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AME 513 Principles of Combustion. Lecture 11 Non-premixed flames I: 1D flames. Outline. Flat flames Liquid droplets Stretched flames. “ Non-premixed ” or “ diffusion ” flames. Inherently safer – no mixing of fuel and oxidant except at time/place combustion is desired
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AME 513Principles of Combustion Lecture 11 Non-premixed flames I: 1D flames
Outline • Flat flames • Liquid droplets • Stretched flames AME 513 - Fall 2012 - 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 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) AME 513 - Fall 2012 - Lecture 11 - Nonpremixed flames I
≈ ()1/2 AME 513 - Fall 2012 - 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. 1999-01-0509. AME 513 - Fall 2012 - Lecture 11 - Nonpremixed flames I
1D planar steady nonpremixed flame • 1D flame, convection from left to right, unknowns Tf, xf • ru = const. (mass conservation); assume rD & k/CP = const. AME 513 - Fall 2012 - Lecture 11 - Nonpremixed flames I
1D planar steady nonpremixed flame • Fuel, 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 two • Also 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 AME 513 - Fall 2012 - Lecture 11 - Nonpremixed flames I
1D planar steady nonpremixed flame • For 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: AME 513 - Fall 2012 - Lecture 11 - Nonpremixed flames I
1D planar steady nonpremixed flame • Not solvable for xfin closed form but look at special cases… • Special 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 ∞ AME 513 - Fall 2012 - Lecture 11 - Nonpremixed flames I
1D planar steady nonpremixed flame • Energy equation: • Solutions AME 513 - Fall 2012 - 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 AME 513 - Fall 2012 - Lecture 11 - Nonpremixed flames I
1D planar steady nonpremixed flame • Can solve explicitly for Tf if you’re desperate AME 513 - Fall 2012 - 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 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 AME 513 - Fall 2012 - Lecture 11 - Nonpremixed flames I
1D planar steady nonpremixed flame • Special case #2: LeF = Leox = 1 • When LeF = Leox = 1, convection (contained in Pe = uL/a) does not affect Tf at all! AME 513 - Fall 2012 - Lecture 11 - Nonpremixed flames I
1D planar steady nonpremixed flame • Super 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: AME 513 - Fall 2012 - Lecture 11 - Nonpremixed flames I
1D planar steady nonpremixed flame • Special case 3: Pe ∞ • As Pe (convection effects) increase, effects of LeF & Leox on flame temperatures decrease AME 513 - Fall 2012 - Lecture 11 - Nonpremixed flames I
1D planar steady nonpremixed flame • Much of our understanding of nonpremixed flames is contaminated by the facts that • Leox (O2 in air)≈ 1 • We 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 LeF • Consider low Pe: for CH4/air • Similar trend for Pe -∞ (homework problem…) AME 513 - Fall 2012 - Lecture 11 - Nonpremixed flames I
Basic structure of nonpremixed flame • The inevitable Excel spreadsheet … (Pe = 3, S = 1 shown) AME 513 - Fall 2012 - Lecture 11 - Nonpremixed flames I
Droplet combustion • Heat 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. location • As fuel burns, droplet diameter dd(t) decreases until dd= 0 or droplet may extinguish before reaching dd= 0 • Experiments typically show dd(0)2 - dd(t)2 ≈ Kt AME 513 - Fall 2012 - Lecture 11 - Nonpremixed flames I
Droplet combustion Marchese et al. (1999), space experiments, heptane in O2-He AME 513 - Fall 2012 - 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 Yf, Yox and T AME 513 - Fall 2012 - Lecture 11 - Nonpremixed flames I
Droplet combustion • Unknowns • Flame temperature Tf and flame location rf (as with flat flame) • Fuel mass flux mdot = ruA = rdud(4πrd2) 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) AME 513 - Fall 2012 - Lecture 11 - Nonpremixed flames I
Droplet combustion • Fuel 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 AME 513 - Fall 2012 - Lecture 11 - Nonpremixed flames I
Droplet combustion • Oxygen side (r ≥rf) AME 513 - Fall 2012 - Lecture 11 - Nonpremixed flames I
