1. AME 514 Applications of Combustion
Paul D. Ronney
Spring 2015

2. AME 514 - Basic information
Instructor: Paul Ronney
Office: OHE 430J; Phone: (213) ; Fax: (213)
Office hours: 9:00 am – 12:00 pm Thursdays, other times by
appointment
Website:
Schedule: 1 lecture per week, Tuesdays 6:40 - 9:20 pm, RTH 109
Lectures: On campus, also webcast through the USC Distance Education Network
Credit: 3 units
Prerequisite: AME 513 or equivalent or permission of instructor
Textbook: none required, but a good general text on combustion is S. R. Turns, "An Introduction to Combustion"
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3. AME 514 - Basic information
Grading: 5 homework assignments,1 for each section of the course (60%), final exam (40%)
Each homework will consist of
(1) report on a seminal paper in the field chosen from a list provided by PDR (others OK with approval in advance from me)
(2) usual analytical / numerical problems
Final exam will consist of 6 problems (1 per section, plus one "anything goes"), choose 4/6
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4. Helpful handy hints
I'll hand out printed copies of lectures so you can annotate them, but for best results, download and use PowerPoint files (includes color, movies, hyperlinks, embedded spreadsheets, etc.)
If you don't have PowerPoint, you can download a free PowerPoint viewer from Microsoft's website (but then you won't be able to use the embedded spreadsheets, etc.)
Please ask questions in class - the goal of the lecture is to maintain a 2-way dialogue on the subject of the lecture
Bringing your laptop allows you to download files from my website as necessary and play along in the studio audience
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5. Tentative outline 1) Advanced fundamental topics (3 lectures)
Flammability and extinction
Ignition
Emissions formation and remediation
2) Microscale reacting flows and power generation (3 lectures)
i) Scaling considerations
ii) Microscale internal combustion engines
iii) Microscale gas turbine and rocket propulsion
iv) Thermoelectrics
v) Fuel cells
3) Turbulent combustion (3 lectures)
i) Premixed-gas flames
ii) Non premixed flames
iii) Edge flames
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6. Tentative outline 4) Advanced propulsion systems (3 lectures)
i) Hypersonic propulsion
ii) Pulse detonation engines
5) Emerging needs & technologies (3 lectures)
i) Applications of combustion (aka "chemically reacting flow") knowledge to other fields
Frontal polymerization
Bacteria growth
Inertial confinement fusion
Astrophysical combustion
ii) New technologies
Transient plasma ignition
HCCI engines
Microbial fuel cells
iii) Future needs in combustion research
Optional after-class "field trips" to combustion labs (Egolfopoulos, Ronney)
Other topics (for example, optical diagnostics) may be substituted by request of a majority of registered students
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7. Alternative topics 1) Microgravity combustion (3 lectures)
i) Premixed-gas flames
ii) Particle-laden flames
iii) Droplets
iv) Flame spread over solid fuel beds
2) Optical diagnostics (3 lectures)
Quantum physics of gases
Absorption / transmission techniques (absorption spectroscopy, shadowgraphy, schlieren, interferometry)
Scattering techniques (Rayleigh, Raman, Mie, LDV)
Fluorescence techniques
3) Computational methods in combustion (3 lectures)
Governing equations
Numerical methods
Applications
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8. Assignment
By Friday 1/16
(Optional) me your schedule to me so I can choose office hours (default: 12:30 - 3:30 Weds.)
(Optional) send suggestions to me for other lecture topics and what could be deleted (if I don't hear from you I assume you approve the currently proposed syllabus)
If you want to add a unit, you must state what unit should be removed
(Optional) review material on premixed flames
Turns Chapters 8 & 15
Egolfopoulos's AME 513 notes
My AME 513 notes ( lectures 8 & 9)
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9. Advanced fundamental topics (3 lectures)
Why study combustion? (0.1 lectures)
Quick review of AME 513 concepts (0.2 lectures)
Flammability & extinction limits
(1.2 lectures)
Ignition (0.5 lectures)
Emissions formation & remediation
(1 lecture)

