Norman,
Okay, so just to clear up where you're coming from:
(1) you already accept that the conservation of momentum isn't violated by the
commonly adopted De Laval thrust equation for *exceeding* Ve x q
As we've already demonstrated in detail why thrust is optimised for the given
conditions for when Pe=Pa which provides us with F=Ve x q. Any other expansion
for a given Pa provides us with F < Ve x q
But (2) you're questioning why, say over expansion in an atmosphere should
provide us with less thrust than ideal expansion from the point of view of
momentum transfer alone given that (again neglecting flow separation) the Ve
equations tells us that the more we expand, the faster our exhaust will be
accelerated and because it's travelling at >M1 it can't be slowed or affected
within the confines of the nozzle from atmosphere. So, therefore, it stands to
reason that thrust should only be proportional to the level of expansion as it
is within a vacuum.
If I have your point correct (and that's a bit of an *if*) this is my answer to
it:
The problem here is that we're ultimately dealing with 2 separate systems here
: inside the nozzle and outside the nozzle. Our momentum equations can balance
the 2 systems adapting them to 1 system *only* in the case of ideal expansion.
Why is this? Three words : MASS FLOW RATE or Mdot
Only in conditions where Pe = Pa can the Mass Flow Rates of the inside of the
nozzle and outside consider to be balanced.
Inside the nozzle, our mass flow rate is obviously fixed - it's whatever
propellant flow is being fed into our chamber and the more we expand it, the
more density of the exhaust drops inversely proportionally to that.
However, the same is not true on the other side of the nozzle wall. The more
nozzle area we have on our expansion bell, the more mass flow we have
interacting it on the outside of the wall. The Mass Flow Rate on this side is
proportional to the area of the wall.
So, as you can see, as we expand beyond ideal (Pe=Pa) the :
(1) density of exhaust on the *inside* is falling with extra expansion yet
staying constant on the outside
coz
(2) the mass flow rate on the inside is constant, but increasing proportionally
with area on the outside
So, it's misleading to look specifically at velocity to highlight these
imbalances although that's exactly how propulsion equations adjust Ve into c
But again, only for convenience.
Troy.
-----Original Message-----
From: arocket-bounce@xxxxxxxxxxxxx [mailto:arocket-bounce@xxxxxxxxxxxxx] On
Behalf Of Norman Yarvin
Sent: Saturday, 5 November 2016 4:04 PM
To: arocket@xxxxxxxxxxxxx
Subject: {Spam?} [AR] Re: {Spam?} Re: Pressure Thrust and Momentum Thrust
On Thu, Nov 03, 2016 at 08:34:50AM +1100, Troy Prideaux wrote:
shooting out fast. So if you take this "pressure thrust"
All that's fine enough, as far as it goes. (Your model is certainly
less confusing than the one in the textbook.) But I think this
"pressure thrust" accounts for 100% of the thrust in this scenario --
because where would any other force on the cylinder come from? Yet
in this scenario there's still plenty of momentum being created: the gas is
and add the thrust from momentum to it, you'd be double counting.
Likewise, if you calculate the momentum of the exhaust gases and
compute the thrust that way, you shouldn't add the "pressure thrust" to it.
Yup, good points.
The thing is, that's what the textbook equation does in this scenario.
It takes hole area times pressure, and adds to that the mass flow times the
exhaust velocity. So it gets exactly twice the correct result.
For rocket engines with a nozzle, there is not complete double counting, just
a
bit of it.
In any case, if purporting to explain something to me, you might
deign to actually address the main point I was making, namely that if
one takes the textbook equation and compares it under vacuum
conditions to the conservation of momentum, there's an extra term
("pressure
thrust") on one side of the textbook equation, so the two equations
disagree.
And when you take out the scales to judge credibility and put all the
rocket textbooks with that equation on one pan of the scale and the
conservation of momentum on the other, that resounding "thunk" you
hear is the conservation of momentum establishing its authority and
launching the textbooks past escape velocity.
