Laminar diffusion Flames

8/3/2019 Laminar diffusion Flames

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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 1304LG


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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



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Candle Flame.



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Diffusion Flame Structure.



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.

Diffusion Flame Regimes.



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A Simple Approach For simple laminar diffusion flames on circularnozzles (similar to a candle flame), flame height is

mostly 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.



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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 tube

radius.

Approximating y2 by R2 yieldst = R2/2D



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- Since t = Lf/v

then,

Lf vR2

2D

- Volume flow rate

QF = vR2

so that

Lf QFD



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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 inversely

proportional to pressure, the height of the

flame is independent of pressure at givenmass flow rate.

- Flame height is proportional to volume flowrate of fuel.



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Methane diffusion flames at high pressures.



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Nonreacting Constant-Density Laminar Jet

Physical Description:

Analysis presented in the previous section is very

crude 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 of

a fluid flowing into an infinite reservoir.

Important points: basic flow and diffusional pro-cesses.



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Potential core: the effects of viscous shear and

molecular diffusion are not in effect yet; so the ve-

locity and nozzle-fluid (fuel) mass fraction remain

unchanged from their nozzle-exit values and are

uniform in this region.

In the region between the potential core and the

jet edge, both the velocity and fuel concentrationdecrease monotonically to zero at the jet edge.

Beyond the potential core the viscous shear and

diffusion effects are active across whole field ofthe jet.



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Initial jet momentum is conserved throughout the

entire flowfield.

As the jet moves into surroundings, some of the

momentum 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:



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2

0

(r, x)v2x(r, x)rdr

Momentum flow ofthe jet at any x,J

= ev2eR

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 (convection

and diffusion of fuel mass).



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Distribution of fuel mass fraction,

YF(r, x), should

be similar to dimensionless velocity distribution,

vx(r, x)/ve.

Fuel molecules diffuse radially outward accordingto Ficks law.

The effect of moving downstream is to increase

time 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:



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2 0

(r, x)vx(r, x)YF(r, x)rdr = eveR2YF,e

(8.2)

where YF,e = 1. To determine the velocity and mass fraction fields

we need to make some asumptions.

Assumptions:

1. M We = M W. P =const. T = const.: Uniform

density field.2. Species transport is by Ficks diffusion law.



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3. Momentum and species diffusivities are constant

and equal, i.e. the Schmidt Number is unity,

Sc

/D = 1

4. Diffusion is considered only in radial direction;

axial diffusion is neglected.(This may not be a good asumption very near to

the nozzle exit; since near the exit it is expected

that the axial diffusion will be significant in com-parison with the downstream locations.)



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Conservation Laws (Boundary-layer equations):

Mass Conservation:vx

x+

1

r

(vrr)

r= 0 (8.3)

Axial Momentum Conservation:

vxvxx + vr

vxr =

1

r

r rvxr (8.4) Species Conservation: For the jet fluid (fuel)

vx YF

x+ vr Y

F

r= D1

rr

rv

x

r

(8.5)



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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 and

YF, and

- one boundary condition for vr.



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Along the jet centreline, r = 0,

vr(0, x) = 0 (8.7a)

vx

r (0, x) = 0 (8.7b)YFr

(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)



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At the jet exit, x = 0, we assume uniform ax-ial velocity and fuel mass fraction, and zero else-

where:vx(r

R, 0) = ve

vx(r > R, 0) = 0 (8.7f)

YF(r

R, 0) = YF,e = 1

YF(r > R, 0) = 0 (8.7g)

Solution:

Velocity field can be obtained by assuming theprofiles to be similar.



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Intrinsic shape of the velocity profiles is the sameeverywhere in the flowfield.

Radial distribution of vx(r, x), when normalized

by the local centreline velocity vx(0, x), is a uni-versal function that depends only on the similarity

variable r/x.

- Solutions for axial and radial velocities:

vx

=3

8

Je

x 1 +2

4 2

(8.8)



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vr =

3Je

16e

1/2 1x

34

1 + 2

4 2

(8.9)

where Je is the jet initial momentum flow,

Je = ev2

eR2

(8.10)

and,

=3eJe

161/2 1

rx

(8.11)



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Axial velocity distribution in dimensionless form(substitute Eqn.8.10 into 8.8),

vxve = 0.375

eveR

xR1 1 + 24

2

(8.12)

Dimensionless centreline velocity decay obtainedby setting r = 0 (= 0),

vx,0ve = 0.375

eveR

xR1 (8.13)



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Velocity decays inversely with axial distance, andproportional to the jet Reynolds number,

Rej eveR

From Eqn.8.13, we see that the solution is not

valid near the nozzle;

- at small values of x, the dimensionless cen-

terline velocity becomes larger than unity,which is not physically correct.



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Other parameters used to characterize jets are thespreading rate and spreading angle, .

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 setting

vx/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.



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r1/2/x = 2.97

eveR = 2.97Rej1 (8.14)

tan1(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., = D.



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Then the functional form of the solution for YF isidentical to that for vx/ve,

YF =3

8

QFDx

1 + 24 2

(8.16)

where QF = veR2, volumetric flow rate of fuel.