10. Why study combustion?
> 80% of world energy production results from combustion of fossil fuels
Energy sector accounts for 9% of US Gross Domestic Product
Our continuing habit of burning things and our quest to find more things to burn has resulted in
Economic booms and busts
Political and military conflicts
Global warming (or the need to deny its existence)
Human health issues
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11. US energy flow, 2010, units 1015 BTU/yr
Each 1015 BTU/yr = 33.4 gigawatts
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12. What do we do with combustion?
Power generation (coal, natural gas)
Transportation (land, air, sea vehicles)
Weapons (rapid production of high-pressure gas)
Heating
Lighting
Cooking (1/3 of the world’s population still uses biomass-fueled open fires)
Hazardous waste & chemical warfare agent destruction
Production of new materials, e.g. nano-materials
(Future?) Portable power, e.g. battery replacement
Unintended / undesired consequences
Fires and explosions (residential, urban, wildland, industrial)
Pollutants – NOx (brown skies, acid rain), CO (poisonous), Unburned HydroCarbons (UHCs, catalyzes production of photochemical smog), formaldehyde, particulates, SOx
Global warming from CO2 & other products
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13. What do we want to know? From combustion device
Power (thermal, electrical, shaft, propulsive)
Efficiency (% fuel burned, % fuel converted to power)
Emissions
From combustion process itself
Rates of consumption
Reactants
Intermediates
Rates of formation
Products
Global properties
Rates of flame propagation
Rates of heat generation (more precisely, rate of conversion of chemical enthalpy to thermal enthalpy)
Temperatures
Pressures
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14. Why do we need to study combustion?
Chemical thermodynamics only tells us the end states - what happens if we wait “forever and a day” for chemical reaction to occur
We need to know how fast reactions occur
How fast depends on both the inherent rates of reaction and the rates of heat and mass transport to the reaction zone(s)
Chemical reactions + heat & mass transport = combustion
Some reactions occur too slowly to be observed, e.g.
2 NO  N2 + O2
has an adiabatic flame temperature of 2650K but no one has ever made a flame with NO because reaction rates are too slow!
Chemical reaction leads to gradients in temperature, pressure and species concentration
Results in transport of energy, momentum, mass
Combustion is the study of the coupling between thermodynamics, chemical reaction and transport processes
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15. Types of combustion
Premixed - reactants are intimately mixed on the molecular scale before combustion is initiated; several flavors
Deflagration
Detonation
Homogeneous reaction
Nonpremixed - reactants mix only at the time of combustion - have to mix first then burn; several flavors
Gas jet (Bic lighter)
Liquid fuel droplet
Liquid fuel jet (e.g. candle, Diesel engine)
Solid (e.g. coal particle, wood)
Type
Chemical reaction
Heat / mass transport
Momentum transport
Thermo-dynamics
Deflagration ✔ ✗
Detonation
Homogeneous reaction
Nonpremixed flames
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16. Deflagrations
Subsonic propagating front sustained by conduction of heat from the hot (burned) gases to the cold (unburned) gases which raises the temperature enough that chemical reaction can occur; since chemical reaction rates are very sensitive to temperature, most of the reaction is concentrated in a thin zone near the high-temperature side
May be laminar or turbulent
Temperature increases in “convection-diffusion zone” or “preheat zone” ahead of reaction zone, even though no heat release occurs there, due to balance between convection & diffusion
Reactant concentration decreases in convection-diffusion zone, even though no chemical reaction occurs there, for same reason
How can we have reaction at the reaction zone even though reactant concentration is low there? (See diagram…) Because reaction rate is much more sensitive to temperature than reactant concentration, so benefit of high T outweighs penalty of low concentration
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17. Schematic of deflagration
Turbulent premixed flame experiment in a fan-stirred chamber (
Flame thickness () ~ /SL
( = thermal diffusivity)
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18. Structure of deflagration
Outside of the thin reaction zone, only convection and diffusion of enthalpy are present, thus energy conservation can be written as, for 1D steady flow from right to left (in -x direction, as in diagram on previous page)
with boundary conditions
T = Tf at x = 0 (flame front)
T  T∞ as x  ∞ (far upstream of flame)
T  Tf as x  -∞ (far downstream of flame)
noting that due to mass conservation
U = ∞SL = constant
and assuming k and CP are constant,
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19. Structure of deflagration
Thus, the temperature profile is an exponential with decay length = flame thickness /SL
Reactant concentration profile is essentially a mirror image of the temperature profile, at least for Lewis number  /D = 1
D = reactant diffusivity
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20. Premixed flames - detonation
Supersonic front sustained by heating of gas by shock wave
After shock front, need time (thus distance = time x velocity) before reaction starts to occur ("induction zone")
After induction zone, chemical reaction & heat release occur
Pressure & temperature behavior coupled strongly with supersonic/subsonic gasdynamics
Ideally only M3 = 1 "Chapman-Jouget detonation" is stable
(M = Mach number = Vc; V = velocity, c = sound speed = (RT)1/2 for ideal gas)
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21. Premixed flames - homogeneous reaction
Model for knock in premixed-charge engines
Fixed mass (control mass) with uniform (in space) T, P and composition
No "propagation" in space but propagation in time
In laboratory, we might heat the chamber to a certain T and see how long it took to react; in engine, compression of mixture (increases P & T, thus reaction rate) will initiate reaction
Fuel + O2
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22. "Non-premixed" or "diffusion" flames
Reactants mix at the time of combustion - mix then burn - only subsonic
Many types - gas jet (Bic lighter), droplet, liquid fuel (e.g. Kuwait oil fire, candle), solid (e.g. coal particle, wood)
Reaction zone must lie where fuel & O2 fluxes in stoichiometric proportion
Generally assume "mixed is burned" - mixing slower than chemical reaction
No inherent propagation rate (flame location determined by stoich. location) or thickness ( depends on mixing layer thickness ~ (/)1/2) ( = strain rate) - unlike premixed flames with characteristic propagation rate SL and thickness  ~ /SL that are almost independent of 
Candle
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23. "Non-premixed" or "diffusion" flames
Candle
Kuwait
Oil fire
Forest fire
Diesel engine
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24. AME Spring Lecture 1

