AE4451 Propulsion
Spring 2002
Assignment 1: Tradeoff between shocks and heat addition losses in the design of engines for high-speed flight.
Deadlines:
In
this assignment we will use concepts from your Introduction to AE, and
Thermodynamics and Compressible Flow courses to investigate the design of
engines for flight at very high speeds.
In
any jet propulsion system, there are 4 steps:
Lets
focus on steps 1 and 2 for the case of an engine which must operate on a vehicle
flying at very high speed (Mach numbers well above 2, and going up to, say, 8).
Designers can either declerate the air entering the engine, to a low subsonic
Mach number, and then add the heat – or they can decide to keep the flow
supersonic and add heat to the supersonic flow. Ramjet engines which use the latter approach are called supersonic-combustion ramjets or
“scramjets”. We are going to see some of the arguments for and against each
approach, and see which approach works
better in each Mach number and altitude range. All the reasoning below is based
on concepts and methods which you
studied in AE1350 and AE3450.
You
know from AE1350 and 3450 that
a)
if
supersonic flow is slowed down to subsonic speeds, shocks are encountered. The “strength” of each shock depends on
the “normal Mach number” upstream of each shock, and through each such shock,
there is a loss in stagnation pressure – which translates to drag.
b)
Because
of the above, engine designers try to slow down the flow through several
oblique shocks followed by the inevitable final normal shock
c)
If
the aircraft only had to fly at one altitude, speed and angle of attack, it
might be possible to design a supersonic inlet where the flow gets decelerated
entirely “isentropically” – the equivalent of an infinite number of shocks,
each of essentially zero strength. However, such an inlet would perform
terribly at other conditions.
d)
Thus,
engines might have inlets which create at best, 2 oblique shocks plus one
normal shock. The geometry of the inlet may be variable so that the normal Mach
number upstream of each shock is kept the same – the first oblique shock, which
occurs at the flight Mach number, is at a shallow slope, the next a bit closer
to normal, etc.
e)
One
solution to the above drag-creation, might be to NOT slow the flow down in
subsonic speeds, but add heat (which is turned into work by the engine)
directly to the supersonic flow. There
are severe difficulties with igniting and
holding a flame lit in a supersonic flow, but we’ll ignore those for
now, and assume that heat can be added somehow.
f)
If
heat is added to a moving fluid, there is a loss in stagnation pressure. This
is calculated using the “Rayleigh Line Flow” theory which you studied in
AE3450. The loss increases as the Mach number increases.
g)
As
heat is added to a flow, the Mach number goes towards 1.0, according to the
Rayleigh Line theory. So, supersonic flow comes down in Mach number towards 1.0
h)
The
amount of heat that can be added to a supersonic flow is determined by the
choking condition: If the Mach number reaches 1.0 inside a duct, that’s
obviously the limit for supersonic heat addition – any more heat addition will
probably cause a normal shock.
i)
On
the other hand, if the flow is decelerated to subsonic speeds, heat can be
added until the duct “chokes” – if the material can stand the temperature.
j)
Thus
the other limitation on heat addition, whether the heat addition is subsonic or
supersonic, is the wall temperature. In both subsonic and supersonic flow
cases, we will assume that the limiting temperature is the static temperature
of the flow. If you can’t add any heat without exceeding the temperature limit
or choking the flow, then you can’t fly at this condition because the engine
cannot produce any thrust there, as explained in AE1350.
Your assignment is to develop a model to compare the
options of subsonic heat addition, and supersonic heat addition.
1.
In
each case, assume that heat is added, until the flow chokes (Mach number
reaches 1.0 by the Rayleigh Line calculation) or the temperature limit is
reached.
2.
Assume
that the molecular weight of the air is unchanged from its standard value,
but the ratio of specific heats is
1.33.
3.
Pick
a combination of flight Mach number (e.g. 3.0) and altitude (e.g. 15,000
meters). Pick the case of subsonic combustion. Compute the drop in stagnation
pressure due to shocks, and find the stagnation pressure and temperature at the
start of heat addition. For subsonic combustion, assume that the Mach number is
0.2 at the beginning of heat addition.
4.
Use
the Rayleigh Line Flow expressions to calculate the amount of heat that can be
added – and decide whether it is limited by reaching the temperature limit, or
by choking of the flow. Calculate the stagnation pressure drop during the heat
addition.
5.
Find
the total drop in stagnation pressure as a percentage of the freestream
stagnation pressure.
6.
Repeat
these calculations for different values of Mach number at each altitude
(ranging from 2.0 to 8.0), and then for several altitudes (sea-level to 50,000
meters). Use standard atmospheric conditions.
7.
Plot
each of the quantities calculated – Stagnation pressure drops due to shock
losses, due to heat addition, and the total drop – against flight Mach number,
and repeat this for several altitudes on the same chart. Mark the zones where
heat addition is limited by the temperature limit, and by choking.
8.
Also see how this region changes when you
change the limiting temperature (see above for what temperature to use). Explore
limiting temperatures in the range from 1500K to 4000K.
9.
Repeat
the above for supersonic combustion. In this case, the Mach number is brought
down from the flight Mach number to 2.0 through 3 oblique shocks, each having
the same normal Mach number. Heat is then added until the temperature limit of
the combustor is reached, or the duct chokes.
10.
On
a plot, show where supersonic combustion becomes
a)
more
efficient than subsonic combustion (lower stagnation pressure drop).
b)
More
powerful in that more heat can be added before the temperature limit is
reached.
Discuss how much better supersonic combustion
becomes, and how the temperature limits affect this decision.
The
region of the jet engine where heat is added, can be modeled as a constant-area
duct. The duct is 3m long and 0.3m in diameter (I don’t see where that matters
in this assignment, but it may matter later).
Note
on procedure for shock loss calculation.
a)
Subsonic
combustion case:
For
a given flight condition (flight speed and altitude), guess the Mach number in
front of the final normal shock (e.g., 1.4).
The ideal design is one where each of the 3 shocks in the system (2
oblique + 1 normal) have the same normal Mach number. Lets call the shocks OS1
(the first oblique shock, where the upstream flow is the freestream), OS2, and
NS1 (normal shock).
Having
guessed the normal Mach number, and calculated M3, the Mach number downstream
of NS1 (subsonic), you know 3 things aboutOS2: Normal Mach number downstream of
OS2 (same as M3), total Mach number downstream of OS2 ( = guessed value) and
normal Mach number upstream of OS2 ( = guessed value). From this you can calculate the inclination
of OS2, and the full Mach number M2 ahead of OS2. Given M2 and the guessed
normal Mach number, you can solve for OS1.
Knowing also that the Mach number ahead of OS1 is the freestream Mach
number, you can keep changing your guess of normal Mach number until everything
matches.
Then
calculate the drop in stagnation pressure across each shock, and find the
overall % drop in stagnation pressure.
Assume
that in the subsonic case, the flow is isentropically slowed / speeded up to
Mach 0.2 prior to heat addition. Heat addition can proceed until the flow is
choked, or the temperature limit is reached.
b) In the supersonic heat addition case,
the flow is decelerated through 3 oblique
shocks, to Mach 2.0 at the beginning of heat addition. Same process as before –
normal Mach number ahead of each shock is the same.