Droplet combustion • Temperature (rd ≤ r ≤ rf) • Temperature (r ≥rf) AME 513 - Fall 2012 - 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 YF,d are unknown) (and of course flame location rf is unknown) • Special case: LeF = Leox = 1 AME 513 - Fall 2012 - 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: LeF = 1 AME 513 - Fall 2012 - 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 AME 513 - Fall 2012 - Lecture 11 - Nonpremixed flames I
Droplet combustion • 4 equations for 4 unknowns: AME 513 - Fall 2012 - Lecture 11 - Nonpremixed flames I
Droplet combustion AME 513 - Fall 2012 - Lecture 11 - Nonpremixed flames I
Droplet combustion • So finally we can calculate the mass burning rate (Pe) in terms of known properties AME 513 - Fall 2012 - Lecture 11 - Nonpremixed flames I
Droplet combustion • Comments • (8k/rdCP)ln(1+B) is called the burning rate constant – units length2/time • k/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 Tfhardly 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, rdor 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. QR • Can also use this formula for mdot even if no combustion (just evaporation of a cold droplet in a hot atmosphere) – set QR = 0 • Nothing in expression for Pe, Tf, rf or YF,d depend on pressure AME 513 - Fall 2012 - Lecture 11 - Nonpremixed flames I
Droplet combustion • What about flame radius rf? • df/dd is constant and doesn’t 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 (see example), YF,d ≈ B/(1+B) which is only slightly less than 1 AME 513 - Fall 2012 - Lecture 11 - Nonpremixed flames I
Droplet combustion • Comment on T and Yox profiles for r ∞ • This is identical to pure diffusion in spherical geometry: so diffusion dominates convection at large r AME 513 - Fall 2012 - Lecture 11 - Nonpremixed flames I
Droplet combustion • Example for typical fuel (heptane, C7H16) in air AME 513 - Fall 2012 - Lecture 11 - Nonpremixed flames I
Droplet combustion • What if Le ≠ 1? AME 513 - Fall 2012 - Lecture 11 - Nonpremixed flames I
Droplet combustion AME 513 - Fall 2012 - Lecture 11 - Nonpremixed flames I
Droplet combustion • Same as previous results when Leox = 1 • LeFdoesn’t affect burning rate (Pe), rf or Tf at all, only YF,d! • For decreasing Leox • B’ (thus Pe) increases, but not much because of ln(1+B’) term • rf decreases because of Leox term; increasing B’ inside ln( ) term has less effect • Tf increases because of (1/Le) exponent AME 513 - Fall 2012 - Lecture 11 - Nonpremixed flames I
Droplet combustion • The d2-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 Red = Droplet Reynolds number = ud(t)/ Nu = Nusselt number based on droplet diameter u= droplet velocity relative to gas Pr = Prandtl number = / = kinematic viscosity a = thermal diffusivity = k/Cp • Reduces to the previous result for u = 0 (thus Re = 0) AME 513 - Fall 2012 - Lecture 11 - Nonpremixed flames I
1D stretched flame • Simple counterflow, fuel at x = +∞, oxidant at x = -∞, u = -Sx, again assume rD & k/CP = constant • Stagnation plane (u = 0) at x = 0, but flame may be on either side of x = 0 depending on S, LeF & Leox • Somewhat similar to plane unstretched case but this configuration is easy to obtain experimentally • Model for local behavior of flame in turbulent flow field (“laminar flamelet” model) Fuel + inert x = 0 Oxidant + inert AME 513 - Fall 2012 - Lecture 11 - Nonpremixed flames I
1D stretched flame • Species conservation: AME 513 - Fall 2012 - Lecture 11 - Nonpremixed flames I
1D stretched flame • Energy equation: AME 513 - Fall 2012 - Lecture 11 - Nonpremixed flames I
1D stretched flame • Stoichiometric balance condition at flame sheet is the same as always AME 513 - Fall 2012 - Lecture 11 - Nonpremixed flames I
1D stretched flame • Energy balance condition is the same as always AME 513 - Fall 2012 - Lecture 11 - Nonpremixed flames I
1D stretched flame • For S = 1, LeF = Leox = 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 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 AME 513 - Fall 2012 - 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 Tf • For 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 Tfbut opposite trend for Leox S = 1 S = 15 AME 513 - Fall 2012 - Lecture 11 - Nonpremixed flames I