25. Diesel engine combustion
Two extremes
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
P. F. Flynn, 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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26. Temperatures of non-premixed flames
Adiabatic temperature of premixed flame, simplest approximation (const. CP, no dissociation, complete combustion, const. pressure):
Tf = T∞ + YF,0QR/CP
(YF,0 = fuel mass fraction far from front, QR = fuel heating value)
Non-premixed not as simple, depends on transport of reactants to front & heat/products away from front
Simplest approximation: diffusion dominated, no convection
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27. Temperatures of non-premixed flames
Mass flux (per unit cross-section area of flame) of fuel to flame front = DF(∂YF/∂x) = DF(0 - YF,0)/(xf - 0) = DFYF,0/xf
Heat generation rate per unit area of flame front = QRDFYF,0/xf
Mass flux of O2 to flame front = DoxYox,/( - xf)
Heat conducted away from flame front per unit area
= k(∂T/∂x)left + k(∂T/∂x)right = k(Tf - TF,0)/xf + k(Tf - Tox,)/( - xf)
Unknowns Tf & xf
Equations
Heat generation rate = heat conduction rate away from front
QRDFYF,0/xf = k(Tf - TF,0) + k(Tf - Tox,)
Mass flux of O2 / fuel = stoichiometric O2 / fuel mass ratio = 
DoxYox,/( - xf) / DFYF,0/xf = 
Combine to obtain
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28. Temperatures of non-premixed flames
Implications - temperature
Increasing either YF,0 or Yox, increases flame temperature Tf
Increasing TF,0 or Tox, increases Tf
Decreasing LeF or Leox increases Tf
Above results very different from premixed flames
LeF & Leox don't affect adiabatic Tf
Only increasing Y of stoichiometrically deficient reactant increases Tf - increasing Y of other reactant decreases Tf
If TF,0 = Tox, = T∞ AND LeF = Leox = 1, then
Tf = T∞ + fstoichQR/CP
where fstoich = YF,0/(1+) is the mass fraction of fuel in a stoichiometric mixture of fuel + inert (fuel mass fraction YF,0) and oxygen + inert (O2 mass fraction Yox,)
Very much unlike premixed flames, where Tf is essentially independent of LeF & Leox, and only depends on Y of stoichiometrically deficient reactant
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29. Temperatures of non-premixed flames
Implications - flame position
Increasing YF,0 or decreasing LeF moves flame AWAY from fuel source
Increasing Yox, or decreasing Leox moves flame AWAY from ox. source
Since Yox,/YF,0 << 1 for fuel-air mixtures (≈ for CH4-air), flame lies very close to air side
Since Yox,/YF,0 << 1, Leox affects Tf much more than LeF), but since Leox ≈ 1 for O2 in N2, Tf is hardly affected by fuel type even though LeF varies greatly between fuels
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30. Law of Mass Action (LoMA)
First we need to describe rates of chemical reaction
For a chemical reaction of the form
AA + BB  CC + DD
e.g. 1 H2 + 1 I2  2 HI
A = H2, A = 1, B = I2, B = 1, C = HI, C = 2, D = nothing, D = 0
the Law of Mass Action (LoMA) states that the rate of reaction
[ i ] = concentration of molecule i (usually moles per liter)
kf = "forward" reaction rate constant
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31. Law of Mass Action (LoMA)
How to calculate [ i ]?
According to ideal gas law, the total moles of gas per unit volume (all molecules, not just type i) = P/T
Then [ i ] = (Total moles / volume)*(moles i / total moles), thus
[ i ] = (P/T)Xi (Xi = mole fraction of i)
Minus sign on d[A]/dt and d[B]/dt since A & B are being depleted
Basically LoMA states that the rate of reaction is proportional to the number of collisions between the reactant molecules, which in turn is proportional to the concentration of each reactant
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32. Comments on LoMA AME 514 - Spring 2015 - Lecture 1
The reaction rate constant kf is usually of the Arrhenius form
Z = pre-exponential factor, n = another (nameless) constant, E = "activation energy" (cal/mole);  = gas constant; working backwards, units of Z must be (moles per liter)1-A-vB/(K-nsec)
With 3 parameters (Z, n, E) any curve can be fit!
The exponential term causes extreme sensitivity to T for E/ >> T!
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33. Comments on LoMA "Diary of a collision"
Boltzman (1800's): fraction of molecules in a gas with translational kinetic energy greater than E is proportional to exp(-E/T), thus E represents the "energy barrier" that must be overcome for reaction to occur
E has no relation to enthalpy of reaction hf (or heating value QR); E affects reaction rates whereas hf & QR affect end states (e.g. Tad), though hf & QR affect reaction rates indirectly by affecting T
"Diary of a collision"
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34. Comments on LoMA The full reaction rate expression is then
H2 + I2  2HI is one of few examples where the actual conversion of reactants to products occurs in a single step; most fuels of interest go through many intermediates during oxidation; even for the simplest hydrocarbon (CH4) the "standard" mechanism ( includes 53 species and 325 individual reactions!
The only likely reactions in gases, where the molecules are far apart compared to their size, are 1-body, 2-body or 3-body reactions, i.e. A  products, A + B  products or A + B + C  products
In liquid or solid phases, the close proximity of molecules makes n-body reactions plausible
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35. Comments on LoMA AME 514 - Spring 2015 - Lecture 1
Recall that the forward reaction rate is
Similarly, the rate of the reverse reaction can be written as
kb = "backward" reaction rate constant
At equilibrium, the forward and reverse rates must be equal, thus
This ties reaction rate constants (kf, kb) and equilibrium constants (Ki's) together
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36. Deflagrations - burning velocity
Since the burning velocity (SL) << sound speed, the pressure across the front is almost constant
How fast will the flame propagate? Simplest estimate based on the hypothesis that
Rate of heat conducted from hot gas to cold gas (i) =
Rate at which enthalpy is conducted through flame front (ii) =
Rate at which heat is produced by chemical reaction (iii)
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37. Deflagrations - burning velocity
Estimate of i
Conduction heat transfer rate = -kA(T/)
k = gas thermal conductivity, A = cross-sectional area of flame
T = temperature rise across front = Tproducts - Treactants
 = thickness of front (unknown at this point)
Estimate of ii
Enthalpy flux through front = (mass flux) x Cp x T
Mass flux = VA ( = density of reactants = ∞, V = velocity = SL)
Enthalpy flux = ∞CpSLAT
Estimate of iii
Heat generated by reaction = QR x (d[fuel]/dt) x Mfuel x Volume
Volume = A
QR = CPT/f
[F]∞ = fuel concentration in the cold reactants
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38. Deflagrations - burning velocity, thickness
Combine (i) and (ii)
 = k/CpSL = /SL ( = flame thickness)
 = k/Cp = thermal diffusivity (units length2/time)
For air at 300K & 1 atm,  ≈ 0.2 cm2/s
For gases  ≈  ( = kinematic viscosity)
For gases  ~ P-1T1.7 since k ~ P0T.7,  ~ P1T-1, Cp ~ P0T0
For typical stoichiometric hydrocarbon-air flame, SL ≈ 40 cm/s, thus  ≈ /SL ≈ cm (!) (Actually when properties are temperature-averaged,  ≈ 4/SL ≈ 0.02 cm - still small!)
Combine (ii) and (iii)
SL = {w}1/2
w = overall reaction rate = (d[fuel]/dt)/[fuel]∞ (units 1/s)
With SL ≈ 40 cm/s,  ≈ 0.2 cm2/s, w ≈ 1600 s-1
1/w = characteristic reaction time = 625 microseconds
Heat release rate per unit volume = (enthalpy flux) / (volume)
= (CpSLAT)/(A) = CpSL/k)(kT)/ = (kT)/2
= (0.07 W/mK)(1900K)/( m)2 = 3 x 109 W/m3 !!!
Moral: flames are thin, fast and generate a lot of heat!
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39. Deflagrations - burning velocity
More rigorous analysis (Zeldovich, 1940)
Tad = adiabatic flame temperature; T∞ = ambient temperature
Note same form SL ~ (aw)1/2 as simple estimate, where w ~ Z[F]∞-1e-b
Still more rigorous (Bush and Fendell, 1970, n = 1)
Note results are same to leading order for n = 1, Bush and Fendell added next order in expansion in powers of 1/b(1-e)
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40. Deflagrations - burning velocity
