AME 513 Principles of Combustion Lecture 12 Non-premixed flames II: 2D flames, extinction

AME 513 - Fall 2012 - Lecture 12 - Nonpremixed flames II Outline Jet flames Simple models of nonpremixed flame extinction AME 513 - Fall 2012 - Lecture 12 - Nonpremixed flames II

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 513 - Fall 2012 - Lecture 12 - Nonpremixed flames II

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 513 - Fall 2012 - Lecture 12 - Nonpremixed flames II

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 513 - Fall 2012 - Lecture 12 - Nonpremixed flames II

Nonpremixed-gas flames - laminar gas-jet flames AME 513 - Fall 2012 - Lecture 12 - Nonpremixed flames II

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 513 - Fall 2012 - Lecture 12 - Nonpremixed flames II

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 513 - Fall 2012 - Lecture 12 - Nonpremixed flames II

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 513 - Fall 2012 - Lecture 12 - Nonpremixed flames II

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 513 - Fall 2012 - Lecture 12 - Nonpremixed flames II

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 513 - Fall 2012 - Lecture 12 - Nonpremixed flames II

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 513 - Fall 2012 - Lecture 12 - Nonpremixed flames II

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 513 - Fall 2012 - Lecture 12 - Nonpremixed flames II

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 513 - Fall 2012 - Lecture 12 - Nonpremixed flames II

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

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 513 - Fall 2012 - Lecture 6 - Chemical Kinetics III

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 513 - Fall 2012 - Lecture 6 - Chemical Kinetics III