Cesar Olmedo
October-February 2010
Laboratory Work
Since December 2009 the supersonic combustion team has progressed in
the developed of prototype and mathematical models. I have contributed to
development of the mixing theory by developing models mixing length. These
calculations are based on the Prandtl mixing theory. In the design of the super
sonic diffuser our team will use a pre mixed design to ignite the hydrogen fuel to
produce combustion.
In supersonic combustion the combustion takes place at supersonic
speeds this make any type of deflagration very difficult if not impossible. Ignition
and flame holding or stabilization of flame is the primary problems to solve. In
this type of combustion; ignition is mainly dependent on how good your fuel and
air mix at molecular level. That means that the fuel and air mixture must be close
to its stoichiometric ratio as possible and have an equivalence ratio of .2 to 2.
Mixing at supersonic speed is very difficult due to the high velocities that
are encountered. To get a better understanding of how air and fuel mix we can
simplify the concept of mixing, if we breakdown mixing of two streams ;one of air
and the other of fuel in a tube if the streams are separated by a splitter plate in
the middle of the tube as shown in . Fig 1 we can see that a mixing layer will
develop further down the tube.
U1 Air Flow
U2 Fuel Flow
Mixing
Layer
δm
Fig 1
If we assume that both air and fuel co-flow in the tube the mixing effect is due to
lateral transport between air and fuel boundary layer or molecular diffusion of
both species. In FIG 2 it can be seen how diffusion occurs between the fuel and
air.
Ya
U1 Air
U2
Ya
δm
Fuel
Fig2
Because the molecules of air and fuel have different sizes and different densities,
this allows them to diffuse into each other which produce a diffusion flame.
Hydrogen which is what we use for fuel in our supersonic diffuser is 14
times less dense than air which is what allows hydrogen to diffuse in the air. As
both air and fuel co flow down the pipe the diffusion of air into fuel continues until
a balance is reached and the fuel will begin to diffuse back into the fuel source
stream. This is why in many diffusion combustion the flame only burns where the
air and fuel mixture have achieved the correct stoichiometric and equivalence
ratio.
When air and fuel have diffused into each other, to a near stoichiometric point
and the mixture can ignite. The distance it took to mix to that point is called Xm
which is the distance required to reach the stoichiometric point. Fig 3
Ya
Ya
U1
δm
U2
Fig 3
X=0
Xm
When air and fuel streams are no longer co flowing and if the velocity of air is
much greater than that hydrogen. Then both streams will become turbulent and
produce vortices which will engulf both stream and steam roll both streams into
each other.
Xm
Fig 4
These vortices are turbulent and will not form exactly in front of the splitter
but rather occur downstream. This turbulent flow and vortices help reduce the
mixing length using these concepts we can re draw a new mixing layer for
different flowing streams and turbulent vortices flow. Fig 4
Now that there is a concept that describes turbulent flow we need to incorporate
compressible flow.
We will consider Prandtl’s mixing theory approximate solution for the time mean
growth rate velocity profile in a turbulent shear layers.
This assumption assumes equal density and constant pressure flow and
velocities, using these assumptions the time-mean velocity profile is given
approximately by
3
u
 1  r   y   y  
1 
3   4  
uc
 1  r        
Where  the local shear layer width, which varies as
1 r 
x
1 r 
  6B2 
B is an empirical constant that can only be determine by experiments
To improve results based on experimental results and testing another equation
can be used that contains the same form but different constant, this constant has
a range which makes calculation a little more controlled. The constant ranges
from .25 to .45
1 r 
x
1 r 
  C 
This equation was developed by Dimotakis, “High speed flight propulsion
systems”
If we consider the effects of different densities on the shear layer growth, we
assume different densities that are constant then we can develop a ratio s and
s
2
1
use the stagnation pressure equation to express static and dynamic pressure we
obtain
p1 
u1  u1  uc
1
1
1 (u1 ) 2  p 2   2 (u1 ) 2
2
2
u2  uc  u2
and if we solve for the ratio of densities we obtain
s1 / 2
1

u
 2  1
1 u2
Solving for uc
uc 
u1  s 2 u2
1
1 s2
if we substitute dynamic pressure with Mach speed we obtain
1
2
 1 1
  1
2   1 1
2   2 1
M C1 
 1  2
M C2 
1 
2
2




U 1  U c
U  U 2
M C1 
M C2  c
a1
a2
We can determine the effects of compressibility on the shear layer width. Since
know we can incorporate the compressibility effect into the shear layer equation
1 r 
x
1 r 
  C 
Shear width without compressibility effects

1  s1 / 2 r 
1/ 2

m
 1  r  1  s r 
1  s1 / 2 r  Shear with compressibility effects


 C 
1




1/ 2


x
 1  s r  2  1  1.291  r  

1  r  
when experimental test were performed using this equations it was determine
that shear width would be compress to 0 at mach speed of 2, which can’t be true
since hypersonic flight has reached higher speeds. Dimotakis developed and
empirical compressibility correct factor
f ( M c1 )  .2  .8e 3 M c1
2

1  s1 / 2 r 
1/ 2

m
 1  r  1  s r 
1  s1 / 2 r  this is the correct


 .49 f ( M c1 )C 
1




1/ 2


x
 1  s r  2  1  1.291  r  
1  r  

compressibility shear width layer equation. Using this equation we can determine
a location for an igniter since we can calculate mixing length based on our mach
speed. Substitution of shear layer width in the mean velocity will allow us to
determine the shear stress when we multiply it by dynamic viscosity.
3
u
 1  r   y   y  
1 
3   4  
uc
 1  r        
3
  y  y 

 1  r 
3

4
     uc
 1  r 
     

   u 1 
Using this equation we can determine the mixing efficiency if we can calculate
the mixing length by using the following equations
LmF
 0.179Cm e1.720 , 0  1
b
Lm A
 3.333Cm e 1.2040 , 0  1
b
These equations will determine the efficiency of the mixture of the mixture based
on the mixing length developed in shear compressibility equation.
Future work for this project
1. Flame holder design refinement currently have one design with CFD
profiles , but there is still further designs that need to be tested
2. Fabrication of prototype using stainless steel sheets of 1/16th thickness
3. Finalize diffuser dimensions and received Dr Wu Approval
4. Fix supersonic wind tunnel
Vibration
For the vibrations project, there are two possible engines which could be used
to extract data. The mini-turbine produces significant vibration, but this is mainly
due to the turbo machinery – the compressor and the turbine, not the combustion
or pressure fluctuations produced by the gas exhaust flow. The combustion
chamber has no such turbo machinery, but has much less vibration. These two
options will be weighed to decide the future of the vibration project.
Direction from faculty includes capturing the vibration response for the
mini-turbine. During testing, significant vibration was recorded from the miniturbine and the correlation between resonant frequencies and angular velocity of
the turbine was apparent. Validation on this data is needed, however.