How does SL vary with pressure? Define order of reaction (n) = A+ B; since
Thus SL ~ {w}1/2 ~ {P-1Pn-1}1/2 ~ P(n-2)/2
For typical n = 2, SL independent of pressure
For "real" hydrocarbons, working backwards from experimental results, we find typically SL ~ P-0.4, thus n ≈ 1.2
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41. Deflagrations - temperature effect
Since Zeldovich number () >> 1
For typical hydrocarbon-air flames, E ≈ 40 kcal/mole
 = cal/mole, Tf ≈ 2200K if adiabatic
  ≈ 10, at T close to Tf, w ~ T10 !!!
 Thin reaction zone concentrated near highest temp.
 In Zeldovich (or any) estimate of SL, overall reaction rate  must be evaluated at Tad, not T∞
How can we estimate E? If reaction rate depends more on E than concentrations [ ], SL ~ {w}1/2 ~ {exp(-E/T)}1/2 ~ exp(E/2T) - Plot of ln(SL) vs. 1/Tad has slope of -E/2
If  isn't large, then w(T∞) ≈ w(Tad) and reaction occurs even in the cold gases, so no control over flame is possible!
Since SL ~ w1/2, SL ~ (T)1/2 ~ T5 typically!
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42. Deflagrations - summary
These relations show the effect of Tad (depends on fuel & stoichiometry),  (depends on diluent gas (usually N2) & P), w (depends on fuel, T, P) and pressure (engine condition) on laminar burning rates
Re-emphasize: these estimates are based on an overall reaction rate; real flames have 1000's of individual reactions between 100's of species - but we can work backwards from experiments or detailed calculations to get these estimates for the overall reaction rate parameters
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43. Deflagrations
Schematic of flame temperatures and laminar burning velocities
Real data on SL (Vagelopoulos & Egolfopoulos, 1998)
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44. Advanced fundamental topics
Flammability & extinction limits
Description of flammability limits
Chemical kinetics of limits
Time scales
Mechanisms of limits
Buoyancy effects - upward & downward
Conduction heat loss to tube walls
(Sidebar) more about flames in tubes
Radiation heat loss
Optically thin limit
(Sidebar) reabsorption effects
Aerodynamic stretch
Chemical fire suppressants
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45. Flammability and extinction limits
Reference: Ju, Y., Maruta, K., Niioka, T., "Combustion Limits," Applied Mechanics Reviews, Vol. 53, pp (2001)
Too lean or too rich mixtures won't burn - flammability limits
Even if mixture is flammable, still won't burn in certain environments
Small diameter tubes
Strong hydrodynamic strain or turbulence
High or low gravity
High or low pressure
Understanding needed for combustion engines & industrial combustion processes (leaner mixtures  lower Tad  lower NOx); fire & explosion hazard management, fire suppression, ...
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46. Flammability limits - basic observations
Limits occur for mixtures that are thermodynamically flammable - theoretical adiabatic flame temperature (Tad) far above ambient temperature (T∞)
Limits usually characterized by finite (not zero) burning velocity at limit
Models of limits due to losses - most important prediction: burning velocity at the limit (SL,lim) - better test of limit predictions than composition at limit
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47. Premixed-gas flames – flammability limits
2 limit mechanisms, (1) & (2), yield similar fuel % and Tad at limit but very different SL,lim
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48. Flammability limits in vertical tubes
Most common apparatus - vertical tube (typ. 5 cm in diameter)
Ignite mixture at one end of tube, if it propagates to other end, it's "flammable"
Limit composition depends on orientation - buoyancy effects
Upward propagation Downward propagation
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49. Chemical kinetics of limits
Lean hydrocarbon-air flames: main branching reaction (promotes combustion) is
H + O2  OH + O; d[O2]/dt = [H][O2]T-0.8e-16500/RT
[ ]: mole/cm3; T: K; R: cal/mole-K; t: sec
Depends on P2 since [ ] ~ P, strongly dependent on T
Why important? Only energetically viable way to break O=O bond (120 kcal/mole), even though [H] is small
Main H consumption reaction
H + O2 + M  HO2 + M; {M = any molecule}
d[O2]/dt = [H][O2][M]T0e+1000/RT for M = N2
(higher rate for CO2 and especially H2O)
Depends on P3, nearly independent of T
Why important? Inhibits combustion by replacing H with much less active HO2
Branching or inhibition may be faster depending on T and P
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50. Chemical kinetics of limits
Rates equal ("crossover") when
[M] = 101.5T-0.8e-17500/RT
Ideal gas law: P = [M]RT thus
P = 103.4T0.2e-17500/RT (P in atm)
 crossover at 950K for 1 atm, higher T for higher P
…but this only indicates that chemical mechanism may change and perhaps overall W drop rapidly below some T
Computations show no limits without losses – no purely chemical criterion (Lakshmisha et al., 1990; Giovangigli & Smooke, 1992) - for steady planar adiabatic flames, burning velocity decreases smoothly towards zero as fuel concentration decreases (domain sizes up to 10 m, SL down to 0.02 cm/s)
…but as SL decreases, d increases - need larger computational domain or experimental apparatus
Also more buoyancy & heat loss effects as SL decreases ….
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51. Chemical kinetics of limits
Ju, Masuya, Ronney (1998)
Ju et al., 1998
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52. Aerodynamic effects on premixed flames
Aerodynamic effects occur on a large scale compared to the transport or reaction zones but affect SL and even existence of the flame
Why only at large scale?
Re on flame scale ≈ SL/ ( = kinematic viscosity)
Re = (SL/)() = (1)(1/Pr) ≈ 1 since Pr ≈ 1 for gases
Reflame ≈ 1  viscosity suppresses flow disturbances
Key parameter: stretch rate ()
Generally  ~ U/d
U = characteristic flow velocity
d = characteristic flow length scale
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53. Aerodynamic effects on premixed flames
Strong stretch ( ≥ w ~ SL2/ or Karlovitz number Ka  /SL2 ≥ 1) extinguishes flames
Moderate stretch strengthens flames for Le < 1
Buckmaster &
Mikolaitis, 1982a (Ze = b in my notation), cold reactants against adiabatic products
SL/SL(unstrained, adiabatic flame)
ln(Ka)
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54. Lewis number tutorial
Le affects flame temperature in curved (shown below) or stretched flames
When Le < 1, additional thermal enthalpy loss in curved/stretched region is less than additional chemical enthalpy gain, thus local flame temperature in curved region is higher, thus reaction rate increases drastically, local burning velocity increases
Opposite behavior for oppositely curved flames
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55. TIME SCALES - premixed-gas flames
See Ronney (1998)
Chemical time scale
tchem ≈ /SL ≈ (a/SL)/SL ≈ a/SL2
a = thermal diffusivity [typ. 0.2 cm2/s],
SL = laminar flame speed [typ. 40 cm/s]
Conduction time scale
tcond ≈ Tad/(dT/dt) ≈ d2/16a
d = tube or burner diameter
Radiation time scale
trad ≈ Tad/(dT/dt) ≈ Tad/(L/rCp) (L = radiative heat loss per unit volume)
Optically thin radiation: L = 4sap(Tad4 – T∞4)
ap = Planck mean absorption coefficient [typ. 2 m-1 at 1 atm]
L ≈ 106 W/m3 for HC-air combustion products
trad ~ P/sap(Tad4 – T∞4) ~ P0, P = pressure
Buoyant transport time scale
t ~ d/V; V ≈ (gd(Dr/r))1/2 ≈ (gd)1/2
(g = gravity, d = characteristic dimension)
Inviscid: tinv ≈ d/(gd)1/2 ≈ (d/g)1/2 (1/tinv ≈ Sinv)
Viscous: d ≈ n/V Þ tvis ≈ (n/g2)1/3 (n = viscosity [typ cm2/s])
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56. Time scales (hydrocarbon-air, 1 atm)
Conclusions
Buoyancy unimportant for near-stoichiometric flames
(tinv & tvis >> tchem)
Buoyancy strongly influences near-limit flames at 1g
(tinv & tvis < tchem)
Radiation effects unimportant at 1g (tvis << trad; tinv << trad)
Radiation effects dominate flames with low SL
(trad ≈ tchem), but only observable at µg
Small trad (a few seconds) - drop towers useful
Radiation > conduction only for d > 3 cm
Re ~ Vd/n ~ (gd3/n2)1/2 Þ turbulent flow at 1g for d > 10 cm
AME Spring Lecture 1

