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

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Presentation on theme: "AME 513 Principles of Combustion Lecture 11 Non-premixed flames I: 1D flames."— Presentation transcript:

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

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

3 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)

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

5 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

6 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.

7 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

8 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 :

9 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  ∞

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

11 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

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

13 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

14 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!

15 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:

16 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

17 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…)

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

19 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

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

21 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

22 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)

23 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

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

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

26 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

27 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

28 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

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

30 30 AME Fall Lecture 11 - Nonpremixed flames I Droplet combustion

31 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

32 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

33 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

34 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

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

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

37 37 AME Fall Lecture 11 - Nonpremixed flames I Droplet combustion

38 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

39 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)

40 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)

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

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

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

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

45 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

46 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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