©2011 by Moshe Matalon. This material is not to be sold, reproduced or distributed without the prior written permission of the owner, M. Matalon. 1

Lecture 12 The Structure of diffusion flames

Mixture fraction formulation

1

!cp(!T!t

+ v "#T )$#" (!#T ) =Q!

"(!YF!t

+ v "#YF )$#" (!DF#YF ) = $! FWF!

"(!YO!t

+ v "#YO )$ #" (!DO#YO ) = $!OWO!

! = B !YFWF

!YOWO

e-E/RT

!T = P0W /R

Governing Equations

+ mass and momentum equations

For simplicity of presentation we will treat the properties constant, but the results can be easily generalized if they are temperature-dependent.

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Non-dimensionalization. Choose, U as a unit of speed, lD = λ/ρ0cpU and lD /U as units of length and time. Use Q/cpνFwF and P0 to nondimensionalize temperature and pressure and scale ρ accordingly.

!(!T!t

+ v "#T )$#2T =!

"(!YF!t

+ v "#YF )$ LeF$1#2YF = $!

"(!YO!t

+ v "#YO )$ LeO$1#2YO = $!"

" = D#2YFYOe$ /Ta$$ /T

#T =1

We are already familiar with all the parameters appearing in these equations. Note, in particular, that

D ~ Dth /U2

(Be!E / R !Ta )!1! =

Q/cpR 2/ E Ta is the (dimensionless) adiabatic temperature

3

YF = 0

reaction zone YO = 0

The structure of the Burke-Schumann solution

on either side of the reaction sheet

!(!T!t

+ v "#T )$#2T = 0

!(!YF!t

+ v "#YF )$ LeF$1#2YF = 0

!(!YO!t

+ v "#YO )$ LeO$1#2YO = 0

D!" # YFYO~ 0

The chemical reaction is confined to a sheet F(x,t) = 0

n

ξ2 ξ1

n

! = n ""n

+!#

Introduce an orthogonal intrinsic coordinate system (ξ1, ξ2, n), then where n is a coordinate along the normal, and n is a unit normal to the surface, given by

n = !F|!F |

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!1!"2""#2

~ D! 2yFyO

"2

"#2(" + LeF

!1yF ) ="2

"#2[" + ($LeO )

!1yO ]= 0

The solution inside the reaction zone is slightly modified from the values along the sheet and, on is obtained by stretching the normal coordinate and determining the appropriate balance between the various terms in the conservation laws.

!

n = " #

Y = " yF (#) +!, YO = " yO (#) +!T =Ta +"$ (#) +!

!"2!""2

~ D#3e!$ /TaO(1)

!"# $# yFyO

# ~ D!1/3

•  The reaction zone is diffusive-reactive, with convection negligibly small •  The thickness of the reaction zone for large D is ~ D-1/3 •  The reaction proceeds at a rate that depends on the local fuel and oxidizer concentrations, but is independent of temperature (nothing has been assumed about the Zel’dovich number, so that exp(β/Ta ‒ β/Τ) ∼ 1 when T ~ Ta ) 5

! +! -1LeO

!1yO = a! + b

! + LeF!1yF = !c! + d

Matching conditions with the solution on either side of the reaction sheet, yields

b = d = 0 and expressions for a,c

!2!!"2 =C(a" "! )(c" +! )

! # a" as "#"$

! #"c" as "#$

A transformation of the coordinate η and the dependent variable τ yields, after some algebraic manipulations, the canonical problem

!2!!" 2 = (! "" )(! +" )

!!!"

#±1 as " #±$

ζ

ϕ

The solution always exist!

!2

!!2(" + LeF

"1yF ) =!2

!!2[" + (#LeO )

"1yO ]= 0

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Activation Energy Asymptotics

The previous discussion did not exhibit the extreme sensitivity of the reaction rate to small changes in temperature. The proper expansion parameter is the Zel’dovich, or activation energy parameter β, or ε ∼ 1/β.

