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AME 513 Principles of Combustion Lecture 11 Non-premixed flames I: 1D flames

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2 AME Fall Lecture 11 - Nonpremixed flames I Outline Flat flames Liquid droplets Stretched flames

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3 AME Fall Lecture 11 - Nonpremixed flames I “Non-premixed” or “diffusion” flames Inherently safer – no mixing of fuel and oxidant except at time/place combustion is desired Slower than premixed – need to mix AND burn, not just burn Simplest approach to determining properties: “mixed is burned” - chemical reaction rates faster than mixing rates No inherent propagation rate (unlike premixed flames where S L ~ [ ] 1/2 ) No inherent thickness (unlike premixed flames where thickness ~ /S L ) - 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, L I /u’ for turbulent flow, 1/ 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 ~ exp(-E/RT) is very sensitive to temperature (like premixed flames)

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4 AME Fall Lecture 11 - Nonpremixed flames I ≈ ( ) 1/2

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5 AME Fall Lecture 11 - Nonpremixed flames I Diesel engine combustion Two limiting cases Droplet combustion - vaporization of droplets is slow, so droplets burn as individuals Gas-jet flame - vaporization of droplets is so fast, there is effectively a jet of fuel vapor rather than individual droplets Reality 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

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6 AME Fall Lecture 11 - Nonpremixed flames I 1D planar steady nonpremixed flame 1D flame, convection from left to right, unknowns T f, x f u = const. (mass conservation); assume D & k/C P = const.

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7 AME Fall Lecture 11 - Nonpremixed flames I 1D planar steady nonpremixed flame Fuel, oxidizer mass fractions … but how to determine flame location x f ? Note S is the ratio of mass of oxidizer stream to mass of fuel stream needed to make a stoichiometric mixture of the two Also frequently used in analyses is the stoichiometric mixture fraction Z st = 1/(1+S) = mass fraction of fuel stream in a stoichiometric mixture of fuel and oxidant streams

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8 AME Fall Lecture 11 - Nonpremixed flames I 1D planar steady nonpremixed flame For reaction F Fuel + ox Ox products, ratio of fuel to oxidizer mass fluxes due to diffusion must be in stoichiometric ratio = F M F / ox M ox for (but opposite directions, hence - sign) at x = x f :

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9 AME Fall Lecture 11 - Nonpremixed flames I 1D planar steady nonpremixed flame Not solvable for x f in closed form but look at special cases… Special case #1: weak convection (Pe 0, exp(Pe) ≈ 1 + Pe, throw out terms of order Pe 2 ) Special case 2: Le F = Le ox = 1 Special case 3: Pe ∞

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10 AME Fall Lecture 11 - Nonpremixed flames I 1D planar steady nonpremixed flame Energy equation: Solutions

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11 AME Fall Lecture 11 - Nonpremixed flames I 1D planar steady nonpremixed flame Matching: heat release = (fuel flux to reaction zone) x (fuel heating value) = conductive heat flux away from reaction zone on both sides

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12 AME Fall Lecture 11 - Nonpremixed flames I 1D planar steady nonpremixed flame Can solve explicitly for T f if you’re desperate

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13 AME Fall Lecture 11 - Nonpremixed flames I 1D planar steady nonpremixed flame Special case 1: Pe 0 Dependence on Pe disappears (as expected) Behavior same on fuel and oxidant side except for stoichiometric scaling factor ox M ox / F M F (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

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14 AME Fall Lecture 11 - Nonpremixed flames I 1D planar steady nonpremixed flame Special case #2: Le F = Le ox = 1 When Le F = Le ox = 1, convection (contained in Pe = uL/ ) does not affect T f at all!