57. Flammability limits due to losses
Golden rule: at limit
Why 1/b not 1? T can only drop by O(1/b) before extinction - O(1) drop in T means exponentially large drop in , thus exponentially small SL (could also say heat generation occurs only in /b region whereas loss occurs over  region)
AME Spring Lecture 1

58. Flammability limits due to losses
Heat loss to walls
tchem ~ tcond  SL,lim ≈ (8)1/2a/d at limit
or Pelim  SL,limd/a ≈ (8)1/2 ≈ 9
Actually Pelim ≈ 40 due to temperature averaging - consistent with experiments (Jarosinsky, 1983)
Upward propagation in tube
Rise speed at limit ≈ 0.3(gd)1/2 due to buoyancy alone (same as air bubble rising in water-filled tube (Levy, 1965))
Pelim ≈ 0.3 Grd1/2; Grd = Grashof number  gd3/n2
Causes stretch extinction (Buckmaster & Mikolaitis, 1982b):
tchem ≈ tinv or 1/tchem ≈ Sinv
Note f(Le) < 1 for Le < 1, f(Le) > 1 for Le > 1 - flame can survive at lower SL (weaker mixtures) when Le < 1
AME Spring Lecture 1

59. Difference between S and SL
long flame skirt at high Gr or with small f (low Lewis number, Le)
(but note SL not really constant over flame surface!)
AME Spring Lecture 1

60. Flammability limits due to losses
Downward propagation – sinking layer of cooling gases near wall outruns & "suffocates" flame (Jarosinsky et al., 1982)
tchem ≈ tvis Þ SL,lim ≈ 1.3(ga)1/3
Pelim ≈ 1.65 Grd1/3
Can also obtain this result by equating SL to sink rate of thermal boundary layer = 0.8(gx)1/2 for x = 
Consistent with experiments varying d and a (by varying diluent gas and pressure) (Wang & Ronney, 1993) and g (using centrifuge) (Krivulin et al., 1981)
More on limits in tubes…
AME Spring Lecture 1

61. Flammability limits in vertical tubes
Upward propagation Downward propagation
AME Spring Lecture 1

62. Flammability limits in tubes
Upward propagation - Wang & Ronney, 1993
AME Spring Lecture 1

63. Flammability limits in tubes
Downward propagation - Wang & Ronney, 1993
AME Spring Lecture 1

64. Flammability limits – losses - continued…
Big tube, no gravity – what causes limits?
Radiation heat loss (trad ≈ tchem) (Joulin & Clavin, 1976; Buckmaster, 1976)
What if not at limit? Heat loss still decreases SL, actually 2 possible speeds for any value of heat loss, but lower one generally unstable
AME Spring Lecture 1

65. Flammability limits – losses - continued…
Doesn't radiative loss decrease for weaker mixtures, since temperature is lower? NO!
Predicted SL,lim (typically 2 cm/s) consistent with µg experiments (Ronney, 1988; Abbud-Madrid & Ronney, 1990)
AME Spring Lecture 1

66. Reabsorption effects Is radiation always a loss mechanism?
Reabsorption may be important when aP-1 < d
Small concentration of blackbody particles - decreases SL (more radiative loss)
More particles - reabsorption extend limits, increases SL
Abbud-Madrid & Ronney (1993)
AME Spring Lecture 1

67. Reabsorption effects on premixed flames
Gases – much more complicated because absorption coefficient depends strongly on wavelength and temperature & some radiation always escapes (Ju, Masuya, Ronney 1998)
Absorption spectra of products different from reactants
Spectra broader at high T than low T
Dramatic difference in SL & limits compared to optically thin
AME Spring Lecture 1

68. Stretched flames - spherical
Spherical expanding flames, Le < 1: stretch allows flames to exist in mixtures below radiative limit until flame radius rf is too large & curvature benefit too weak (Ronney & Sivashinsky, 1989)
Adds stretch term (2S/R) (R = scaled flame radius; R > 0 for Le < 1; R < 0 for Le > 1) and unsteady term (dS/dR) to planar steady equation
Dual limit: radiation at large rf, curvature-induced stretch at small rf (ignition limit)
AME Spring Lecture 1

69. Stretched flames - spherical
Theory (Ronney & Sivashinsky, 1989)
Experiment
(Ronney, 1985)
AME Spring Lecture 1

70. Stretched counterflow or stagnation flames
Mass + momentum conservation, 2D, const. density ()
(ux, uy = velocity components in x, y directions)
admit an exact, steady (∂/∂t = 0) solution which is the same with or without viscosity (!!!):
 = rate of strain (units s-1)
Similar result in 2D axisymmetric geometry:
Very simple flow characterized by a single parameter , easily implemented experimentally using counter-flowing round jets…
AME Spring Lecture 1

71. Stretched counterflow or stagnation flames
S = duz/dz – flame located where uz = SL
Increased stretch pushes flame closer to stagnation plane - decreased volume of radiant products
Similar Le effects as curved flames z
AME Spring Lecture 1

72. Premixed-gas flames - stretched flames
Stretched flames with radiation (Ju et al., 1999): dual limits, flammability extension even for Le >1, multiple solutions (which ones are stable?)
AME Spring Lecture 1

