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8. Laminar Diffusion Flames(Laminar Non-Premixed Flames)
• In a diffusion flame combustion occurs at the in-terface between the fuel gas and the oxidant gas,and the burning depends more on rate of diffu-sion of reactants than on the rates of chemical pro-cesses involved.
• It is more difficult to give a general treatment ofdiffusion flames, largely because no simple, mea-surable parameter, analogous to the burning veloc-ity in premixed flames, can be defined.
8. Laminar Diffusion (Non-Premixed) Flames 1 AER 1304–ÖLG
• Used in certain applications (e.g., residential gasappliances).- mostly partially-premixed flames
• Used in fundamental flame research.• Primary concern in design is the flame geometry.• Parameters that control the flame shape,
- Fuel flow rate- Fuel type- Other factors
8. Laminar Diffusion (Non-Premixed) Flames 2 AER 1304–ÖLG
A Simple Approach• For simple laminar diffusion flames on circularnozzles (similar to a candle flame), flame height ismostly used to characterize the flame.
• For simple treatments, reaction zone is defined asthe region where the fuel and air mixture is stoi-chiometric. This assumption is, of course, clearlyincorrect as reaction will be occuring over an ex-tremely wide range of fuel/air ratios.
• Diffusion process is rate-determining so that rateof reaction is directly related to the amounts offuel and oxidant diffusing into the reaction zone.
8. Laminar Diffusion (Non-Premixed) Flames 6 AER 1304–ÖLG
• For a simple conical laminar diffusion flame,molecular diffusion is considered only in radialdirection.
• Average square displacement (Einstein diffusionequation) is given by
y2 = 2Dt• Height of the flame is taken as the point where theaverage depth of penetration is equal to the tuberadius.
• Approximating y2 by R2 yieldst = R2/2D
8. Laminar Diffusion (Non-Premixed) Flames 7 AER 1304–ÖLG
- Sincet = Lf/v
then,
Lf ≈ vR2
2D- Volume flow rate
QF = vπR2
so thatLf ≈ QF
πD
8. Laminar Diffusion (Non-Premixed) Flames 8 AER 1304–ÖLG
• Although very crude, this approximation permitscertain predictions:- At a given flow rate, flame height is indepen-dent of the burner diameter.
- Since the diffusion coefficent D is inverselyproportional to pressure, the height of theflame is independent of pressure at givenmass flow rate.
- Flame height is proportional to volume flowrate of fuel.
8. Laminar Diffusion (Non-Premixed) Flames 9 AER 1304–ÖLG
Methane diffusion flames at high pressures.
8. Laminar Diffusion (Non-Premixed) Flames 10 AER 1304–ÖLG
Nonreacting Constant-Density Laminar Jet
Physical Description:• Analysis presented in the previous section is verycrude and provides only very qualitative featuresof laminar diffusion flames.
• To develop an understanding of the reacting lami-nar jet, we start with a nonreacting laminar jet ofa fluid flowing into an infinite reservoir.
• Important points: basic flow and diffusional pro-cesses.
8. Laminar Diffusion (Non-Premixed) Flames 11 AER 1304–ÖLG
• Potential core: the effects of viscous shear andmolecular diffusion are not in effect yet; so the ve-locity and nozzle-fluid (fuel) mass fraction remainunchanged from their nozzle-exit values and areuniform in this region.
• In the region between the potential core and thejet edge, both the velocity and fuel concentrationdecrease monotonically to zero at the jet edge.
• Beyond the potential core the viscous shear anddiffusion effects are active across whole field ofthe jet.
8. Laminar Diffusion (Non-Premixed) Flames 13 AER 1304–ÖLG
• Initial jet momentum is conserved throughout theentire flowfield.
• As the jet moves into surroundings, some of themomentum is transferred to air, decreasing thevelocity of the jet.
• Along the jet increasing quantities of air are en-trained into the jet as it proceeds downstream.
- We can express this mathematically using an inte-gral form of momentum conservation:
8. Laminar Diffusion (Non-Premixed) Flames 14 AER 1304–ÖLG
2π∞
0
ρ(r, x)v2x(r, x)rdr
Momentum flow ofthe jet at any x,J
= ρev2eπR
2
Momentum flow issuingfrom the nozzle,Je
(8.1)
where subscript e specifies the nozzle exit condi-tions.
