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Re: turbojet mechanical advantage



At 10:52 AM 1/2/00 -0500, I wrote:

At the most basic physics level, what makes a turbojet engine
go the right way?


Short answer: MECHANICAL ADVANTAGE !!!!!!!!!!!!!!!!!!!!!!!!!!!!
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!



Longer answer: I asked what's wrong with this picture:

-------------------------------------
| | | xxx | | |
jet out / / / / / / air in
<---- ==/=/=/================/=/=/== <--
/ / / / / /
| | | xxx | | |
------------------------------------- (what's wrong with
^ ^ ^ this picture?)
turbine flame compressor


One thing that's wrong is that it suggests that the turbine is "just like"
the compressor. It's not. The pitch of the turbine blades is roughly
twice the pitch of the compressor blades. It's hard to illustrate that
using ASCII art, and I've not yet found a picture on the web that
illustrates it properly, but if you go to an aviation museum and look at
the blades on a cut-away engine, this fact will jump out at you (now that
you know what to look for).

-----

VOLUME BALANCE: In particular, suppose we take an engine and rather than
burning fuel in it, we just pump some gas (e.g. argon) into it using a
hose. No temperature changes or pressure differences occur in this
case. (Also, for simplicity, we are ignoring all forms of leakage and
slippage.)


-------------------------------------
| | | |
mixture _/ / / / air pumped in
out =_/ ===================/=/=/== by "compressor"
<-- / / / / <--
| | | |
------------------ ----------------
| |
| |
| |
^
argon injected


So in the steady state, if we pump in one liter per second of argon, we get
out 2 liters per second of mixed gas (one liter/sec of air mixed with one
liter/sec of argon). This two liters of mixed gas turns the shaft just
enough to pump one liter/sec of pure air in the front.

In general, if the mechanical advantage is k, the ratio of output volume
(mixed gas) to injected volume (argon) is
ratio = 1 over (1 - 1/k)
which you may notice has the same form as an equation that showed up in the
analysis of upwind sailing. If you try to set the mechanical advantage to
unity, as suggested by figure 1, then the thing definitely does not work.

------

PRESSURE BALANCE: Now we need to think about the pressure. In normal
operation, there is a large pressure difference across the compressor
stage. There is a k-fold smaller pressure difference across the turbine
stage. This is a good thing, because we *want* some left-over pressure to
make thrust. The whole point of the turbojet engine is to produce an
unbalanced force!

So we see that another thing that is wrong with figure 1 is that the nozzle
is missing. A better drawing would be

-\ /-------------------------------------
\-/ | xxx | | |
jet _/ / / / air in
out =_/ ===================/=/=/== <--
<-- / / / /
/-\ | xxx | | |
-/ \-------------------------------------
^ ^ ^ ^
nozzle turbine flame compressor



Obviously we have:
-- atmospheric pressure far ahead of the engine,
-- high pressure in the guts of the engine, and
-- atmospheric pressure far aft of the engine.

If there is (say) a 100 PSI pressure difference across the compressor
stage, there is only a 50 PSI pressure drop across the turbine stage
(assuming k=2). The rest of the pressure drop occurs across the
nozzle. Note that this latter pressure drop does not occur because the gas
is pushing on the nozzle -- it occurs because the gas is pushing on itself,
converting pressure (potential energy) to velocity (kinetic energy).

-----

ENERGY BALANCE: We have seen that the compressor stage involves a
relatively small volume of gas with a big pressure difference, while the
turbine stage involves a relatively large volume of gas with a smallish
pressure difference. Since the energy to drive the compressor comes from
the turbine, it is important to verify the energy-balance equation:
pressure(turbine) * volume(turbine) = pressure(comp) * volume(comp)

We see that the operation depends on the fact that the volume of the
exhaust gasses is much larger than the volume of air taken in through the
compressor.

(The *mass* is practically the same, but the volume is larger because of
the temperature increase, and it is the volume that enters into the energy
equation above. For homework, give three reasons why the mass of the fuel
is insignificant compared to the mass of the air.)

-----

STARTING: Have you ever wondered how big a battery it takes to start the
engines on a DC-10?

Well, I've seen the starting-battery on a DC-10 and it's only slightly
bigger than the battery in my car. The trick is that they don't use it for
starting the main engines. There's an APU (Auxiliary Power Unit) in the
tail, and the battery is only big enough to start the APU. The APU is a
tiny little gas turbine (about 400 HP if I recall) and once it is going,
you can bleed some high-pressure air from it to spin up the main
engines. It's not done using exactly the method suggested in the "volume
balance" section above, but it is done pneumatically.