73. Premixed-gas flames - stretched flames
Dual limits & Le effects seen in µg experiments, but evidence for multivalued behavior inconclusive
Guo et al. (1997)
AME Spring Lecture 1

74. Chemical fire suppressants
Key to suppression is removal of H atoms
H + HBr  H2 + Br
H + Br2  HBr + Br
Br + Br + M  Br2 + M
H + H  H2
Why Br and not Cl or F? HCl and HF too stable, 1st reaction too slow
HBr is a corrosive liquid, not convenient - use CF3Br (Halon 1301) - Br easily removed, remaining CF3 very stable, high CP to soak up heat
Problem - CF3Br very powerful ozone depleter - banned!
Alternatives not very good; best ozone-friendly chemical alternative is probably CF3CH2CF3 or CF3H
Other alternatives (e.g. water mist) also being considered
AME Spring Lecture 1

75. Chemical fire suppressants
AME Spring Lecture 1

76. References AME 514 - Spring 2015 - Lecture 1
Abbud-Madrid, A., Ronney, P. D., "Effects of Radiative and Diffusive Transport Processes on Premixed Flames Near Flammability Limits," Twenty Third Symposium (International) on Combustion, Combustion Institute, 1990, pp
Abbud-Madrid, A., Ronney, P. D., "Premixed Flame Propagation in an Optically-Thick Gas," AIAA Journal, Vol. 31, pp (1993).
Buckmaster, J. D. (1976). The quenching of deflagration waves, Combust. Flame 26,
Buckmaster, J. D., Mikolaitis, D. (1982a). The premixed flame in a counterflow, Combust. Flame 47,
Buckmaster, J. D., Mikolaitis, D. (1982b). A flammability-limit model upward propagation through lean methan-air mixtures in a standard flammability tube. Combust. Flame 45, pp
Giovangigli, V. and Smooke, M. (1992). Application of Continuation Methods to Plane Premixed Laminar Flames, Combust. Sci. Tech. 87,
Guo, H., Ju, Y., Maruta, K., Niioka, T., Liu, F., Combust. Flame 109: (1997).
Jarosinsky, J. (1983). Flame quenching by a cold wall, Combust. Flame 50, 167.
Jarosinsky, J., Strehlow, R. A., Azarbarzin, A. (1982). The mechanisms of lean limit extinguishment of an upward and downward propagating flame in a standard flammability tube, Proc. Combust. Inst. 19,
Joulin, G., Clavin, P. (1976). Analyse asymptotique des conditions d'extinction des flammes laminaries, Acta Astronautica 3, 223.
Ju, Y., Masuya, G. and Ronney, P. D., "Effects of Radiative Emission and Absorption on the Propagation and Extinction of Premixed Gas Flames" Twenty-Seventh International Symposium on Combustion, Combustion Institute, Pittsburgh, 1998, pp
Ju, Y., Guo, H., Liu, F., Maruta, K. (1999). Effects of the Lewis number and radiative heat loss on the bifurcation of extinction of CH4-O2-N2-He flames, J. Fluid Mech. 379,
AME Spring Lecture 1

77. References
Krivulin, V. N., Kudryavtsev, E. A., Baratov, A. N., Badalyan, A. M., Babkin, V. S. (1981). Effect of acceleration on the limits of propagation of homogeneous gas mixtures, Combust. Expl. Shock Waves (Engl. Transl.) 17,
Lakshmisha, K. N., Paul, P. J., Mukunda, H. S. (1990). On the flammability limit and heat loss in flames with detailed chemistry, Proc. Combust. Inst. 23,
Levy, A. (1965). An optical study of flammability limits, Proc. Roy. Soc. (London) A283, 134.
Ronney, P.D., "Effect of Gravity on Laminar Premixed Gas Combustion II: Ignition and Extinction Phenomena," Combustion and Flame, Vol. 62, pp (1985).
Ronney, P.D., "On the Mechanisms of Flame Propagation Limits and Extinction Processes at Microgravity," Twenty Second Symposium (International) on Combustion, Combustion Institute, 1988, pp
Ronney, P. D., "Understanding Combustion Processes Through Microgravity Research," Twenty-Seventh International Symposium on Combustion, Combustion Institute, Pittsburgh, 1998, pp
Ronney, P.D., Sivashinsky, G.I., "A Theoretical Study of Propagation and Extinction of Nonsteady Spherical Flame Fronts," SIAM Journal on Applied Mathematics, Vol. 49, pp (1989).
Wang, Q., Ronney, P. D. (1993). Mechanisms of flame propagation limits in vertical tubes, Paper no. 45, Spring Technical Meeting, Combustion Institute, Eastern/Central States Section, March 15-17, 1993, New Orleans, LA.
AME Spring Lecture 1

78. Advanced fundamental topics
End of flammability limits notes - sidebar topics from here on …

79. Effects of radiative emission and absorption on the propagation and extinction of premixed gas flames
Yiguang Ju and Goro Masuya
Department of Aeronautics & Space Engineering
Tohoku University, Aoba-ku, Sendai 980, Japan
Paul D. Ronney
Department of Aerospace & Mechanical Engineering
University of Southern California
Los Angeles, CA
Published in Proceedings of the Combustion Institute, Vol. 27, pp (1998)

80. Background
Microgravity experiments show importance of radiative loss on flammability & extinction limits when flame stretch, conductive loss, buoyant convection eliminated – experiments consistent with theoretical predictions of
Burning velocity at limit (SL,lim)
Flame temperature at limit
Loss rates in burned gases
…but is radiation a fundamental extinction mechanism? Reabsorption expected in large, "optically thick" systems
Theory (Joulin & Deshaies, 1986) & experiment (Abbud-Madrid & Ronney, 1993) with emitting/absorbing blackbody particles
Net heat losses decrease (theoretically to zero)
Burning velocities (SL) increase
Flammability limits widen (theoretically no limit)
… but gases, unlike solid particles, emit & absorb only in narrow spectral bands - what will happen?
AME Spring Lecture 1

81. Background (continued)
Objectives
Model premixed-gas flames computationally with detailed radiative emission-absorption effects
Compare results to experiments & theoretical predictions
Practical applications
Combustion at high pressures and in large furnaces
IC engines: 40 atm - Planck mean absorption length (LP) ≈ 4 cm for combustion products ≈ cylinder size
Atmospheric-pressure furnaces - LP ≈ 1.6 m - comparable to boiler dimensions
Exhaust-gas or flue-gas recirculation - absorbing CO2 & H2O present in unburned mixture - reduces LP of reactants & increases reabsorption effects
AME Spring Lecture 1