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15 AME Fall Lecture 11 - Nonpremixed flames I 1D planar steady nonpremixed flame Super special case 2a: Le F = Le ox = 1 AND T F,0 = T ox,0 = T ∞ : To interpret the Y F,0 /(…) term, consider stoichiometric mixture of fuel and oxidizer streams:

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16 AME Fall Lecture 11 - Nonpremixed flames I 1D planar steady nonpremixed flame Special case 3: Pe ∞ As Pe (convection effects) increase, effects of Le F & Le ox on flame temperatures decrease

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17 AME Fall Lecture 11 - Nonpremixed flames I 1D planar steady nonpremixed flame Much of our understanding of nonpremixed flames is contaminated by the facts that Le ox (O 2 in air) ≈ 1 We live in a concentrated fuel / diluted oxidizer world (S >> 1); we already showed that for Le ox ≈ 1, at high Pe, flame temperature is unaffected by Pe or Le F Consider low Pe: for CH 4 /air Similar trend for Pe -∞ (homework problem…)

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18 AME Fall Lecture 11 - Nonpremixed flames I Basic structure of nonpremixed flame The inevitable Excel spreadsheet … (Pe = 3, S = 1 shown)

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19 AME Fall Lecture 11 - Nonpremixed flames I Droplet combustion Heat from flame conducted to fuel surface, vaporizes fuel, fuel convects/diffuses to flame front, O 2 diffuses to flame front from outside, burning occurs at stoich. location As fuel burns, droplet diameter d d (t) decreases until d d = 0 or droplet may extinguish before reaching d d = 0 Experiments typically show d d (0) 2 - d d (t) 2 ≈ Kt

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20 AME Fall Lecture 11 - Nonpremixed flames I Droplet combustion Marchese et al. (1999), space experiments, heptane in O 2 -He

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21 AME Fall Lecture 11 - Nonpremixed flames I Droplet combustion Analysis similar to 1D planar flame with specified mass flux but need to use 1D steady spherical version of convection- diffusion conservation equations for Y f, Y ox and T

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22 AME Fall Lecture 11 - Nonpremixed flames I Droplet combustion Unknowns Flame temperature T f and flame location r f (as with flat flame) Fuel mass flux mdot = uA = d u d (4πr d 2 ) 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 = r f, but the total mass flow (i.e. of inert and products) must be constant out to r = ∞ Fuel concentration at droplet surface Y F,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)

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23 AME Fall Lecture 11 - Nonpremixed flames I Droplet combustion Fuel side (r d ≤ r ≤ r f ) Note similarities to planar case, but now due to r 2 factors in conservation equations we have exp(-Pe/r) terms instead of exp(-Pe*x) terms

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24 AME Fall Lecture 11 - Nonpremixed flames I Droplet combustion Oxygen side (r ≥ r f )

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25 AME Fall Lecture 11 - Nonpremixed flames I Droplet combustion Temperature (r d ≤ r ≤ r f ) Temperature (r ≥ r f )

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26 AME Fall Lecture 11 - Nonpremixed flames I Droplet combustion As 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 Y F,d are unknown) (and of course flame location r f is unknown) Special case: Le F = Le ox = 1

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27 AME Fall Lecture 11 - Nonpremixed flames I Droplet combustion As with flat flame, energy balance at flame sheet is Again looks similar to flat-flame case… Special case: Le F = 1

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28 AME Fall Lecture 11 - Nonpremixed flames I Droplet combustion New 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

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29 AME Fall Lecture 11 - Nonpremixed flames I Droplet combustion 4 equations for 4 unknowns:

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30 AME Fall Lecture 11 - Nonpremixed flames I Droplet combustion

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31 AME Fall Lecture 11 - Nonpremixed flames I Droplet combustion So finally we can calculate the mass burning rate (Pe) in terms of known properties

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32 AME Fall Lecture 11 - Nonpremixed flames I Droplet combustion Comments (8k/ d C P )ln(1+B) is called the burning rate constant – units length 2 /time k/ d C P is NOT the thermal diffusivity because d 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 (Q R ) appears only inside a ln( ), thus changing T f 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 surface In fact since you can’t change k, d or C P significantly in fuel/air combustion, only the droplet diameter affects burning time significantly (time ~ 1/d d 2 ) Flame temperature almost same as plane flame with adjusted enthalpy release Q R – L v vs. Q R Can also use this formula for mdot even if no combustion (just evaporation of a cold droplet in a hot atmosphere) – set Q R = 0 Nothing in expression for Pe, T f, r f or Y F,d depend on pressure

