MAE 5310: COMBUSTION FUNDAMENTALS
Introduction to Laminar Diffusion Flames:
Non-Reacting Constant Density Laminar Jets
Mechanical and Aerospace Engineering Department
Florida Institute of Technology
D. R. Kirk
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LAMINAR DIFFUSION FLAME OVERVIEW
•
Subject of lots of fundamental research
– Applications to residential burners (cooking
ranges, ovens)
– Used to develop an understanding of how soot,
NO2, CO are formed in diffusion burning
– Mathematically interesting: transcendental
equation with Bessel functions (0th and 1st order)
• Introduce concept of conserved scalar (very
useful in various aspects of combustion and
introduced here)
•
Desire to understand flame geometry (usually desire
short flames)
– What parameters control flame size and shape
– What is the effect of different types of fuel
– Arrive at useful (simple) expression for flame
lengths for circular-port and slot burners
CO2 production in diffusion flame
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LAMINAR DIFFUSION FLAME OVERVIEW (LECTURE 1)
•
•
•
Reactants are initially separated, and reaction occurs only at the interface between fuel and
oxidizer (mixing and reaction taking place)
Diffusion applies strictly to molecular diffusion of chemical species
In turbulent diffusion flames, turbulent convection mixes fuel and air macroscopically, then
molecular mixing completes the process so that chemical reactions can take place
Orange
Blue
Full range of f
throughout
reaction zone
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NON-REACTING CONSTANT DENSITY LAMINAR JETS
• Examine non-reacting laminar jet of fluid (fuel) issuing into a infinite reservoir of
quiescent fluid (oxidizer)
– Why? Simpler case to develop understanding of basic flow field
• Physical description of jet (Reference picture on next slide)
– Potential core: effects of viscous shear and diffusion have yet to be felt
• Both the velocity profile and nozzle-fluid mass fraction remain unchanged
from their nozzle-exit values and are uniform in this region
• Similar to developing pipe flow, except that in a pipe conservation of mass
requires uniform flow to accelerate
– Between potential core and jet ‘edge’, both velocity and fuel concentration
(mass fraction) decrease monotonically to zero at edge of jet
• Beyond potential core (x > xc), effects of viscous shear and mass diffusion
are active across whole width of jet
– Initial jet momentum is conserved through entire flow field
• Jet momentum flow at any x, J = momentum flow issuing from nozzle, Je
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NON-REACTING CONSTANT DENSITY LAMINAR JETS
Centerline
velocity
decay
•
•
•
•
Radial
velocity
decay
Processes that control velocity field (convection and diffusion of momentum) are similar to processes
that control fuel concentration field (convection and diffusion of mass)
Distribution of YF(r,x) similar to distribution of ux(r,x)/ue
Because of high concentration of fuel in center of jet, fuel molecules diffuse radically outward in
accordance with Fick’s law (see assumptions page)
Effect of moving downstream is to increase time available for diffusion to take place
– Width of the region containing fuel molecules grows with axial distance, x, and centerline fuel
concentration decays
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DETAILED ANALYSIS: NON-REACTING LAMINAR JETS
• Assumptions
– Jet velocity profile is uniform at tube exit (r ≤ R)
– Molecular weights of jet and reservoir fluid are equal (MWfuel=MWair),
constant T and P, ideal gas, constant r
– Species molecular transport is by binary diffusion governed by Fick’s law
– Momentum and species diffusivities are constant and equal
• Schmidt, Sc = n/D = 1 (Recall Le = a/D)
– Only radial diffusion of momentum and species is important
• Axial diffusion is neglected
• Implies that solution only applies some distance downstream of nozzle exit
since near exit axial diffusion is quite important
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GOVERNING EQUATIONS AND BOUNDARY CONDITIONS
u x 1  u r r 

0
x r r
u
u
1   u x 
u x x  vr x  n
r

x
r
r r  r 
Boundary Layer Equations (see Schlichting or White)
Conservation of mass
Momentum
Boundary Conditions:
vr 0, x   0
u x
0, x   0
r
YF
0, x   0
r
u x , x   0
YF , x   0
u x r  R,0   ue
u x r  R,0   0
YF r  R,0   YF ,e  1
YF r  R,0   0
Along the jet centerline (r = 0)
No sources of sinks of fluid along axis
Symmetry
At large radii (r → ∞)
At jet exit (x =0) axial velocity and fuel mass
fraction are uniform
Everywhere else they are zero
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FLOW FIELD RESULTS: SIMILARITY SOLUTION
3 Je   2 
ux 
1  
8 x 
4
 3J e 

vr  
 16re 
1
2
The velocity field can be obtained by assuming the profiles to
be similar. The idea of similarity is that the intrinsic shape
of the profile is the same everywhere in the flow field
For this problem implication is that radial distribution ux(r,x),
when normalized by local centerline velocity ux(0,r), is a
function that depends only on similarity variable, r/x
3
2