• The process that control the diffusion and convec-tion of momentum are similar to the processes thatcontrol the fuel concentration field (convectionand diffusion of fuel mass).
8. Laminar Diffusion (Non-Premixed) Flames 15 AER 1304–ÖLG
• Distribution of fuel mass fraction, YF(r, x), shouldbe similar to dimensionless velocity distribution,vx(r, x)/ve.
• Fuel molecules diffuse radially outward accordingto Fick’s law.
• The effect of moving downstream is to increasetime available for diffsuion.
• The width of the region containing fuel growswith x and centerline fuel concentration decays.
• The mass of fluid issuing from nozzle is con-served:
8. Laminar Diffusion (Non-Premixed) Flames 16 AER 1304–ÖLG
2π∞
0
ρ(r, x)vx(r, x)YF(r, x)rdr = ρeveπR2YF,e
(8.2)
where YF,e = 1.• To determine the velocity and mass fraction fieldswe need to make some asumptions.
Assumptions:1. MWe =MW∞. P =const. T = const.: Uniformdensity field.
2. Species transport is by Fick’s diffusion law.
8. Laminar Diffusion (Non-Premixed) Flames 17 AER 1304–ÖLG
3. Momentum and species diffusivities are constantand equal, i.e. the Schmidt Number is unity,
Sc ≡ ν/D = 1
4. Diffusion is considered only in radial direction;axial diffusion is neglected.(This may not be a good asumption very near tothe nozzle exit; since near the exit it is expectedthat the axial diffusion will be significant in com-parison with the downstream locations.)
8. Laminar Diffusion (Non-Premixed) Flames 18 AER 1304–ÖLG
Conservation Laws (Boundary-layer equations):• Mass Conservation:
∂vx∂x
+1
r
∂(vrr)
∂r= 0 (8.3)
• Axial Momentum Conservation:vx∂vx∂x
+ vr∂vx∂r
= ν1
r
∂
∂rr∂vx∂r
(8.4)
• Species Conservation: For the jet fluid (fuel)vx∂YF∂x
+ vr∂YF∂r
= D1r
∂
∂rr∂vx∂r
(8.5)
8. Laminar Diffusion (Non-Premixed) Flames 19 AER 1304–ÖLG
• In addition, we have,
YOx = 1− YF (8.6)
Boundary Conditions:• To solve Eqns.8.3-8.5 for the unknown functions:
- vx(r, x), vr(r, x), and YF(r, x)requires,- three boundary conditions each for vx andYF, and
- one boundary condition for vr.
8. Laminar Diffusion (Non-Premixed) Flames 20 AER 1304–ÖLG
• Along the jet centreline, r = 0,vr(0, x) = 0 (8.7a)
∂vx∂r(0, x) = 0 (8.7b)
∂YF∂r
(0, x) = 0 (8.7c)
where the last two result from symmetry.• At large radii (r →∞),
vx(∞, x) = 0 (8.7d)
YF(∞, x) = 0 (8.7e)
8. Laminar Diffusion (Non-Premixed) Flames 21 AER 1304–ÖLG
• At the jet exit, x = 0, we assume uniform ax-ial velocity and fuel mass fraction, and zero else-where:
vx(r ≤ R, 0) = vevx(r > R, 0) = 0
(8.7f)
YF(r ≤ R, 0) = YF,e = 1YF(r > R, 0) = 0
(8.7g)
Solution:• Velocity field can be obtained by assuming theprofiles to be similar.
8. Laminar Diffusion (Non-Premixed) Flames 22 AER 1304–ÖLG
• Intrinsic shape of the velocity profiles is the sameeverywhere in the flowfield.
• Radial distribution of vx(r, x), when normalizedby the local centreline velocity vx(0, x), is a uni-versal function that depends only on the similarityvariable r/x.