Linan, Acta Astro. 1974; Cheatham & Matalon, JFM 2000

n = !"Y = ! yF (!)+!, YO = ! yO (!)+!

T = Ta +!" (#)+!

! = D"2YFYO e# /Ta!# /T = D"2YFYO e

# (T!Ta )/TTa ~ D"a2 ($yF )($yO )e

#$% /Ta2

! ! Ta2 /"Let

!1!"2""#2

= D̂! 2"a2 yFyOe

! /Ta2

!"2!""2

= D#3$a2

D̂ = O(1)!"# yFyOe

!

D = (! 3 /Ta4 )D̂ distinguished limit

7

!2!!" 2 = (! "" )(! +" ) exp{#"1/3(! +$" )}

!!!"

#±1 as " #±$

! +! -1LeO

!1yO = a! + b

! + LeF!1yF = !c! + d

!2

!!2(" + LeF

"1yF ) =!2

!!2[" + (#LeO )

"1yO ]= 0

The coupling functions provide linear relations between yF, yO and τ which, when substituted into the equation for the temperature yields, as before, after an appropriate transformation of the variables and some manipulations, a canonical problem

but with two auxiliary parameters γ and δ determined from the “outer solution”, namely the solution outside the reaction zone. Note that δ ~ D is effectively a “reduced Damköhler number”, which extend the solution to large but not necessarily infinite D. It is evident that when δ = ∞ the problem reduces to the structure of the BS solution.

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γ = 0

two solutions exists for δ > δc = 0.84954and none for δ < δc

9

two solutions exists for δ > δc = 0.8281and none for δ < δc

γ = 0.4

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  The solution of the BVP determines

where S1 and S2 are the leakage functions, that determine the degree of leakage of fuel and/or oxidizer (SF and SO ) through the reaction zone.

  The parameter -1< γ < 1 fixes the amount of heat needed for given conditions and depends strongly on the mixture strength φ. Specifically it is the excess of heat conducted to one side of the reaction sheet, and is given by   The parameter δ measures the intensity of the reaction; it controls how much heat can be generated for given conditions, and depends on the excess/deficiency in available enthalpy in the reaction zone.

 The nonexistence of solutions for δ < δc , or equivalently for D below a critical value is associated with extinction. Linan (1974) provided the approximation

! = !1"

#T /#n( )n=0+

+ #T /#n( )n=0!

{ } ! = " #T/#n[ ] > 0where

! =4LeOLeFTa

4D" 3!2

ehf /2 hf = (Tf !Ta )! +1!"2 SF +

1+"2 SO

lim!!"(" ## ) = $1/3S1 and lim!!#"(" +# ) = $1/3S2

!c ! e " (1# ! )# (1# ! )2 + 0.26(1# ! )3 + 0.055(1# ! )4{ } 11

Dext

Tf

D

BS limit

Flam

e te

mpe

ratu

re

Damköhler number

aT

T f ~

Ta +

θ-1τ f

The critical δc determines Dext

two solutions for δ > δc

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S1 S2

SF =

�S1 0 � γ < 1S2 −1 < γ � 0

SO =

�S2 0 � γ < 1S1 −1 < γ � 0 .

The leakage functions

Cheatham & Matalon, JFM 2000

13

[YF ] = [YO] = [T ] = 0

1

Q

�λ∂T

∂n

�= −

�ρD

F

νFWF

∂YF

∂n

�= −

�ρD

O

νOWO

∂YO

∂n

�

n

Reaction sheetF(x, t) = 0

Fuel

Oxidizer

The new conditions across the reaction zone constitute a nontrivial generaliza-tion of the Burke-Schumann solution, extending its validity for the limit of com-plete combustion down to extinction; i.e., for the whole range Dext < D < ∞.