1
4
x   2 2
1  
4 

Solution for axial velocity
Solution for radial velocity
 3r J 
  e e 
 16 
1
2

1 r
x
1
 r eue R  x    2 
ux
  1  
 0.375
ue

4

 R  
 r u R  x 
 0.375 e e  
ue
   R 
u x,0
contains similarity variable, r/x
2
Axial velocity in dimensionless form
1
Dimensionless centerline velocity
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CENTERLINE VELOCITY DECAY FOR LAMINAR JETS
•
•
•
•
Velocity decays inversely with axial distance and is directly proportional to jet Reynolds number, Re j
Solution is not valid near nozzle
Decay is more rapid with lower Re jets
– As Re is decreased, relative importance of initial jet momentum becomes smaller in comparison
with viscous shearing action, which slows the jet
Figure also represents decay of centerline mass fraction, YF (see next slides)
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SPREADING RATE, SPREADING ANGLE, JET HALF WIDTH
•
Other parameters are frequently used to
characterize jets
– Jet half-width, r1/2
• Radial location where jet velocity has
decayed to 1/2 of centerline value
– Spreading rate
• Ratio of the jet half-width to the axial
distance, x,
– Spreading angle, a
• Angle whose tangent is the spreading rate
•
•
•
High Reynolds number jets are narrow
Low Reynolds number jets are wide
Consistent with Reynolds number dependence of
velocity decay
r1
   2.97
 
 2.97
x
 r eue R  Re j
 r1 
1 
2 
a  tan
 x 
 
2
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CONCENTRATION FIELD SOLUTION AND RESULTS
ux
u x
u
1   u x 
 vr x  n
r

x
r
r r  r 
ux
YF
Y
1   YF 
 vr F  D
r

x
r
r r  r 
YF  Yox  1
If n/D = 1 (Lewis number unity), function form
of solution for YF is identical to what for ux/ue
3 QF   2 
YF 
1  
8 Dx 
4
QF  ueR 2
2
QF is volumetric flow rate from nozzle
1
x   
YF  0.375 Re j   1  
4
R 
YF , 0
Solution of concentration field is
mathematically similar to governing equation
for momentum conservation
x
 0.375 Re j  
R
1
2
2
Written with Rej as controlling parameter
Centerline expression
Solutions can only be applied far from nozzle
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EXAMPLES: NON-REACTING LAMINAR JETS
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•
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Part 1
– A jet of ethylene (C2H4) exits a 10 mm diameter nozzle into still air at 300 K, and 1 atm.
– Compare spreading angles and axial location where jet centerline mass fraction drops to
stoichiometric value
– Initial jet velocities of 10 cm/s and 1 cm/s, ethylene at 300 K is 1.023x10-9 N s/m2
– Answer comment:
• Low-velocity jet is much wider
• Fuel concentration of low-velocity jet decays to same value as high-velocity jet in 1/10th
distance
Part 2
– Using 1 cm/s as a baseline case, determine what nozzle exit radius is required to maintain same
flow rate if exit velocity is increased by a factor of 10 to 10 cm/s
Part 3
– Determine axial location for YF,0 = YF,stoichiometric for condition in Part 2 and compare with baseline
– Answer comment:
• The distance calculated in Part 3 is identical to the 1 cm/s case in Part 1
• Spatial fuel mass-fraction distribution depends on initial volumetric flow rate, Q, for a given
fuel (n = /r = constant)
Problem #4-40 from F. White, Viscous Fluid Flow:
– Air at 20 °C and 1 atm issues from a circular hole and forms a round laminar jet. At 20 cm
downstream of the hole the maximum jet velocity is 35 cm/s. Estimate, at this position (a) the 1%
jet thickness, (b) the jet mass flow, and (c) an appropriate Reynolds number for the jet
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LOOK AGAIN AT BUNSEN BURNER
Secondary diffusion flame
Results when CO and H
products from rich inner flame
encounter ambient air
Fuel-rich pre-mixed
inner flame
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•
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What determines shape of flame? (velocity profile, flame speed, heat loss to tube wall)
Under what conditions will flame remain stationary? (flame speed must equal speed of normal
component of unburned gas at each location)
What factors influence laminar flame speed and flame thickness (f, T, P, fuel type)
How to characterize blowoff and flashback
Most practical devices (Diesel-engine combustion) has premixed and diffusion burning
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