- Solutions for axial and radial velocities:
vx =3
8π
Jeµx
1 +ξ2
4
−2(8.8)
8. Laminar Diffusion (Non-Premixed) Flames 23 AER 1304–ÖLG
vr =3Je16πρe
1/21
x
ξ − ξ3
4
1 + ξ2
4
2 (8.9)
where Je is the jet initial momentum flow,
Je = ρev2eπR
2 (8.10)
and,
ξ =3ρeJe16π
1/21
µ
r
x(8.11)
8. Laminar Diffusion (Non-Premixed) Flames 24 AER 1304–ÖLG
• Axial velocity distribution in dimensionless form(substitute Eqn.8.10 into 8.8),
vxve= 0.375
ρeveR
µ
x
R
−11 +
ξ2
4
−2
(8.12)
• Dimensionless centreline velocity decay obtainedby setting r = 0 (ξ = 0),
vx,0ve
= 0.375ρeveR
µ
x
R
−1(8.13)
8. Laminar Diffusion (Non-Premixed) Flames 25 AER 1304–ÖLG
• Velocity decays inversely with axial distance, andproportional to the jet Reynolds number,
Rej ≡ ρeveR
µ
• From Eqn.8.13, we see that the solution is notvalid near the nozzle;- at small values of x, the dimensionless cen-terline velocity becomes larger than unity,which is not physically correct.
8. Laminar Diffusion (Non-Premixed) Flames 26 AER 1304–ÖLG
• Other parameters used to characterize jets are thespreading rate and spreading angle, α.
• We introduce jet half-width, r1/2.- Half-width: radial location where jet velocity hasdecayed to one-half of its centreline value.
• An expression for r1/2 can be derived by settingvx/vx,0 to be one half and solving for r.
• Jet spreading rate= r1/2/x.• Jet spreading angle is the angle whose tangent isthe spreading rate.
8. Laminar Diffusion (Non-Premixed) Flames 28 AER 1304–ÖLG
r1/2/x = 2.97µ
ρeveR= 2.97Rej
−1 (8.14)
α ≡ tan−1(r1/2/x) (8.15)
• High-Rej jets are narrow, while low-Rej jets arewide.
• Comparing Eqns.8.4 and 8.5, we see that YFplays the same mathematical role as vx/ve, if theSchmidt number is unity, i.e., ν = D.
8. Laminar Diffusion (Non-Premixed) Flames 30 AER 1304–ÖLG
• Then the functional form of the solution for YF isidentical to that for vx/ve,
YF =3
8π
QFDx 1 +
ξ2
4
−2(8.16)
where QF = veπR2, volumetric flow rate of fuel.• By applying Sc = 1 to Eqn.8.16,
YF = 0.375Rejx
R
−11 +
ξ2
4
−2(8.17)
8. Laminar Diffusion (Non-Premixed) Flames 31 AER 1304–ÖLG
• Centreline values of mass fraction,
YF,0 = 0.375 Rejx
R
−1(8.18)
• Again, it should be noted that the solutions arevalid far from the nozzle. The dimensionless dis-tance downstream where the solution is valid mustexceed the jet Reynolds number, that is,
(x/R) ≥ 0.375 Rej (8.19)
8. Laminar Diffusion (Non-Premixed) Flames 32 AER 1304–ÖLG
Jet Flame Physical Description
• The burning laminar fuel jet has much in commonwith our previous discussion of the non-reactingjet.
- As the fuel flows along the flame axis, it diffusesradially outward, while the oxidizer diffuses radi-ally inward.
- The “flame surface” can be defined as,
Flame Surface ≡ Locus of points whereΦ equals unity
(8.20)
8. Laminar Diffusion (Non-Premixed) Flames 33 AER 1304–ÖLG
• The products formed at the flame surface diffuseradially both inward and outward.
• An overventilated flame is where there is morethan enough oxidizer in the immediate surround-ings to continuously burn the fuel.
• Underventilated flame is the opposite of above.• Flame length for an overventilated flame is deter-mined at the axial location where,
Φ(r = 0, x = Lf ) = 1 (8.21)
8. Laminar Diffusion (Non-Premixed) Flames 35 AER 1304–ÖLG
• Chemical reaction zone is quite narrow (but signif-icantly larger than laminar flame thickness).
