# Lecture 25

Mechanics

Feb 22, 2014 (7 years and 8 months ago)

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Turbulent
Combustion, Conservation
Equations
Closure

Turbulence and its effects on mass, momentum, species
and energy transport

Practical devices involve turbulent flows specifically promoted by the
design engineer to obtain efficient mixing, preheating and volumetric
reaction rates.

Practical device sizes and speeds desired by human beings and their
environment automatically lead to turbulent flows.

First principle solution of turbulence and turbulent combustion is an
unsolved grand challenge problem.

Turbulent Non
-
premixed Flames

Length and time scales of turbulence and their influence
on combustion

Turbulent Premixed Flames

See Figure 11.1 pg. 429 of Turns.
v
X

is shown as a

function of time. Imagine that in a three dimensional flow

v
Y

and
v
Z

would also fluctuate with time but together the

three components and density must satisfy the

conservation of mass equation and the equation of state.

Now consider the simpler problem of a two dimensional boundary layer over a flat plate (see pp. 438
-
449 of Turns).

Eq. 11.13 is the axial momentum equation with term
(1) : transient mass flux in the axial
direction, (2) : advection of axial mass flux by the axial
velocity, and term (4): the effect of molecular viscosity on the
axial momentum.

3/20

2
2
(1) (2) (3) (4)
x
x x x y x
v
v v v v v
t x y y
   

  
  
   
Turbulence

Equation 11.14 splits the axial velocity into its mean
and fluctuating components but does not recognize
the fluctuations in density that are omnipresent in
combustion. Therefore, equation 11.14 is highly
limiting.

In class, we will derive a version that recognizes the
fluctuating density and the fact that its product with
fluctuating velocity may not necessarily have a zero
mean (If you get a chance, practice this exercise on

Ignoring the density fluctuations and defining a
stationary flow (time derivative of mean velocity is
zero), the highly limiting equation 11.14 results.

Turbulence

2
2
(')
(') (')(') (')(')
'(')(')(')''2'''''
(')(')(')'''''''''
0;'
x x
x x x x x x y y x x
x x x x x x x x x x x x
y y x x y x y x y x x y y x
x
v v
v v v v v v v v v v
t x y y
v v v v v v v v v v v v
v v v v v v v v v v v v v v
v v
   
        
      
 
 
  
       
   
         
       

'0;'0;''0,''0;'''0
x x x x y x y x
v v v v v v v etc
 
 
    
2
2
''''
''''''''''''''
x
x x x y x x x y x
x x x x y x x y x x y x
v
v v v v v v v v v
t x y y x x
v v v v v v v v v v v v
x x y y x y
     
     

    
    
     
     
     
     
Take averages and substitute zero and non zero values

In variable density flows, density weighted or Favre averaging is used to avoid the

Complications arising from correlations involving density fluctuations.

Turbulence

The second and the third term of equation 11.13
upon separation of stream
-
wise (x) and cross
-
stream
(y) velocity components into their mean and
fluctuating values and averaging yield terms that
-
zero quantities called
“Reynolds Stresses.”

Reynolds Stress involving stream
-
wise velocity
fluctuations multiplied by themselves will generally
have positive and negative values cancelling each
other in magnitude and therefore “tau
-
xx” defined in
equation 11.18a is negligible compared to “tau
-
xy

defined in equation 11.18b resulting in equation
11.19. Turns mentions the nine Reynolds stress
components out of which only one is retained in an
axisymmetric jet mixing problem.

Turbulence

Turbulence

.0
';'(')(')''
..''0
V
t
V V V V V V V
V V
t

      

 

 

        

  

2
..0
.
V V
t
P
RT P Const T
RT

 
 

    

     
Coupling between conservation of mass and conservation of energy

Continuity equation involving average density and average velocity has a source

Turbulence and Combustion

6/20

Equation 11.20 describes momentum transfer with an
axisymmetric (fuel or premixed fuel and air) jet
problem with boundary conditions allowing for a co
-
flow of air or stagnant atmospheric air or “stagnant,”
products surrounding the axisymmetric jet.

For describing any reacting flow (premixed or non
-
premixed) corresponding conservation of mass,
species and energy equations are necessary.

The conservation of mass equation results in transient
of mean density being driven by divergence of mean
mass flux and the divergence of fluctuating mass flux
represented by the density
-
velocity correlation. The
density
-
velocity correlation is a very important
quantity in turbulent combustion studied over a couple
of decades with models like Bray Moss Libby (BML)
model covered in ME 625.

Turbulence and Combustion

7/20

The conservation of species equation results in the
transient of mean mass fraction of every species “
i

being driven by the mean and fluctuating advection
flux, mean and fluctuating diffusion flux, and mean and
fluctuating components of the reaction rate.

The conservation of energy equation results in the
transient of the mean chemical plus sensible energy
being driven by the mean and fluctuating advection,
mean and fluctuating thermal conduction/diffusion flux,
and mean and fluctuating radiation heat loss.

Fluctuating temperature and species concentrations
loss being different than the quantities calculated
based on mean temperature and mean species
concentrations.

The significance of these effects is still discussed.

