AME513-F12-lecture12 - University of Southern Californiaronney.usc.edu/AME513F12/Lecture12/AME513-F12-lecture12.pdfLow Re: depends Froude number (Fr = u e ... of lecture notes

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AME 513

Principles of Combustion"

Lecture 12 Non-premixed flames II: 2D flames, extinction

2 AME 513 - Fall 2012 - Lecture 12 - Nonpremixed flames II

Outline"  Jet flames   Simple models of nonpremixed flame extinction

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Axisymmetric jet – boundary layer flow "  Assumptions: steady, axisymmetric, constant density, zero

mean axial (x) pressure gradient   Boundary layer approximation – convective transport (of

momentum or species) only in axial (x) direction, diffusion only in radial (r) direction

  Jet momentum J = constant (though kinetic energy is not, nor is mass flow since entrainment occurs)

Continuity :!!""u = 0 # $ux

$x+

1r$ rur( )$r

= 0;Momentum : !!u!t+!u "!!( ) !u = %

!!P"+!g+#!2 !u

# x %momentum :ux$ux$x

+ur$ux$r

+#r$$r

r $ux$r

&

'(

)

*+= 0;Fuel :ux

$YF$x

+ur$YF$r

+DF

r$$r

r $YF$r

&

'(

)

*+= 0

Let $ = Crx

;C = 3J16%# 2"

;J = !mue = "ueA( )ue = "ue2 4%re2 (Axial momentum at jet exit)

$$x

=$$$x

$$$

=%Crx2

$$$

=%$x

$$$

, $$r=$$$r

$$$

=Cx$$$


Axisymmetric jet – boundary layer flow "

  On the axis (r = 0 thus η = 0), both ux & YF decay as 1/x   Off axis, the jet spreads ~ η ~ r/x, i.e. linearly   This allows us to use “simple” scaling to estimate flame

lengths…

Continuity : !ux!x

+1r! rur( )!r

= 0 " !ux!!

=C!2

! !ur( )!!

x #momentum :ux!ux!x

+ur!ux!r

+"r!!r

r !ux!r

$

%&

'

()= 0 " ur #

!uxC

$

%&

'

()!ux!!

+C!x

!!!

!!ux!!

$

%&

'

()= 0

Eventually leads to :

ux (x, r) =3J

16#"$1x

1#!2

4$

%&

'

()

#2

"ux (x, r)ue

=38

Re xre

$

%&

'

()

#1

1#!2

4$

%&

'

()

#2

;Re * 2uere"

ur =3J

16#$1x

! 1#!2 / 4( )1+!2 / 4( )

2 "ur (x, r)ue

=34

xre

$

%&

'

()

#1! 1#!2 / 4( )1+!2 / 4( )

2

YF (x, r) = 3re ue2D

xre

$

%&

'

()

#1

1#!2

4$

%&

'

()

#2

"YF (x, r)YF,e

=34

Re xre

$

%&

'

()

#1

1#!2

4$

%&

'

()

#2

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Nonpremixed-gas flames - laminar gas-jet flames"  Flame height (Lf) scaling estimated by equating time for

diffusion of O2 to jet centerline tD

tD ~ d2/Dox, d = stream tube diameter, Dox = oxygen diffusivity

to convection time (tC) for fuel to travel from jet exit to end of flame at Lf

tC ~ Lf/u

  The problem arises that d is not necessarily the same as the jet exit diameter de = 2re – if the flow accelerates or decelerates (i.e. u changes), to conserve mass d must change

  For the simplest case of constant u, d:

d2/D ~ Lf/ue ⇒ Lf ~ ued2/D or Lf/de ~ Ude/D

Gases: D ≈ ν ⇒ Lf/de ~ Ude/ν = Red

  Which is consistent with experimental data for laminar, momentum-controlled jets


Nonpremixed-gas flames - laminar gas-jet flames"

1

10

100

1 10 100 1000

Sunderland et al. (1g)Sunderland et al. (µg, 1 atm)Sunderland et al. (µg, 0.5 atm)Sunderland et al. (µg, 0.25 atm)Cochran and Masica (µg)Bahadori et al. (µg)Bahadori & Stocker (µg)