• Flame temperature distribution exhibits an annularshape until the flame tip is reached.
• In the upper regions, the bouyant forces are impor-tant.
• As a result, the jet accelerates narrowing theflame.
• The narrowing of the flow increases the fuel con-centration gradients, dYF/dr, thus enhancing dif-fusion.
8. Laminar Diffusion (Non-Premixed) Flames 36 AER 1304–ÖLG
• By ignoring the effects of heat released by re-action, Eqn.8.16 provides a crude description offlame boundaries when YF = YF,stoic.
YF =3
8π
QFDx 1 +
ξ2
4
−2(8.16)
- When r equals zero, we get a flame length,
Lf ≈ 3
8π
QFDYF,stoic (8.22)
8. Laminar Diffusion (Non-Premixed) Flames 39 AER 1304–ÖLG
• Flame length is proportional to volumetric flowrate of fuel.
• Flame length is inversely proportional to the stoi-chiometric fuel mass fraction.
• Since QF = veπR2, various combinations of ve
and R can yield the same flame length.• Since the diffusion coefficent D is inversely pro-portional to pressure, the height of the flame isindependent of pressure at given mass flow rate.
8. Laminar Diffusion (Non-Premixed) Flames 40 AER 1304–ÖLG
Historical Theoretical Formulations:• Burke and Schumann (1928)
- constant velocity field parallel to flame axis.- reasonable predictions of Lf for round burn-ers.
• Roper and Roper et al (1977)- relaxed single constant velocity assumption.- provides extremely good predictions.- matched by experimental results/correlations.- round and slot-burners.
8. Laminar Diffusion (Non-Premixed) Flames 41 AER 1304–ÖLG
Roper’s Solutions and Correlations:Circular Port:
Lf,thy =QF(T∞/TF)
4πD∞ ln(1 + 1/S)T∞Tf
0.67
(8.59)
Lf,expt = 1330QF(T∞/TF)ln(1 + 1/S)
(8.60)
where S is stoichiometric molar oxidizer-fuel ra-tio, D∞ mean diffusion coefficient of oxidizer atT∞, TF and Tf are fuel stream and mean flametemperatures, respectively.
8. Laminar Diffusion (Non-Premixed) Flames 42 AER 1304–ÖLG
Square Port:
Lf,thy =QF(T∞/TF)
16D∞ {inverf[(1 + S)−0.5]}2T∞Tf
0.67
(8.61)
Lf,expt = 1045QF(T∞/TF)
{inverf[(1 + S)−0.5]}2 (8.62)
where inverf is the inverse of error function Erf,
Erfw =2√π
w
0
e−t2
dt
8. Laminar Diffusion (Non-Premixed) Flames 43 AER 1304–ÖLG
Slot Burner–Momentum Controlled:
Lf,thy =bβ2QF
hID∞YF,stoicT∞TF
2TfT∞
0.33
(8.63)
Lf,expt = 8.6 · 104 bβ2QFhIYF,stoic
T∞TF
2
(8.64)
where b is the slot width and h is the length, and,
β =1
4× inverf[1/(1 + S)]
8. Laminar Diffusion (Non-Premixed) Flames 44 AER 1304–ÖLG
I is the ratio of actual initial momentum flowfrom the slot to that of uniform flow,
I =Je,actmFve
For uniform flow I = 1. For a fully developedflow, assuming parabolic exit velocity, I = 1.5.
• Equations 8.63 and 8.64 are anly applicable toconditions where the oxidizer is stagnant.
8. Laminar Diffusion (Non-Premixed) Flames 45 AER 1304–ÖLG
Slot Burner–Buoyancy Controlled:
Lf,thy =9β4Q4FT
4∞
8D2∞ah4T 4F
1/3TfT∞
2/9
(8.65)
Lf,expt = 2 · 103 β4Q4FT4∞
ah4T 4F
1/3
(8.66)
where a is the mean buoyant acceleration,
a ∼= 0.6g TfT∞− 1 (8.67)
and g is the gravitational acceleration.