Non
-
premixed
Jet Flame Heights:

Turbulent Flame Analog

For a laminar jet, the flame length is determined by the
time that it takes for the co
-
flow fluid to diffuse to the jet
centerline:

v
f
e
L
t

The mean squared displacement of the co
-
flow molecules
due to
turbulent
diffusion
during that time must be equal
to the square of the jet radius:

2
2 2
2 2
2
v
v
f j e
F
j f
e
L r
Q
t r L

     
D D
D D
8/20

Non
-
premixed
Jet Flame
Heights

For turbulent flows, the molecular
viscosity
is replaced by the
turbulent eddy viscosity (see discussion of turbulent mixing
length, Turns, p.
427
-
450):

2 2
,
,
,
v v
v
v v
j e j e
f t
t m rms
m j rms e
f t j
r r
L
r
L r

 

 
 
9/20

Non
-
premixed Turbulent Jet
Flame Heights:

Effects of Buoyancy

For
buoyant turbulent
flames, relative importance of initial jet
momentum and buoyancy
is characterized
by Froude number:

3/2
1/4 1/2
v
Fr
e stoich
f
f j
e
f
T g d
T

 

   
   
   
where:

v
j f
stoich
e
d jet exit diameter T flame temperature rise
f stoich mixture fraction
uniform exit velocity
  

10/20

Nonpremixed

Turbulent Jet Flame
Heights:
Kalghati

Correlations

Nonpremixed

Turbulent Jet Flame
Heights:
Kalghatgi

Correlations

*
1/2
*
*
/
f stoich f stoich
j
j e
j
L f L f
L
d
d
d momentum diameter
 

 

2/5
*
1/5
2
*
13.5
5
1 0.07
23 5
f
f
f
f
Fr
L for Fr
Fr
L for Fr
 

 

The dimensionless flame length
can be correlated with the
Froude number:

Non
-
premixed
Turbulent Jet Flame

Non
-
premixed flames are often used in furnaces and the
parameter.

integrated over the load surface area yields the energy
received by the load that can be divided by the fuel flow rate
multiplied by the heating value of the fuel (see denominator of
equation 13.13) to yield the radiant fraction (see LHS of
equation 13.13).

The energy received by the load is equal to the net energy
emitted by the flame and this energy is used to define the
Planck mean absorption coefficient, effective radiation
temperature and effective radiating volume of the flame in
equations 13.14 and 13.15.

Non
-
premixed
Turbulent Jet Flame Liftoff
and Blowout:
Kalghatgi

Correlations

Kalghatgi view of turbulent jet diffusion flame
stabilization
:

u
max
(
x=h
)

u
(
r
)

x

=
h

= liftoff
height

Fuel

Oxidizer

Oxidizer

r

S
T
= u(r
stoich
)

S
T

Rich Limit

Rich Limit

Lean Limit

Lean Limit

r
stoich

r
stoich

Non
-
premixed
Turbulent Jet Flame Liftoff :
Kalghatgi

Correlation

For flame liftoff, Kalghatgi (CST
41
, 17
-
29, 1984) has
developed the following correlation:

1.5
,max
,max
v
50 Re
e L
e e
h
e L
S h
S

 

 
 
 
 
 
 
 
 
where
h

= flame liftoff height

Non
-
premixed
Turbulent Jet Flame
Blowout:
Kalghatgi

Correlation

For flame blowout, Kalghatgi (CST
26
, 233
-
239, 1981)
proposes the following correlation:

where

1.5
6
,max
v
0.017 Re 1 3.5 10 Re
e e
H H
L
S

 
 
  
 
 
 
 
 
,max
1/2
,
,
Re
4 5.8
e L
H
e
F e
e
j
F stoich
S H
Y
H d
Y

 
 
 
 
 
 
 
 
Non
-
premixed
Turbulent Jet Flame
Blowout:
Kalghatgi

Correlation

From isothermal turbulent jet mixing theory,
H

is the distance
along the jet centerline where the mixture fraction drops to the
stoichiometric value
f
stoich
. Flame blowout occurs when the
liftoff height
h

is approximately equal to 0.7
H
.

Also see Pitts (Combust. Flame
76
, 197
-
212, 1989).

Turbulent Premixed Flames

Laminar premixed flames are limited by the amount of fuel +
air mixture they can burn per unit area. Turbulence increases
the area over which the mixture can be burnt by wrinkling the
flame.

The increase in the mass burning rate by flame wrinkling
leads to increase in the volumetric combustion rate even if the
flame speed remains fixed at the value given by the laminar
flame speed.

The increase in flame surface area per unit volume is one
way to keep track of the effect of turbulence on the premixed
flame fuel consumption and heat release rates.

Another way is to define an equivalent turbulent flame speed
over a fixed mean surface area (see equation 12.1 on page
457 and Example 12.1 on page 458).

Turbulent Premixed Flame Regimes

Premixed jet flames wrinkled by turbulence (see Figure 12.6
page 459) define a flame brush.

In spark ignition engine, a spark kernel may be small and
laminar at the time of ignition but may grow to be turbulent
and have significantly higher surface area than that of an
equivalent spherical surface and may lead to very high
propagation speed allowing high engine RPM.

Visually compare the mass burned by the flame in the first 10
pictures and the last ten pictures in Figure 12.7. At higher
speeds the wrinkling of the flame may finally lead to breaking
into many spherical wrinkled pockets that burn with even
higher overall speed and lead to complete combustion.

Turbulent Premixed Flame Regimes

Flame thickness less than Kolmogorov scale leads to
wrinkled laminar flame regime. (Reactions are very fast and
the flame is so thin that any fluctuation in local properties
does not affect the flames effects on the reactants, its speed,
its
t
emperature etc
.
)

Flame thickness greater than Kolmogorov scale but less than
integral scale leads to flamelets in eddies. Flame thickness is
greater than the scales of the smallest fluctuations but less
than the scale of the larger fluctuations. The preheating and
reaction rate processes inside of the flame are affected by the
scalar fluctuations and a correction to the overall volumetric
reaction rate is needed on the basis of the effects of these
fluctuations.

Flame thickness greater than integral scale leads to
distributed reactions (Reactions are very slow)