L f/dj

Reynolds number (Re)

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Nonpremixed-gas flames - laminar gas-jet flames"  For buoyancy-controlled jets, the flow accelerates

u ~ (gLf)1/2, g = acceleration of gravity

and to conserve volume flow, the stream tube diameter must decrease

u(πd2/4) = constant = ue(πde2/4), thus d ~ de(ue/u)1/2 (round jet)

ud = constant = uede, thus d ~ de(ue/u) (slot jet)

and thus for laminar, buoyancy-controlled round-jet flames tC ~ Lf/u, tD ~ d2/Dox, u ~ (gLf)1/2, d ~ de(ue/u)1/2 Lf/u ~ d2/Dox, thus Lf/(gLf)1/2 ~ de

2(ue/u)/Dox ~ de2(ue/(gLf)1/2)/Dox

Lf ~ uede2/Dox – same as momentum-controlled, consistent with

experiments shown on previous slide

but for laminar, buoyancy-controlled slot-jet flames tC ~ Lf/u, tD ~ d2/Dox, u ~ (gLf)1/2, d ~ de(ue/u)1 Lf/u ~ d2/Dox, thus Lf/(gLf)1/2 ~ de

2(ue/u)2/Dox ~ de(ue2/(gLf)1)/Dox

Lf ~ (ue4de

4/gDox2)1/3 – very different from momentum-controlled!


Nonpremixed-gas flames - gas-jet flames"  Also if jet is turbulent, D ≠ constant, instead D ~ u’LI ~ ud   Example for round jets, momentum controlled

tC ~ Lf/u, tD ~ d2/Dox, Dox ~ ud Lf/u ~ d2/Dox, thus Lf/u ~ de

2/(ud) Lf ~ de– flame length doesn’t depend on exit velocity at all – consistent with experiments shown on next slide

 Also high ue ⇒ high u’ ⇒ Ka large - flame “lifts off” near base  Still higher ue - more of flame lifted  When lift-off height = flame height, flame “blows

off” (completely extinguished)

Lifted flame (green = fuel; blue = flame)

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Nonpremixed-gas flames - gas-jet flames"  Summary of jet flame scaling

 Always equate diffusion time to convection time »  Diffusion time ~ d2/Dox, d = stream tube diameter, Dox = oxygen

diffusivity) »  Convection time ~ Lf/u

 Volume conservation (2 choices) » uede

2 ~ u(Lf)d(Lf)2 (round jet) » uede ~ u(Lf)d(Lf) (slot jet)

 Buoyancy effects (2 choices) » Buoyant flow: u(Lf) ~ (gLf)1/2 » Nonbuoyant: u(Lf) = u(0) = constant

  Turbulence effects (2 choices) » Laminar: Dox = molecular diffusivity= constant » Turbulent: Dox ~ u’LI ~ u(Lf)d(Lf)

  Total of 2 x 2 x 2 possibilities!


Burke-Schumann (1928) solution"  Axisymmetric flow of a fuel & annual oxidizer with equal u   Overventilated if ratio of mass flow of fuel to oxidizer is <

stoichiometric ratio, otherwise underventilated   Burke-Schumann solution is essentially a boundary layer

approximation – assume convection only in streamwise direction, diffusion only in radial direction – valid at high Pe = urj/D

  Solution rather complicated (Eq. 9.55)

but flame height involves only  Dimensionless coordinate ~ xD/urj

2  Stoichiometric coefficient to identify flame location

!u!YF!x

"1r!!r

r!D !YF!r

#

$%

&

'(= 0

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Flame widths at 1g and µg"

 Note Lf ≈ same at 1g or µg (microgravity) for round jet, but flame width greater at µg because tjet larger

 µg flame width ~ (Dtjet)1/2 - greater difference at low Re due to axial diffusion (not included in aforementioned models) & stronger buoyancy effects

