AME 513 - Principles of Combustion

AME 513

Principles of Combustion

Lecture 11Non-premixed flames I: 1D flames#1AME 513 - Fall 2012 - Lecture 11 - Nonpremixed flames IOutlineFlat flamesLiquid dropletsStretched flames#2AME 513 - Fall 2012 - Lecture 11 - Nonpremixed flames INon-premixed or diffusion flamesInherently safer no mixing of fuel and oxidant except at time/place combustion is desiredSlower than premixed need to mix AND burn, not just burnSimplest approach to determining properties: mixed is burned - chemical reaction rates faster than mixing rates No inherent propagation rate (unlike premixed flames where SL ~ [w]1/2)No inherent thickness (unlike premixed flames where thickness ~ /SL) - in nonpremixed flames, determined by equating convection time scale = /u = to diffusion time scale 2/ ~ ()1/2 where is a characteristic flow time scale (e.g. d/u for a jet, where d = diameter, u = velocity, LI/u for turbulent flow, 1/S for a counterflow etc.)Burning must occur near stoichiometric contour where reactant fluxes are in stoichiometric proportions (otherwise surplus of one reactant)Burning still must occur near highest T since w ~ exp(-E/RT) is very sensitive to temperature (like premixed flames)#3AME 513 - Fall 2012 - Lecture 11 - Nonpremixed flames I

()1/2#4AME 513 - Fall 2012 - Lecture 11 - Nonpremixed flames IDiesel engine combustionTwo limiting casesDroplet combustion - vaporization of droplets is slow, so droplets burn as individualsGas-jet flame - vaporization of droplets is so fast, there is effectively a jet of fuel vapor rather than individual dropletsReality is in between, but in Diesels usually closer to the gas jet with extras regions of premixed combustion

Flynn, P.F, R.P. Durrett, G.L. Hunter, A.O. zur Loye, O.C. Akinyemi, J.E. Dec, C.K. Westbrook, SAE Paper No. 1999-01-0509.

#5AME 513 - Fall 2012 - Lecture 11 - Nonpremixed flames I1D planar steady nonpremixed flame1D flame, convection from left to right, unknowns Tf, xfru = const. (mass conservation); assume rD & k/CP = const.

#6AME 513 - Fall 2012 - Lecture 11 - Nonpremixed flames I1D planar steady nonpremixed flameFuel, oxidizer mass fractions

but how to determine flame location xf?Note S is the ratio of mass of oxidizer stream to mass of fuel stream needed to make a stoichiometric mixture of the twoAlso frequently used in analyses is the stoichiometric mixture fraction Zst = 1/(1+S) = mass fraction of fuel stream in a stoichiometric mixture of fuel and oxidant streams

#7AME 513 - Fall 2012 - Lecture 11 - Nonpremixed flames I1D planar steady nonpremixed flameFor reaction nFFuel + noxOx products, ratio of fuel to oxidizer mass fluxes due to diffusion must be in stoichiometric ratio = nFMF/noxMox for (but opposite directions, hence - sign) at x = xf:

#8AME 513 - Fall 2012 - Lecture 11 - Nonpremixed flames I1D planar steady nonpremixed flameNot solvable for xf in closed form but look at special casesSpecial case #1: weak convection (Pe 0, exp(Pe) 1 + Pe, throw out terms of order Pe2)

Special case 2: LeF = Leox= 1

Special case 3: Pe

#9AME 513 - Fall 2012 - Lecture 11 - Nonpremixed flames I1D planar steady nonpremixed flameEnergy equation:

Solutions

#10AME 513 - Fall 2012 - Lecture 11 - Nonpremixed flames I1D planar steady nonpremixed flameMatching: heat release = (fuel flux to reaction zone) x (fuel heating value) = conductive heat flux away from reaction zone on both sides

#11AME 513 - Fall 2012 - Lecture 11 - Nonpremixed flames I1D planar steady nonpremixed flameCan solve explicitly for Tf if youre desperate

#12AME 513 - Fall 2012 - Lecture 11 - Nonpremixed flames I1D planar steady nonpremixed flameSpecial case 1: Pe 0

Dependence on Pe disappears (as expected)Behavior same on fuel and oxidant side except for stoichiometric scaling factor noxMox/nFMF (also expected)Decreasing Le has same effect as increasing reactant concentration (!) completely unlike premixed flame where planar steady adiabatic flame temperature is independent of Le

#13AME 513 - Fall 2012 - Lecture 11 - Nonpremixed flames I1D planar steady nonpremixed flameSpecial case #2: LeF = Leox = 1

When LeF = Leox = 1, convection (contained in Pe = uL/a) does not affect Tf at all!

