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### AME 514Applications of Combustion

### AME 514Lecture 2Sidebar topic: Effects of radiative emission and absorption on the propagation and extinction of premixed gas flames

Lecture 2: Combustion Theory I

Outline

- Ignition revisited
- “Flame ball” solutions
- Short tutorial on radiative transfer in gases
- Lewis number effects on spherical flames
- Dynamic analysis of spherical flames
- Sidebar topic (if time permits…): effects of reabsorption on flame propagation

AME 514 - Spring 2013 - Lecture 2 - Combustion Theory I

Ignition - basic concepts from AME 513

- Experiments (Lewis & von Elbe, 1987) show that a minimum energy (Emin) (not just minimum T or volume) required to ignite a flame
- Emin lowest near stoichiometric (typ. 0.2 mJ) but minimum shifts to richer mixtures for higher HCs (why? Stay tuned…)
- Prediction of Emin relevant to energy conversion and fire safety applications

Minimum ignition energy (mJ)

AME 514 - Spring 2013 - Lecture 2 - Combustion Theory I

Basic concepts from AME 513

- Emin related to need to create flame kernel with dimension () large enough that chemical reaction (w) can exceed conductive loss rate (/2), thus > (/w)1/2 ~ /(w)1/2 ~ /SL ~
- Emin ~ energy contained in volume of gas with T ≈ Tad and radius ≈ ≈ 4/SL

AME 514 - Spring 2013 - Lecture 2 - Combustion Theory I

Predictions from AME 513

- Since ~ P-1, Emin ~ P-2 if SL is independent of P
- Emin ≈ 100,000 times larger in a He-diluted than SF6-diluted mixture with same SL, same P (due to and differences)
- Stoichiometric CH4-air @ 1 atm: predicted Emin ≈ 0.010 mJ ≈ 30x times lower than experiment (due to chemical kinetics, heat losses, shock losses …)
- … but need something more (Lewis number effects):
- 10% H2-air (SL ≈ 10 cm/sec): predicted Emin ≈ 0.3 mJ = 2.5 times higher than experiments
- Lean CH4-air (SL ≈ 5 cm/sec): Emin≈ 5 mJ compared to ≈ 5000mJ for lean C3H8-air with same SL - but prediction is same for both

AME 514 - Spring 2013 - Lecture 2 - Combustion Theory I

Ignition revisited

- Assumptions: 1D spherical (only velocity (u) in radial direction); ideal gases; adiabatic (except for ignition source Q(r,t)); 1 limiting reactant (e.g. very lean); 1-step overall reaction; D, l, CP, etc. constant; low Mach #; no body forces
- Governing equations for mass, energy & species conservations (YF= limiting reactant mass fraction; QR = its heating value)

AME 514 - Spring 2013 - Lecture 2 - Combustion Theory I

More rigorous approach

- Non-dimensionalize (note Tad = T∞ + Y∞QR/CP)

leads to, for mass, energy and species conservation

with boundary conditions

(Initial condition: T = T∞, Y = Y∞,

U = 0 everywhere)

(At infinite radius, T = T∞, Y = Y∞,

U = 0 for all times)

(Symmetry condition at r = 0 for all times)

AME 514 - Spring 2013 - Lecture 2 - Combustion Theory I

Steady (?!?) flame ball solutions

- If reaction is confined to a thin zone near r = RZ (large b)
- This is a flame ball solution - note for Le < > 1, T* > < Tad; for Le = 1, T* = Tad and RZ = d
- Generally unstable
- R < RZ: shrinks & extinguishes; R > RZ: expands & develops steady flame
- RZ related to requirement for initiation of steady flame
- … but stable for a few carefully (or accidentally) chosen mixtures…

AME 514 - Spring 2013 - Lecture 2 - Combustion Theory I

Steady (?!?) solutions

- How can a spherical flame not propagate???

Space experiments show ~ 1 cm

diameter flame balls possible

Movie: 500 sec elapsed time

AME 514 - Spring 2013 - Lecture 2 - Combustion Theory I

Droplet combustion

- Comment on T and Yox profiles for r ∞
- Identical to pure diffusion in spherical geometry:

so diffusion dominates convection at large r

AME 514 - Spring 2013 - Lecture 2 - Combustion Theory I

Flame balls - history

- Zeldovich, 1944– 1st prediction of flame balls (T ~ C1 + C2/r)
- Not possible for cylinder (T ~ C1+C2ln(r)) or plane (T ~ C1+C2r)
- Zeldovich, 1944; Joulin, 1985; Buckmaster, 1985: adiabatic flame balls are unstable
- Ronney (1990): stable(?) flame balls accidentally discovered in very lean H2-air mixtures in drop-tower experiment
- Farther from limit - expanding cellular flames

Far from limit

Close to limit

AME 514 - Spring 2013 - Lecture 2 - Combustion Theory I

Flame balls - history

- Buckmaster, Joulin, Ronney (1990, 1991): window of stable conditions with radiative loss, near-limit & low Lewis number
- Radiative loss (important for near limit mixtures) needed, otherwise no “penalty” for ball growing larger (loss ~ r3, generation ~ r1), larger = weaker
- Flame ball: a tiny dog wagged by an enormous tail
- Low Le needed, otherwise no “benefit” to flame curvature
- 2 solutions
- Small flame balls nearly adiabatic (small volume), always unstable
- Large flame balls stable, but if too large then unstable to 3D (cellular) instability