1

10

10 100 1000

Sunderland et al. (1g)Sunderland et al. (µg, 1 atm)Sunderland et al. (µg, 0.5 atm)Sunderland et al. (µg, 0.25 atm)Cochran and MasicaBahadori et al.

w/d

o

Reynolds number (Re)


Flame lengths at 1g and µg"

do = 3.3 mm, Re = 21 ! ! !d = 0.42 mm, Re = 291!Sunderland et al. (1999) - C2H6/air

 Low Re: depends Froude number (Fr = ue2/gde)

 1g (low Fr): buoyancy dominated, teardrop shaped  µg (Fr = ∞): nearly diffusion-dominated, morelike a spherical

droplet flame  High Re: results independent of Fr

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Nonpremixed flame extinction"

  Up till now, we’ve assumed “mixed is burned”(infinitely fast chemical reaction) – but obviously nonpremixed flames can be extinguished due to finite-rate chemistry

  Generic estimate of extinction condition   Temperature in reaction zone is within 1/β of Tf  Reactant concentration in reaction zone is 1/β of ambient value   Thickness of reaction zone is 1/β of transport zone thickness

Reaction time scale ~ Fuel[ ]!"d Fuel[ ]dt

#

$%

&

'(

!1

= Fuel[ ]!" ZFuel[ ]!"!

Ox[ ]+"!

e!E /)Tf#

$%

&

'(

!1

= ZOx[ ]+"! 2 e!E /)Tf

#

$%

&

'(

!1

Residence time ~ 1!

1*+ !* ~ Z

Ox[ ]+"! 2 e!E /)T ; Ox[ ]+" =

!Yox,+"

Mox

+ Z*

!Yox,+"

Mox"3 e

!E /)Tf ,1 at extinction


Nonpremixed flame extinction"

  For a stretched counterflow flame, for a reaction that is first order in fuel and O2, Liñán (1974) showed that the Damköhler number (Da) at extinction (which contains the stretch rate Σ) is given by

which has the same functional form as estimated on the previous slide (in particular the Z/Σ and e-β/β3 terms)

Daext ! e 1"#( )" 1"#( )2 + 0.26 1"#( )3 + 0.055 1"#( )4$%

&'

Da ( 8!ex f2 TfTox,")

*

+,,

-

.//

3YF,")

S 1+ S( )2!Mox

Z0e""

" 3;" ( Ea

RTf# (

S "1S +1

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15 AME 513 - Fall 2012 - Lecture 6 - Chemical Kinetics III

Final exam" December 17, 11:00 am – 1:00 pm, ZHS 159  Cumulative but primarily covering lectures 7 - 12  Open books / notes / calculators  Laptop computers may be used ONLY to view .pdf versions

of lecture notes – NOT .pptx versions  Note .pdf compilation of all lectures:

http://ronney.usc.edu/AME513F12/AME513-F12-AllLectures.pdf  GASEQ, Excel spreadsheets, CSU website, etc. NOT ALLOWED

  Homework #4 must be turned in by Friday 12/14 at 12:00 noon (NOT 4:30 pm!), solutions will be available at that time where you drop off homework


Midterm exam – topics covered"  Conservation equations

 Mass  Energy  Chemical species  Momentum

  Premixed flames  Rankine-Hugoniot relations  Detonations  Deflagrations

» Propagation rates » Flammability limits, instabilities, ignition

  Nonpremixed flames  Plane unstretched  Droplet  Counterflow   Jet  Extinction

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Midterm exam – types of problems"  Premixed flames (deflagrations and/or detonations)

  Flame temperature  Propagation rates   Ignition or extinction properties

  Nonpremixed flames – mixed is burned in first approximation   Flame temperature   Flame location   Jet flame length scaling  Extinction limit

  General - how would burning rate, flame length, extinction limit, etc. be affected by  Ronney Fuels, Inc. – new fuel or additive  Planet X – different atmosphere (pressure, temperature, etc.)

Documents

AME513-F12-lecture12 - University of Southern Californiaronney.usc.edu/AME513F12/Lecture12/AME513-F12-lecture12.pdfLow Re: depends Froude number (Fr = u e ... of lecture notes