For a general surface F (x, t) = 0 the conditions that mimic the finite-rate chem-istry details inside the reaction zone are

YF |n=0+ = β−1SF (γ, δ) YO|n=0− = β−1SO(γ, δ)

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The Mixture Fraction

In premixed combustion the fuel-to-oxidizer ratio, or equivalence ratio, is fixedand remains constant everywhere. Consequently, the progress of combustion canbe described by a variable measuring the extent of completion of the reaction.In diffusion flames, the fuel-to-oxidizer ratio is not constant and an additionalvariable is needed. A convenient one is the mixture fraction defined as Z = 0in the oxidizer stream and Z = 1 in the fuel stream.

The mixture fraction Z = Z(x, t) is the fraction of the total amount ofmaterial present that came originally from the fuel stream.

Z =νYF − (YO − YO2)

νYF 1 + YO2

Consider a two-stream problem, having uniform properties on one portion of theboundary - the fuel stream, (denoted with subscript 1) and different uniformproperties over the rest of the boundary - the oxidizer stream (denoted withsuscript 2). There is no fuel in the oxidizer boundary and no oxidizer in the fuelboundary.

ν =νOWO

νFWF

15

For example: suppose the fuel stream contains CH4 and the oxidizer streamcontains air. Fuel atoms appear in CH4, CO, OH, H2O, etc . . . .

The element mass fraction of C and H is

16

16YCH4 +

12

28YCO +

1

17YOH +

2

18YH2O

which can be normalized such that Z = 1.

A more general way, good for more involved chemistry, is to use the local massfractions of atoms.

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From the species equations

ρDZ

Dt− ρD∇2Z = 0

Expressions relating the mass fractions to Z are easily obtained. From thedefinition

Z =νYF − (YO − YO2)

νYF 1 + YO2⇒

Zst =YO2

νYF 1 + YO2

YF = YO = 0

defines the stoichiometric surface

ρDZ

Dt− ρDF∇2Z = (ρDF − ρDO)∇2YO

ρDYF

Dt− ρDF∇2YF = −νFWF ω ρ

DYO

Dt− ρDO∇2YO = −νFWO ω

we see that only for equal diffusivities, Z satisfies a reaction-free equation (i.e.,it is a conserved scalar, which is a quantity that is neither created or destroyedby chemical reaction)

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Z 0 1

T∞

T-∞

YF 1

Zst

YO2

Since on one side of the stoichiometric surface YO = 0 and on the other sideYF = 0, we have

For 0 < Z < Zst YO = YO2

�1− Z

Zst

�

For Zst < Z < 1 YF = YF 1

�Z − Zst

1− Zst

�

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DT

Dt=

dT

dZ

DZ

Dt; ∇2T =

dT

dZ∇2Z

ρDT

Dt− λ

cp∇2T =

�ρDZ

Dt− λ

cp∇2Z

�dT

dZ= 0

ρDT

Dt− λ

cp∇2T = Qω

Since in the fast chemistry limit, the temperature also satisfies a reaction-freeequation on either side of the stoichiometric surface, i.e., ω = 0, the temperatureT may also be expressed as a linear combination of Z. For, if T = T (Z(x, t))

which is clearly satisfied, provided, the Lewis number is one (λ/cp = ρD).

T (0) = T2, T (1) = T1

Hence T = T (Z)

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ρDT

Dt− λ

cp∇2T = Qω

ρDYF

Dt− ρDF∇2YF = −νFWF ω

ρDYO

Dt− ρDO∇2YO = −νFWO ω

hF = hF (Z), hO = hO(Z)

We note that, for unity Lewis numbers, the functions

hF = T + (Q/cpνFWF )YF , hO = T + (Q/cpνOWO)YO

are conserved scalars.

hF = T2 + [(QYF 1/cpνFWF ) + T1 − T2]Z

hO = T1 + [(QYO2/cpνOWO) + T2 − T1] (1− Z)

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Z 0 1

YF 1

Zst

YO2

T1

T2

from which we deduce

T =

T2 + [(QYF 1/cpνFWF ) + T1 − T2]Z 0 < Z < Zst

T1 + [T2 + (QYO2/cpνOWO)] (1− Z) Zst < Z < 1

Ta = T2 + [(QYF 1/cpνFWF ) + T1 − T2]Zst

Ta

the adiabatic temperature

21

We have seen that the temperature and mass fractions expressed in terms ofthe mixture fraction Z takes a very simple form. When the combustion fieldis expressed in terms of one coordinate, it is convenient to make a change ofvariables and use Z as the transformed coordinate. This, of course, underthe assumption of unity Lewis numbers. For example, in the counterflow con-figuration, the equation for Z is

This transformation contracts the axial coordinate x from (−∞,∞) to (1, 0).