#14AME 513 - Fall 2012 - Lecture 11 - Nonpremixed flames I1D planar steady nonpremixed flameSuper special case 2a: LeF = Leox = 1 AND TF,0 = Tox,0 = T:

To interpret the YF,0/() term, consider stoichiometric mixture of fuel and oxidizer streams:

#15AME 513 - Fall 2012 - Lecture 11 - Nonpremixed flames I1D planar steady nonpremixed flameSpecial case 3: Pe

As Pe (convection effects) increase, effects of LeF & Leox on flame temperatures decrease

#16AME 513 - Fall 2012 - Lecture 11 - Nonpremixed flames I1D planar steady nonpremixed flameMuch of our understanding of nonpremixed flames is contaminated by the facts thatLeox (O2 in air) 1We live in a concentrated fuel / diluted oxidizer world (S >> 1); we already showed that for Leox 1, at high Pe, flame temperature is unaffected by Pe or LeFConsider low Pe: for CH4/air

Similar trend for Pe - (homework problem)

#17AME 513 - Fall 2012 - Lecture 11 - Nonpremixed flames IBasic structure of nonpremixed flameThe inevitable Excel spreadsheet (Pe = 3, S = 1 shown)

#18AME 513 - Fall 2012 - Lecture 11 - Nonpremixed flames IDroplet combustionHeat from flame conducted to fuel surface, vaporizes fuel, fuel convects/diffuses to flame front, O2 diffuses to flame front from outside, burning occurs at stoich. locationAs fuel burns, droplet diameter dd(t) decreases until dd = 0 or droplet may extinguish before reaching dd = 0Experiments typically show dd(0)2 - dd(t)2 Kt

#19AME 513 - Fall 2012 - Lecture 11 - Nonpremixed flames IDroplet combustionMarchese et al. (1999), space experiments, heptane in O2-He

#20AME 513 - Fall 2012 - Lecture 11 - Nonpremixed flames IDroplet combustionAnalysis similar to 1D planar flame with specified mass flux but need to use 1D steady spherical version of convection-diffusion conservation equations for Yf, Yox and T

#21AME 513 - Fall 2012 - Lecture 11 - Nonpremixed flames IDroplet combustionUnknowns Flame temperature Tf and flame location rf (as with flat flame) Fuel mass flux mdot = ruA = rdud(4rd2) from droplet surface (expressed in Pe in the following analysis) (new)Note that mdot must be constant, but the fuel mass flow is not; the fuel disappears by r = rf, but the total mass flow (i.e. of inert and products) must be constant out to r = Fuel concentration at droplet surface YF,d or stoichiometric parameter S (new)2 more unknowns, so need 2 more equations (total of 4)Reactant diffusive fluxes into flame sheet in stoichiometric proportions (as with flat flame)Fuel enthalpy flux into flame sheet = thermal enthalpy flux out (by heat conduction) (as with flat flame)Energy balance at droplet surface (new)Mass balance at droplet surface (new)#22AME 513 - Fall 2012 - Lecture 11 - Nonpremixed flames IDroplet combustionFuel side (rd r rf)

Note similarities to planar case, but now due to r2 factors in conservation equations we have exp(-Pe/r) terms instead of exp(-Pe*x) terms

#23AME 513 - Fall 2012 - Lecture 11 - Nonpremixed flames IDroplet combustionOxygen side (r rf)

#24AME 513 - Fall 2012 - Lecture 11 - Nonpremixed flames IDroplet combustionTemperature (rd r rf)

Temperature (r rf)