8. Laminar Diffusion (Non-Premixed) Flames 46 AER 1304–ÖLG
Slot Burner–Transition Regime:• Froude Number,
Frf ≡ (veIYF,stoic)2
aLf(8.68)
- Froude number physically represent the ratio ofthe initial jet momentum flow to the buoyant forceexperienced by the flame.
Frf >> 1 Momentum-controlled (8.69a)Frf ≈ 1 Transition (mixed) (8.69b)Frf << 1 Buoyancy-controlled (8.69c)
8. Laminar Diffusion (Non-Premixed) Flames 47 AER 1304–ÖLG
• Note that Lf must be known a priori to establishthe appropriate regime. So it requires a trial anderror approach.
• When Frf ≈ 1,
Lf,T =4
9Lf,M
Lf,BLf,M
3
× 1 + 3.38
Lf,MLf,B
3 2/3
− 1(8.70)
8. Laminar Diffusion (Non-Premixed) Flames 48 AER 1304–ÖLG
Soot Formation in Diffusion Flames:• Fuel type
- Fuel chemical structure and composition• Dilution
- Inert or reactive diluents• Turbulence
- Turbulence time versus chemical time• Temperature• Pressure
8. Laminar Diffusion (Non-Premixed) Flames 49 AER 1304–ÖLG
• Soot does not form in premixed flames exceptwhen Φ ≥ Φcrit
• The details of soot formation process in diffusionflames is elusive
• Conversion of a hydrocarbon fuel with moleculescontaining a few carbon atoms into a carbona-ceous agglomerate containing some millions ofcarbon atoms in a few milliseconds.
• Transition from a gaseous to solid phase• Smallest detectable solid particles are about 1.5nm in diameter (about 2000 amu)
8. Laminar Diffusion (Non-Premixed) Flames 50 AER 1304–ÖLG
• Soot formation involves a series of chemical andphysical processes:- Formation and growth of large aromatic hy-drocarbon molecules leading to soot incep-tion, i.e, transition to first solid particles (pri-mary particles)
- Surface growth and coagulation of primaryparticles to agglomerates
- Growth of agglomerates by picking up growthcomponents from the gas phase
- Oxidation of agglomerates
8. Laminar Diffusion (Non-Premixed) Flames 51 AER 1304–ÖLG
• Smoke Point:- An ASTM standard method to determinesooting tendency of a liquid fuel
- Fuel flow rate is increased until the smokestarts being emitted from the flame tip of alaminar flame on a standard burner
- Greater the fuel flow rate (height of theflame), the lower is the sooting propensity
- Generally used for aviation fuel specifications- Dependent on the fuel chemical composition
8. Laminar Diffusion (Non-Premixed) Flames 52 AER 1304–ÖLG
VinylradicalC2H3
1,3 butadienylradical.
Vinyl acetylene.
Vinyl acetyleneradical.
Linear C6H5
Cyclic C6H5(phenyl radical)
Pyr
olys
is /
oxid
ativ
e py
roly
sis
of F
UE
L
ACETYLENE
HYDROGEN ATOM
MOLECULAR ZONE PARTICLE ZONE
Soot Inception
Soot Surface Growth
CoagulationReaction Time Coordinate
Soot Particle Dia.= 1 nm to 40+ nm.
Allene
Methylacetylene
8. Laminar Diffusion (Non-Premixed) Flames 53 AER 1304–ÖLG
- fuel tube = 11 mm; fuel flow rate = 3.27 cm3/s- air nozzle = 100 mm; air flow rate = 170 L/min- visible flame height = 67 mm (Fuel: C2H4)
8. Laminar Diffusion (Non-Premixed) Flames 54 AER 1304–ÖLG
OXI
DAT
ION
FUELAIRAIR
LUMINOUS FLAMEENVELOPE
PREMIXEDBLUE
FLAME
MolecularZone
ParticleZone
8. Laminar Diffusion (Non-Premixed) Flames 55 AER 1304–ÖLG
0 20 40 60 80 100 120 140 160
Height Above Burner, mm
0
10
20
30
40
Prim
ary
Par
ticle
Dia
met
er, n
m
C2H4 Flame
4.90 ml/min
3.85 ml/min
8. Laminar Diffusion (Non-Premixed) Flames 56 AER 1304–ÖLG
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