⇒

2εxdZ

dx+Dd2Z

dx2= 0

(in dimensional form)

The structure of the reaction zone, which follows the previous discussion pro-vides the limitation D > Dext for the existence of steady solutions, or equiva-lently ε < εext.

Z = 12

�1− erf(

��/D x)

�

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In more general circumstances, the largest changes in the mixture-fraction fieldZ(x, t) occur along the direction of |∇Z| from the surface Z = Zst. A coordinatetransformation (x, y, z) → (x, y, Z(x, y, z)) is then used to describe the reactionzone structure.

− ∂

∂n= |∇Z| ∂

∂Z+

∇Z ·∇T

|∇Z|

where ∇T =

�∂

∂x,∂

∂y

�is the transverse gradient.

− |∇Z|2 ∂2τ

∂η2= D̂ yF yO eτ

This can be easily related to our previous discussion, using that

The equation describing the reaction zone structure becomes

where |∇Z|2 is evaluated here at the stoichiometric surface Zst, and yields thesame canonical problem, as before, with the additional factor 4(1−Zst)2/ |∇Z|2in δ.

23

The dimensional quantity χ = 2D |∇Z|2, in units of 1/s is known as the instan-taneous scalar dissipation rate. Its value at the stoichiometric surface, χst, canbe interpreted as an “inverse diffusion time”. Flame extinction occurs when thescalar dissipation rate χ exceeds a critical value χext

st , which may be associatedwith an extinction time.

The mixture fraction formulation is useful in more complex configurations whenthe profiles depend weakly on the transverse directions x, y and could be con-sidered to depend only on Z. In this case, results of the counterflow problemcould be applied to these more complex problem once the mixture fraction fieldZ(x, t) is known.

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25

Flame instabilities

Instabilities in diffusion flames have been typically observed at high flow rates, ornear-extinction conditions. They are predominately diffusive-thermal in naturewith hydrodynamics playing a secondary role.

The Burke-Schumann solution of complete combustion (corresponding to thelimit D → ∞) is unconditionally stable. Instabilities result only when there issufficient leakage of reactants through the reaction sheet with a flame temper-ature significantly below Ta.

  Instabilities develop when there is significant reactant leakage

i.e. for Dext < D < D*, namely at high flow rates or near-extinction conditions.   Nature of the instability depends on the parameters LF , LO , φ

Dext

Tf

D

BS limit

Flam

e te

mpe

ratu

re

Damköhler number

D* a T

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Planar pulsations

oscillatory cells

1

1

LeF

LeO

high-frequency instability

Stationary cells

Cellular flames - more likely to occur when φ < 1 (lean conditions) cell size ≈ 0.5 - 2 cm

Pulsating flames - more likely to occur when φ > 1 (rich conditions) frequency ≈ 1 - 6 Hz

high-frequency instability

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Nitrogen-diluted H2-O2 Diffusion flame - splitter-plate burner Dongworth & Melvin, 1976

1 cm

Under normal conditions - base remains straight; at high flow rates and at high dilution rates base is cellular. Same behavior when diluted in Ar but not in He.

High flow rate D < D*

in N2 and Ar LF ≈ 0.33 – 0.35

in He LF ≈ 1.02

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Cellular Diffusion Flame H2 & O2 diluted in CO2 (LeF = 0.35, LeO = 0.86, φ = 0.24)

9.1 mm

Robert & Monkewitz - EPFL

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lecture 12 - diffusion flame strcuture...lecture 12 the structure of diffusion flames mixture...

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