AME 514 - Spring 2013 - Lecture 2 - Combustion Theory I

Flame balls - history

- Predictions consistent with experimental observations

AME 514 - Spring 2013 - Lecture 2 - Combustion Theory I

Short tutorial on gas radiation

- More info:

http://ronney.usc.edu/AME517F09/, especially

http://ronney.usc.edu/AME517F09/AME517GasRadiationSummary.pdf

- Because gases vibrate and rotate only at discrete frequencies, they emit and absorb radiation only in narrow bands; this is unlike solid surfaces which have essentially an infinite number of degrees of freedom and so can emit /absorb across the whole spectrum
- Homonuclear diatomic molecules (e.g. O2, N2) cannot radiate
- Other diatomic molecules (e.g. CO) radiate weakly
- Polyatomic gases (e.g. CO2, H2O) radiate more strongly, but not as strongly as particles (e.g. soot)
- The absorption coefficient (a) of gas i is a function of the partial pressure of the gas (Pi), wavelength () and T

AME 514 - Spring 2013 - Lecture 2 - Combustion Theory I

Short tutorial on gas radiation

Double-click plot to open massive spreadsheet

AME 514 - Spring 2013 - Lecture 2 - Combustion Theory I

Short tutorial on gas radiation

- If the length scale (L) of the radiating gas volume is sufficiently small, i.e. L < 1/a() for all , then the optically thin model applies; absorption is negligible and the radiant emission per unit volume () for a gas at temperature Tg with environment temperature T∞ is given by

where aP is the Planck mean absorption coefficient and Ib is the usual Planck function

- aP(T) is tabulated for many gases (next page); in a gas mixture
- If the gas is not optically thin then the analysis is much more complicated; many models have been developed, e.g. the Statistical Narrow Band model (click on spreadsheet on previous slide)

AME 514 - Spring 2013 - Lecture 2 - Combustion Theory I

Short tutorial on gas radiation

- Example: lean CH4-air combustion products at 1 atm

PH2O ≈ 0.1 atm; PH2O ≈ 0.05 atm; T ≈ 1500K

1500K: aP,H2O = 2.2 m-1atm-1, aP,CO2 = 12.2 m-1atm-1,

aP = 2.2*0.1 + 12.2*0.05 = 0.83 m-1

- = 4 * 5.67 x 10-8 W/m2K4 * 1.33 m-1 *[(1500K)4 - (300K)4]

= 9.51 x 105 W/m3 = 0.95 W/cm3

AME 514 - Spring 2013 - Lecture 2 - Combustion Theory I

Lewis number effects

- Energy requirement very strongly dependent on Lewis number!

From computations by Tromans and Furzeland, 1986

AME 514 - Spring 2013 - Lecture 2 - Combustion Theory I

Lewis number effects

- Ok, so why does min. MIE shift to richer mixtures for higher HCs?
- Leeffective = effective/Deffective
- Deff = D of stoichiometrically limiting reactant, thus for lean mixtures Deff = Dfuel; rich mixtures Deff = DO2
- Lean mixtures - Leeffective = Lefuel
- Mostly air, so eff ≈ air; also Deff = Dfuel
- CH4: DCH4 > air since MCH4 < MN2&O2 thus LeCH4 < 1, thus Leeff < 1
- Higher HCs: Dfuel < air, thus Leeff > 1 - much higher MIE
- Rich mixtures - Leeffective = LeO2
- CH4: CH4 > air since MCH4 < MN2&O2, so adding excess CH4 INCREASES Leeff
- Higher HCs: fuel < air since Mfuel > MN2&O2, so adding excess fuel DECREASES Leeff
- Actually adding excess fuel decreases both and D, but decreases more

AME 514 - Spring 2013 - Lecture 2 - Combustion Theory I

Dynamic analysis

- RZ is related (but not equal) to an ignition requirement
- Joulin (1985) analyzed unsteady equations for Le < 1

(, and q are the dimensionless radius, time and heat input)

and found at the optimal ignition duration

which has the expected form

Emin ~ {energy per unit volume} x {volume of minimal flame kernel} ~ {adCp(Tad - T∞)} x {rz3}

AME 514 - Spring 2013 - Lecture 2 - Combustion Theory I

Dynamic analysis

- Joulin (1985)

Radius vs. time Minimum ignition energy

vs. ignition duration

AME 514 - Spring 2013 - Lecture 2 - Combustion Theory I

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 514 - Spring 2013 - Lecture 2 - Combustion Theory I

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 514 - Spring 2013 - Lecture 2 - Combustion Theory I

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) 150 - 9300 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 514 - Spring 2013 - Lecture 2 - Combustion Theory I

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 514 - Spring 2013 - Lecture 2 - Combustion Theory I

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 O2 + .49 N2 + .30 CO2)

AME 514 - Spring 2013 - Lecture 2 - Combustion Theory I

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 1 - 1/√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 514 - Spring 2013 - Lecture 2 - Combustion Theory I

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

AME 514 - Spring 2013 - Lecture 2 - Combustion Theory I

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 514 - Spring 2013 - Lecture 2 - Combustion Theory I

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 514 - Spring 2013 - Lecture 2 - Combustion Theory I

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) ≈ 0.025 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 514 - Spring 2013 - Lecture 2 - Combustion Theory I

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 514 - Spring 2013 - Lecture 2 - Combustion Theory I

Effect of g (continued)

- 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!

Effect of CO2 substitution on flammability limit composition

Effect of CO2 substitution on SL,lim/SL,adiabatic

AME 514 - Spring 2013 - Lecture 2 - Combustion Theory I

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 514 - Spring 2013 - Lecture 2 - Combustion Theory I

Comparison with experiment

- No directly comparable expts., BUT...
- Zhu, Egolfopoulos, Law (1988)
- CH4 + (0.21O2 + 0.79 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 514 - Spring 2013 - Lecture 2 - Combustion Theory I

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 514 - Spring 2013 - Lecture 2 - Combustion Theory I

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