ame 513 principles of combustion

AME 513

Principles of Combustion

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

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

Outline Jet flames Simple models of nonpremixed flame extinction


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)


Axisymmetric jet – boundary layer flow

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

lengths…


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 ≈ Lf/de ~ Ude/ = Red

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


Nonpremixed-gas flames - laminar gas-jet flames


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 flamestC ~ Lf/u, tD ~ d2/Dox, u ~ (gLf)1/2, d ~ de(ue/u)1

Lf/u ~ d2/Dox, thus Lf/(gLf)1/2 ~ de2(ue/u)2/Dox ~ de(ue

2/(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 ~ udLf/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)


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/urj2

Stoichiometric coefficient to identify flame location


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


Flame lengths at 1g and µg

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

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

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

droplet flameHigh Re: results independent of Fr


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/b of Tf Reactant concentration in reaction zone is 1/b of ambient value Thickness of reaction zone is 1/b of transport zone thickness


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 S) is given by

which has the same functional form as estimated on the previous slide (in particular the Z/S and e-b/b3 terms)

15AME 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

http://ronney.usc.edu/AME513F12/AME513-F12-AllLectures.pdf




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


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.)

ame 513 principles of combustion

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