This is where I disagree because of the above point I just made. Thrust
is *fundamentally* derived from *pressure* not momentum.
Fair enough; pressure is force per area, and thrust is force, so the two are
more
closely related than momentum is to force. But that doesn't mean that a
rocket can get away with violating the conservation of momentum. The
conservation of momentum may be written as
dp/dt = 0,
p being the momentum. In rockets in vacuum, that becomes
dp_rocket/dt = - dp_exhaust/dt,
which reduces to
m*a = F = q*Ve,
where m and a are the mass and the acceleration of the rocket, F is the
thrust,
and as before q is the mass flow and Ve is the exhaust velocity.
If instead you write
m*a = F = q*Ve + Pe*Ae,
or just
F = q*Ve + Pe*Ae,
you're purporting to violate the conservation of momentum; you're saying the
rocket is picking up more momentum than it's throwing backwards.
Now, in the atmosphere things are more complicated, since the atmosphere
can pick up momentum too. But it takes a real lack of taste for theory to (as
Huzel and Huang do) start with the conservation of momentum in vacuum,
subtract the correction term (Pa-Pe)*Ae to adapt the equation to atmosphere,
and then then turn around and say that in vacuum Pa=0 so the thrust in vacuum
is given by that last equation above, without realizing that they've just
violated
the law they're basing their whole analysis on.
There is a remaining question, which I sort of think you're driving at; at any
rate, it seems appropriate to raise it as a matter of completeness. The
calculation of thrust can either be done by looking at pressures or by
looking at
momentum. When things are viewed from the point of view of pressures,
adapting the calculation to atmosphere is simple enough: forces inside the
engine are as they are in vacuum (assuming no flow separation), and then
outside the engine there's atmospheric pressure which is pressing inwards and
which delivers a total thrust of -Pa*Ae. (That's not the atmospheric force
pressing on the exit area of the nozzle; it's the total force from everywhere
on
the outside of the engine _except_ on the exit area of the nozzle; the formula
can be derived by realizing that a uniform pressure on the outside of a solid
body must result in no net thrust, and thus if you have one surface missing, a
uniform pressure on the rest must yield a net force opposite to what the force
on that surface would be.) Pe doesn't enter into this adaptation, and I don't
think it really
should: it operates on the inside of the nozzle, and everything on the inside
of
the nozzle we've already covered in the vacuum calculation.
But whether it does or doesn't enter in, there also has to be a corresponding
calculation when viewing things from the point of view of momentum. That is,
some piece of the atmosphere has to pick up momentum to slow the exhaust
down. What is that piece?
It's not the same air that's exerting the pressure on the top and sides of the
engine. That air isn't moving much; it isn't picking up much momentum. Well,
in human terms it may be quite a stiff breeze, or if the rocket is flying
fast it
may even be at hurricane levels, but that still isn't much in a rocket context
where speeds are measured in Mach numbers.
So the air that picks up momentum to slow the exhaust down has got to be air
somewhere underneath the nozzle. And it's not the mere collision of that air
with the escaping exhaust that does it. Such collisions do slow down the
exhaust, but they conserve momentum: every bit of momentum that is
removed from the exhaust is added to the air.
So they don't diminish the momentum of the exhaust; they just spread that
momentum over a wider carrying mass.
What does diminish the momentum is when the exhaust collides with air that's
already moving up towards the nozzle. And when you have low pressure in the
nozzle, air will try to move towards it. This is seen most clearly when air
is
actually sucked up into the nozzle, in cases of flow separation. When you
have
overexpansion but not flow separation, the air is trying to get sucked up into
the nozzle, but the exhaust collides with it and pushes it away before it can
get
in.
Still, it was trying to get in, and the exhaust flow has to lose some of its
momentum just to stop that. This shows below the engine: the the exhaust
stream narrows as the outside air pinches it in; and while that pinch is
mostly
directed inwards, it's partly directed upwards.
This is of course quite a handwaving answer, which doesn't even come close to
delivering any numbers, but the question puzzled me, at any rate, enough to
seem worth mentioning.