82. Numerical model
Steady planar 1D energy & species conservation equations
CHEMKIN with pseudo-arclength continuation
18-species, 58-step CH4 oxidation mechanism (Kee et al.)
Boundary conditions
Upstream - T = 300K, fresh mixture composition, inflow velocity SL at x = L1 = -30 cm
Downstream - zero gradients of temperature & composition at x = L2 = 400 cm
Radiation model
CO2, H2O and CO
Wavenumbers (w) cm-1, 25 cm-1 resolution
Statistical Narrow-Band model with exponential-tailed inverse line strength distribution
S6 discrete ordinates & Gaussian quadrature
300K black walls at upstream & downstream boundaries
Mixtures CH4 + {0.21O2+(0.79-g)N2+ g CO2} - substitute CO2 for N2 in "air" to assess effect of absorbing ambient
AME Spring Lecture 1

83. Results - flame structure
Adiabatic flame (no radiation)
The usual behavior
Optically-thin
Volumetric loss always positive
Maximum T < adiabatic
T decreases "rapidly" in burned gases
"Small" preheat convection-diffusion zone - similar to adiabatic flame
With reabsorption
Volumetric loss negative in reactants - indicates net heat transfer from products to reactants via reabsorption
Maximum T > adiabatic due to radiative preheating - analogous to Weinberg's "Swiss roll" burner with heat recirculation
T decreases "slowly" in burned gases - heat loss reduced
"Small" preheat convection-diffusion zone PLUS
"Huge" convection-radiation preheat zone
AME Spring Lecture 1

84. Flame zone detail Radiation zones (large scale)
Flame structures
Flame zone detail Radiation zones (large scale)
Mixture: CH4 in "air", 1 atm, equivalence ratio (f) = 0.70; g = 0.30 ("air" = 0.21 O N CO2)
AME Spring Lecture 1

85. Radiation effects on burning velocity (SL)
CH4-air (g = 0)
Minor differences between reabsorption & optically-thin
... but SL,lim 25% lower with reabsorption; since SL,lim ~ (radiative loss)1/2, if net loss halved, then SL,lim should be /√2 = 29% lower with reabsorption
SL,lim/SL,ad ≈ 0.6 for both optically-thin and reabsorption models - close to theoretical prediction (e-1/2)
Interpretation: reabsorption eliminates downstream heat loss, no effect on upstream loss (no absorbers upstream); classical quenching mechanism still applies
g = 0.30 (38% of N2 replaced by CO2)
Massive effect of reabsorption
SL much higher with reabsorption than with no radiation!
Lean limit much leaner (f = 0.44) than with optically-thin radiation (f = 0.68)
AME Spring Lecture 1

86. Comparisons of burning velocities
g = 0 (no CO2 in ambient) g = 0.30
Note that without CO2 (left) SL & peak temperatures of reabsorbing flames are slightly lower than non-radiating flames, but with CO2 (right), SL & T are much higher with reabsorption. Optically thin always has lowest SL & T, with or without CO2
Note also that all experiments lie below predictions - are published chemical mechanisms accurate for very lean mixtures?
AME Spring Lecture 1

87. Mechanisms of extinction limits
Why do limits exist even when reabsorption effects are considered and the ambient mixture includes absorbers?
Spectra of product H2O different from CO2 (Mechanism I)
Spectra broader at high T than low T (Mechanism II)
Radiation reaches upstream boundary due to "gaps" in spectra - product radiation that cannot be absorbed upstream
Absorption spectra of CO2 & H2O at 300K & 1300K
AME Spring Lecture 1

88. Mechanisms of limits (continued)
Flux at upstream boundary shows spectral regions where radiation can escape due to Mechanisms I and II - "gaps" due to mismatch between radiation emitted at the flame front and that which can be absorbed by the reactants
Depends on "discontinuity" (as seen by radiation) in T and composition at flame front - doesn't apply to downstream radiation because T gradient is small
Behavior cannot be predicted via simple mean absorption coefficients - critically dependent on compositional & temperature dependence of spectra
Spectrally-resolved radiative flux at upstream boundary for a reabsorbing flame
(πIb = maximum possible flux)
AME Spring Lecture 1

89. Effect of domain size
Limit f & SL,lim decreases as upstream domain length (L1) increases - less net heat loss
Significant reabsorption effects seen at L1 = 1 cm even though LP ≈ 18.5 cm because of existence of spectral regions with L(w) ≈ cm-atm (!)
L1 > 100 cm required for domain-independent results due to band "wings" with small L(w)
Downstream domain length (L2) has little effect due to small gradients & nearly complete downstream absorption
Effect of upstream domain length (L1) on limit composition (o) & SL for reabsorbing flames. With-out reabsorption, o = 0.68, thus reabsorption is very important even for the smallest L1 shown
AME Spring Lecture 1

90. Effect of g (CO2 substitution level)
f = 1.0: little effect of radiation;
f = 0.5: dominant effect - why?
(1) f = 0.5: close to radiative extinction limit - large benefit of decreased heat loss due to reabsorption by CO2
(2) f = 0.5: much larger Boltzman number (defined below) (B) (≈127) than f = 1.0 (≈11.3); B ~ potential for radiative preheating to increase SL
Note with reabsorption, only 1% CO2 addition nearly doubles SL due to much lower net heat loss!
Effect of CO2 substitution for N2 on SL
AME Spring Lecture 1

91. Effect of g (continued)
Effect of CO2 substitution on flammability limit composition
Effect of CO2 substitution on SL,lim/SL,adiabatic
Limit mixture much leaner with reabsorption than optically thin
Limit mixture decreases with CO2 addition even though CP,CO2 > CP,N2
SL,lim/SL,ad always ≈ e-1/2 for optically thin, in agreement with theory
SL,lim/SL,ad up to ≈ 20 with reabsorption!
AME Spring Lecture 1

92. Comparison to analytic theory
Joulin & Deshaies (1986) - analytical theory
Comparison to computation - poor
Better without H2O radiation (mechanism (I) suppressed)
Slightly better still without T broadening (mechanism (II) suppressed, nearly adiabatic)
Good agreement when L(w) = LP = constant - emission & absorption across entire spectrum rather than just certain narrow bands.
Drastic differences between last two cases, even though both have no net heat loss and have same Planck mean absorption lengths!
Effect of different radiation models on SL and comparison to theory
AME Spring Lecture 1

