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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 of diffusion 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
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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

Candle Flame.

8. Laminar Diffusion (Non-Premixed) Flames 3 AER 1304–ÖLG

Diffusion Flame Structure.

8. Laminar Diffusion (Non-Premixed) Flames 4 AER 1304–ÖLG

.

Diffusion Flame Regimes.

8. Laminar Diffusion (Non-Premixed) Flames 5 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

8. Laminar Diffusion (Non-Premixed) Flames 12 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

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

8. Laminar Diffusion (Non-Premixed) Flames 27 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

8. Laminar Diffusion (Non-Premixed) Flames 29 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

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

8. Laminar Diffusion (Non-Premixed) Flames 34 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

8. Laminar Diffusion (Non-Premixed) Flames 37 AER 1304–ÖLG

8. Laminar Diffusion (Non-Premixed) Flames 38 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

QFDx 1 +

ξ2

4

−2(8.16)

- When r equals zero, we get a flame length,

Lf ≈ 3

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

z = 10 mm; r = 5 mm

8. Laminar Diffusion (Non-Premixed) Flames 57 AER 1304–ÖLG

z = 33 mm; r = 0 mm

8. Laminar Diffusion (Non-Premixed) Flames 58 AER 1304–ÖLG

Predicted and Measured Results C2H4-Air Flame

-0.5 0 0.5Predicted

0

1

2

3

4

5

6

7300 1224 2148

-0.5 0 0.5Predicted

0

1

2

3

4

5

6

70.0 4.0 7.9

-0.5 0 0.5Measured

0

1

2

3

4

5

6

70.0 4.0 7.9

Temperature, K Soot volume fraction, ppm

-0.5 0 0.5Measured

0

1

2

3

4

5

6

7300 1224 2148

Temperature, K


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