By applying Sc = 1 to Eqn.8.16,

YF = 0.375Rej xR1 1 + 2

42 (8.17)



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Centreline values of mass fraction,

YF,0 = 0.375 Rej x

R1

(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 must

exceed the jet Reynolds number, that is,

(x/R) 0.375 Rej (8.19)



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Jet Flame Physical Description

The burning laminar fuel jet has much in commonwith our previous discussion of the non-reacting

jet.- As the fuel flows along the flame axis, it diffuses

radially outward, while the oxidizer diffuses radi-

ally inward.

- The flame surface can be defined as,

Flame Surface Locus of points where equals unity

(8.20)



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The products formed at the flame surface diffuseradially both inward and outward.

An overventilated flame is where there is more

than 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,

(r = 0, x = Lf) = 1 (8.21)



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Chemical reaction zone is quite narrow (but signif-icantly larger than laminar flame thickness).

Flame temperature distribution exhibits an annular

shape 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.



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By ignoring the effects of heat released by re-action, Eqn.8.16 provides a crude description of

flame boundaries when YF = YF,stoic.

YF =3

8

QFDx

1 +

2

4

2(8.16)

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

Lf 3

8

QFDYF,stoic (8.22)



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Flame length is proportional to volumetric flowrate of fuel.

Flame length is inversely proportional to the stoi-

chiometric fuel mass fraction. Since QF = veR

2, various combinations of veand R can yield the same flame length.

Since the diffusion coefficent D is inversely pro-portional to pressure, the height of the flame is

independent of pressure at given mass flow rate.



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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.



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Ropers Solutions and Correlations:

Circular Port:

Lf,thy =QF(T/TF)

4D 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.



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Square Port:

Lf,thy =QF(T/TF)

16D {inverf[(1 + S)0.5]}2

TTf

0.67(8.61)

Lf,expt = 1045QF(T/TF)

{inverf[(1 + S)0.5]}2(8.62)

where inverfis the inverse of error function Erf,

Erfw =2

w

0et

2

dt



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Slot BurnerMomentum Controlled:

Lf,thy =b2QF

hIDYF,stoic TTF

2

TfT

0.33

(8.63)

Lf,expt = 8.6 104 b

2QFhIYF,stoic

TTF

2

(8.64)

where b is the slot width and h is the length, and,

=1

4 inverf[1/(1 + S)]



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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 to

conditions where the oxidizer is stagnant.



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Slot BurnerBuoyancy Controlled:

Lf,thy =

94Q4FT

4

8D2

ah4T4F

1/3TfT

2/9(8.65)

Lf,expt = 2 103

4Q4FT

4

ah4T4F

1/3(8.66)

where a is the mean buoyant acceleration,

a= 0.6g TfT 1 (8.67)

and g is the gravitational acceleration.



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Slot BurnerTransition Regime:

Froude Number,

Frf (veIYF,stoic)

2

aLf (8.68)

- Froude number physically represent the ratio of

the initial jet momentum flow to the buoyant force

experienced by the flame.

Frf >> 1 Momentum-controlled (8.69a)Frf 1 Transition (mixed) (8.69b)Frf


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Note that Lf must be known a priori to establishthe appropriate regime. So it requires a trial and

error approach.

When Frf 1,

Lf,T =4

9

Lf,MLf,B

Lf,M3

1 + 3.38Lf,MLf,B 3

2/3

1

(8.70)



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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



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Soot does not form in premixed flames exceptwhen crit

The details of soot formation process in diffusion

flames is elusive Conversion of a hydrocarbon fuel with molecules

containing 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)



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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 growth

components from the gas phase

- Oxidation of agglomerates



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Smoke Point:

- An ASTM standard method to determine

sooting tendency of a liquid fuel

- Fuel flow rate is increased until the smokestarts being emitted from the flame tip of a

laminar 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



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Vinyl

radical

C2H

3

1,3 butadienyl

radical.

Vinyl acetylene.

Vinyl acetylene

radical.

Linear C6H

5

Cyclic C6H

5

(phenyl radical)

Pyrolysis/oxidative

pyr

olysisofFUEL

ACETYLENE

HYDROGEN ATOM

MOLECULAR ZONE PARTICLE ZONE

Soot Inception

Soot Surface Growth

Coagulation

Reaction Time Coordinate

Soot Particle Dia.= 1 nm to 40+ nm.

Allene

Methylacetylene



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- 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)



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OXIDATION

FUELAIRAIR

LUMINOUS FLAMEENVELOPE

PREMIXEDBLUE

FLAME

MolecularZone

ParticleZone



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0 20 40 60 80 100 120 140 160

Height Above Burner, mm

0

10

20

30

40

Prima

ryParticleDiameter,nm

C2H4 Flame

4.90 ml/min

3.85 ml/min



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z = 10 mm; r = 5 mm



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z = 33 mm; r = 0 mm



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Predicted and Measured Results

C2H4-Air Flame

-0.5 0 0.5

Predicted

0

1

2

3

4

5

6

7

300 1224 2148

-0.5 0 0.5

Predicted

0

1

2

3

4

5

6

7

0.0 4.0 7.9

-0.5 0 0.5

Measured

0

1

2

3

4

5

6

7

0.0 4.0 7.9

T t K S t l f ti

-0.5 0 0.5

Measured

0

1

2

3

4

5

6

7

300 1224 2148

T t K

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Laminar diffusion Flames

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