93. Comparison with experiment
No directly comparable expts., BUT...
Zhu, Egolfopoulos, Law (1988)
CH4 + (0.21O CO2) (g = 0.79)
Counterflow twin flames, extrapolated to zero strain
L1 = L2 ≈ 0.35 cm chosen since 0.7 cm from nozzle to stagnation plane
No solutions for adiabatic flame or optically-thin radiation (!)
Moderate agreement with reabsorption
Abbud-Madrid & Ronney (1990)
(CH4 + 4O2) + CO2
Expanding spherical flame at µg
L1 = L2 ≈ 6 cm chosen (≈ flame radius)
Optically-thin model over-predicts limit fuel conc. & SL,lim
Reabsorption model underpredicts limit fuel conc. but SL,lim well predicted - net loss correctly calculated
Comparison of computed results to experiments where reabsorption effects may have been important
AME Spring Lecture 1

94. Conclusions
Reabsorption increases SL & extends limits, even in spectrally radiating gases
Two loss mechanisms cause limits even with reabsorption
(I) Mismatch between spectra of reactants & products
(II) Temperature broadening of spectra
Results qualitatively & sometimes quantitatively consistent with theory & experiments
Behavior cannot be predicted using mean absorption coefficients!
Can be important in practical systems
AME Spring Lecture 1

95. Planck mean absorption coefficient
AME Spring Lecture 1

96. More on flammability limits in tubes
Experiments show that the flammability limits are wider for upward than downward propagation, corresponding to SL,lim,down > SL,lim,up since SL is lower for more dilute mixtures
…but note according to the models, SL,lim,down > SL,lim,up when
Gr < 10,000 f12
but also need Pe > 40 (not in heat-loss limit)
Gr > 18,000
 at high Le (high f) & 18,000 < Gr < 10,000 f12, upward limits may be narrower than downward limits (?!?)
Never observed, but appropriate conditions never tested - high Le, moderate Gr
AME Spring Lecture 1

97. Turbulent limit behavior?
Burned gases are turbulent if Re > 2000
Upward limit: Re ≈ S(r∞/rad-1)d/n  Gr > 300 x 106
Downward limit: Re ≈ SL(r∞/rad-1)d/n  Gr > 40 x not accessible with current apparatus
"Standard" condition (5 cm tube, air, 1 atm):
Gr ≈ 3.0 x 106 : always laminar!
AME Spring Lecture 1

98. Approach Study limit mechanisms by measuring Sb,lim for varying
Tube diameter
 = (diluent, pressure)
Le  = Le(diluent, fuel)
and determine scaling relations (Pelim vs. Gr & Le)
Apparatus
Tubes with 0.5 cm < D < 20 cm; open at ignition end
He, Ne, N2, CO2, SF6 diluents
0.1 atm < P < 10 atm
2 x 102 < Gr < 2 x 109
Absorption tank to maintain constant P during test
Thermocouples
Procedure
Fixed fuel:O2 ratio
Vary diluent conc. until limit determined
Measure Sb,lim & temperature characteristics at limit
AME Spring Lecture 1

99. Results - laminar flames
Upward limit
Low Gr
Pelim ≈ 40 ± 10 at low Gr
Highest T near centerline of tube
High Gr
Pelim ≈ 0.3 Gr1/2 at high Gr
Highest T near centerline (low Le)
Highest T near wall (high Le)
Indicates strain effects at limit
Downward
Pelim ≈ 1.5 Gr1/3 at high Gr
Upward limits narrower than downward limits at high Le & moderate Gr, e.g. lean C3H8-O2-Ne, P = 1 atm, D = 2.5 cm, Le ≈ 2.6, Gr ≈ 19,000: fuel up / fuel down ≈ 0.83
AME Spring Lecture 1

100. Limit regimes - upward propagation
AME Spring Lecture 1

101. Limit regimes - downward propagation
AME Spring Lecture 1

102. Flamelet vs. distributed combustion
Abdel-Gayed & Bradley (1989): distributed if Ka > 0.3
Ka  ReT-1/2U2; ReT  u'LI/n, U  u'/SL
LI  integral scale of turbulence
Estimate for pipe flow
u' ≈ 0.05S(r∞/rad-1); LI ≈ d
SL,lim from Buckmaster & Mikolaitis (1982) model
 Ka ≈ /f2 Gr1/4 ≈ 0.3/f2 at Gr = 700 x 106
Distributed combustion probable at high Gr, moderate Le
Away from limit - wrinkled, unsteady skirt
AME Spring Lecture 1

103. Limit flame - distributed combustion
C3H8-O2-CO2, P = 2.5 atm, d = 10 cm, Le ≈ 1.3, Gr ≈ 6 x 108
AME Spring Lecture 1

104. Farther from limit - wrinkled skirt
C3H8-O2-CO2, P = 2.5 atm, d = 10 cm, Le ≈ 1.3, Gr ≈ 6 x 108
AME Spring Lecture 1

105. Lower Le - boiling tip, no tip opening
C3H8-O2-SF6, P = 2.5 atm, d = 10 cm, Le ≈ 0.7, Gr ≈ 5 x 109
AME Spring Lecture 1

106. Turbulent flame quenching
Why does distributed flame exist at  ≈ 4d, whereas laminar flame extinguishes when  ≈ 1/40 d (Pe = 40)?
Analysis
Nu = hd/k ≈ Re.8 Pr.3 (turbulent heat transfer in pipe)
Qloss ≈ hAT; A = πd; let  = n D (n is unknown)
Qgen ≈ oSbπd2CpT; Sb = 0.3(gd)1/2
Qloss/Qgen ≈ 1/b at quenching limit
 n ≈ 5Gr0.1/b at quenching limit
Gr = 600 x 106,  = 10  n = 3.9 at limit !!!
But low Le  SL low at tip opening  n > 4 at tip opening  distributed flame not observable
AME Spring Lecture 1

107. Conclusions
Probable heat loss & buoyancy-induced limit mechanisms observed
Limit behavior characterized mainly by Lewis & Grashof numbers
Scaling analyses useful for gaining insight
Transition to turbulence & distributed-like combustion observed
High-Gr results may be more applicable to "real" hazards (large systems, turbulent) than classical experiments at low Gr
AME Spring Lecture 1