#25AME 513 - Fall 2012 - Lecture 11 - Nonpremixed flames IDroplet combustionAs with flat flame, stoichiometric balance at flame sheet is

Looks very similar to flat-flame case but again note 1/r terms vs. x in flat-flame case, plus Pe and S are unknowns (since mass flux and YF,d are unknown) (and of course flame location rf is unknown)Special case: LeF = Leox = 1

#26AME 513 - Fall 2012 - Lecture 11 - Nonpremixed flames IDroplet combustionAs with flat flame, energy balance at flame sheet is

Again looks similar to flat-flame caseSpecial case: LeF = 1

#27AME 513 - Fall 2012 - Lecture 11 - Nonpremixed flames IDroplet combustionNew constraint #1 - conductive heat flux to droplet surface = enthalpy needed to vaporize the mass flux of fuel

New constraint #2 - mass balance at droplet surface: mass flow from droplet into gas (fuel only) = rate of fuel convected into gas + rate of fuel diffused into gas

#28AME 513 - Fall 2012 - Lecture 11 - Nonpremixed flames IDroplet combustion4 equations for 4 unknowns:

#29AME 513 - Fall 2012 - Lecture 11 - Nonpremixed flames IDroplet combustion

#30AME 513 - Fall 2012 - Lecture 11 - Nonpremixed flames IDroplet combustionSo finally we can calculate the mass burning rate (Pe) in terms of known properties

#31AME 513 - Fall 2012 - Lecture 11 - Nonpremixed flames IDroplet combustionComments(8k/rdCP)ln(1+B) is called the burning rate constant units length2/timek/rdCP is NOT the thermal diffusivity because rd is the droplet density, not gas density!B is called the Transfer Number ratio of enthalpy generated by combustion to enthalpy need to vaporize fuel; typical values for hydrocarbons 10, much lower for methanol ( 3)Enthalpy release (QR) appears only inside a ln( ), thus changing Tf hardly affects burning rate at all - why? The more rapidly fuel is vaporized, the more rapidly the fuel vapor blows out, thus the harder it is for heat to be conducted back to the fuel surfaceIn fact since you cant change k, rd or CP significantly in fuel/air combustion, only the droplet diameter affects burning time significantly (time ~ 1/dd2)Flame temperature almost same as plane flame with adjusted enthalpy release QR Lv vs. QRCan also use this formula for mdot even if no combustion (just evaporation of a cold droplet in a hot atmosphere) set QR = 0Nothing in expression for Pe, Tf, rf or YF,d depend on pressure#32AME 513 - Fall 2012 - Lecture 11 - Nonpremixed flames IDroplet combustionWhat about flame radius rf?

df/dd is constant and doesnt even depend on transport properties, just thermodynamic properties!As expected, as Yox, decreases (more diluted oxidizer), flame moves farther out (less fuel flux)Also fuel mass fraction at droplet surface YF,d

Since usually YF,d/S 1 (oxidizer more diluted than fuel), flame moves toward oxidizer boundary need steeper gradient of oxidizerS or Zst = 1/(1+S) has significant effect on flame behavior; for flame on oxidizer side, radicals (mostly formed on fuel side because of lower bond strengths of C-H & C-C compared to O=O) are convected away from flame sheet, so flames are weaker even for same Tf

#45AME 513 - Fall 2012 - Lecture 11 - Nonpremixed flames I1D stretched flameTemperature & species profiles are error functionsFor S = 1, profiles are symmetric about x = 0; convection (u) is small & behavior similar to unstretched flame at low Pe, decreasing either Le increases TfFor S > 1, flame lies on oxidizer side of stagnation plane; strong effect of convection - flame temperature is drastically affected by Le, decreasing LeF moves flame closer to x = 0 & increases Tf but opposite trend for Leox

S = 15S = 1#46

Flame temperature T = Tf

Reaction zone x = xf

u

Tox,L

Yox,L

YF,0

TF,0

Temperature (T)

Oxidizer (O)

Fuel (F)

T

Oxidizer boundary x = L

Fuel boundary x = 0

dd = 2rd

df = 2rf

Td

Temperature

[Oxygen]

[Fuel]

Droplet surface

Flame front

T

Tf