1. 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
AME Fall Lecture 12 - Nonpremixed flames II

3. 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)
AME Fall Lecture 12 - Nonpremixed flames II

4. 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…
AME Fall Lecture 12 - Nonpremixed flames II

5. 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
AME Fall Lecture 12 - Nonpremixed flames II

6. Nonpremixed-gas flames - laminar gas-jet flames
AME Fall Lecture 12 - Nonpremixed flames II

7. 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 ~ de2(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 ~ de2(ue/u)2/Dox ~ de(ue2/(gLf)1)/Dox
Lf ~ (ue4de4/gDox2)1/3 – very different from momentum-controlled!
AME Fall Lecture 12 - Nonpremixed flames II

8. 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 ~ de2/(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)
AME Fall Lecture 12 - Nonpremixed flames II

9. 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)
uede2 ~ 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!
AME Fall Lecture 12 - Nonpremixed flames II

10. 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
AME Fall Lecture 12 - Nonpremixed flames II

11. AME 513 - Fall 2012 - Lecture 12 - Nonpremixed flames II
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
AME Fall Lecture 12 - Nonpremixed flames II

12. Flame lengths at 1g and µg
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
do = 3.3 mm, Re = 21 d = 0.42 mm, Re = 291
Sunderland et al. (1999) - C2H6/air
AME Fall Lecture 12 - Nonpremixed flames II

13. 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
AME Fall Lecture 12 - Nonpremixed flames II

14. 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)
AME Fall Lecture 12 - Nonpremixed flames II

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
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:
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
AME Fall Lecture 6 - Chemical Kinetics III

16. 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
AME Fall Lecture 6 - Chemical Kinetics III

17. 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 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 Fall Lecture 6 - Chemical Kinetics III