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33 AME Fall Lecture 11 - Nonpremixed flames I Droplet combustion What about flame radius r f ? d f /d d is constant and doesn’t even depend on transport properties, just thermodynamic properties! As expected, as Y ox,∞ decreases (more diluted oxidizer), flame moves farther out (less fuel flux) Also fuel mass fraction at droplet surface Y F,d Since usually Y F,d /S << 1 (see example), Y F,d ≈ B/(1+B) which is only slightly less than 1

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34 AME Fall Lecture 11 - Nonpremixed flames I Droplet combustion Comment on T and Y ox profiles for r ∞ This is identical to pure diffusion in spherical geometry: so diffusion dominates convection at large r

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35 AME Fall Lecture 11 - Nonpremixed flames I Droplet combustion Example for typical fuel (heptane, C 7 H 16 ) in air

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36 AME Fall Lecture 11 - Nonpremixed flames I Droplet combustion What if Le ≠ 1?

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37 AME Fall Lecture 11 - Nonpremixed flames I Droplet combustion

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38 AME Fall Lecture 11 - Nonpremixed flames I Droplet combustion Same as previous results when Le ox = 1 Le F doesn’t affect burning rate (Pe), r f or T f at all, only Y F,d ! For decreasing Le ox B’ (thus Pe) increases, but not much because of ln(1+B’) term r f decreases because of Le ox term; increasing B’ inside ln( ) term has less effect T f increases because of (1/Le) exponent

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39 AME Fall Lecture 11 - Nonpremixed flames I Droplet combustion The d 2 -law assumes no buoyant or forced convection, but in most applications there is likely to be significant flow; one relation for the effect of flow on burning rate is Re d = Droplet Reynolds number = ud(t)/ Re d = Droplet Reynolds number = ud(t)/ Nu = Nusselt number based on droplet diameter u = droplet velocity relative to gas Pr = Prandtl number = / = kinematic viscosity = kinematic viscosity = thermal diffusivity = k/ C p Reduces to the previous result for u = 0 (thus Re = 0)

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40 Fuel + inert Oxidant + inert x = 0 AME Fall Lecture 11 - Nonpremixed flames I 1D stretched flame Simple counterflow, fuel at x = +∞, oxidant at x = -∞, u = - x, again assume D & k/C P = constant Stagnation plane (u = 0) at x = 0, but flame may be on either side of x = 0 flame may be on either side of x = 0 depending on S, Le F & Le ox depending on S, Le F & Le ox Somewhat similar to plane unstretched case but this unstretched case but this configuration is easy to configuration is easy to obtain experimentally obtain experimentally Model for local behavior of flame in turbulent flow field flame in turbulent flow field (“laminar flamelet” model) (“laminar flamelet” model)

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41 AME Fall Lecture 11 - Nonpremixed flames I 1D stretched flame Species conservation:

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42 AME Fall Lecture 11 - Nonpremixed flames I 1D stretched flame Energy equation:

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43 AME Fall Lecture 11 - Nonpremixed flames I 1D stretched flame Stoichiometric balance condition at flame sheet is the same as always

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44 AME Fall Lecture 11 - Nonpremixed flames I 1D stretched flame Energy balance condition is the same as always

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45 AME Fall Lecture 11 - Nonpremixed flames I 1D stretched flame For S = 1, Le F = Le ox = 1, flame located at stagnation plane For S > 1 (oxidizer more diluted than fuel), flame moves toward oxidizer boundary – need steeper gradient of oxidizer S or Z st = 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 T f

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46 AME Fall Lecture 11 - Nonpremixed flames I 1D stretched flame Temperature & species profiles are error functions For S = 1, profiles are symmetric about x = 0; convection (u) is small & behavior similar to unstretched flame at low Pe, decreasing either Le increases T f For S > 1, flame lies on oxidizer side of stagnation plane; strong effect of convection - flame temperature is drastically affected by Le, decreasing Le F moves flame closer to x = 0 & increases T f but opposite trend for Le ox S = 15 S = 1

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