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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.
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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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