Originally Posted By: paul
whatever the speed of sound is

well Bryan , since the air is limited to the
"speed of sound"

340.29 m/s

Paul, your own source says exactly that. From your NASA page:

The hot exhaust flow is choked at the throat, which means that the Mach number is equal to 1.0 in the throat and the mass flow rate m dot is determined by the throat area

But in your case - 100PSI, the speed of sound is 1/3rd of the value you stated - 131.6m/s (see post #34826 for the calc).

Originally Posted By: paul
the tube size is not important as the tube can be any size according to your reasoning it can never travel faster than the speed of sound.

Right and wrong, depending on what you mean by size.

The radius of your tube will determine the *rate* at which air is released from the tank. And as such, it'll also dictate the force developed. Keep in mind that [Pt-Pe]Ae determines the force derived from pressure; if you double Ae you'll double the force you'll produce.

However, since you have a tank of air the total amount of thrust produced (ISP, measured in newton-seconds) will be the same regardless of the tube's radius - the only difference is a larger diameter tube will produce that thrust in a shorter period of time, while a small-diameter tube will produce that amount of thrust over a long period of time.

Originally Posted By: paul
if I use a large or small tube doesnt matter that much according to your reasoning.

As described above, it depends on what you mean by size. The speed of the air in a tube will be restricted to the speed of sound, regardless of length or diameter. The *mass flow* however will be greater in a larger diameter pipe, but will not be affected by length. Double the cross-sectional area of your pipe, and twice as much mass will flow through it every second. It'll flow at the speed of sound, but there will be more moving

Originally Posted By: paul

I can increase the time that I allow the air to escape from the tank by using a small orifice or tube.

and this will decrease the momentum that is felt by the pipe each second.

Here is where you are missing the implication of choked flow (i.e. flow limited to the speed of sound) - and I think where our entire "conflict" (in regards to your tube) lays.

Forget air for a second, and think of a metal ball in your tube. The 100PSI tank you described earlier produced ~6000N of force through its tube. Pretend for a second that we applied that 6000N of force to the metal ball, but in a way where the source of the force was unconstrained (i.e. not using air confined to the speed of sound). This would accelerate the ball as per the formula F=ma (rearranged to a = F/m to calc the acceleration). The longer the pipe, the more acceleration that ball would experience. Hypothetically speaking (and ignoring relativity) an infinitely long pipe would lead to an infinitely accelerated steel ball.

I believe you will agree with the above description, as that's essentially what you've been saying all along (with the steel ball replaced by a mass of air).

Here is the part you are missing - your flow of air is choked; restricted to a set speed. This is occurring despite the fact that you have ~6000N of force acting on that air. So unlike that steel ball, your air is not continually accelerated - something is holding it back. And, unlike your steel ball, a longer tube won't further accelerate the air - one it gets to the speed of sound it stays there.

The factor that restricts the flow of gas to the speed of sound is the air itself - "real" (as in non-ideal) gases resist their own flow. The faster you push a gas, the more it resists being pushed. When this pushing is pressure-driven, the resistance to flow will equal the force produced by the pressure once the gas reaches the speed of sound. What this means is the momentum of the gas traveling at the speed of sound is perfectly balanced by an opposing force - in this case the backpressure of a non-ideal gas.

So, in your tube you have the situation where your tank initially accelerates the air to the speed of sound - this initial acceleration of the air is generated by the force determined by [Pt-Pe]Ae. At this point you now have a stream of air traveling at the speed of sound, despite the fact the tank's 6000N of force is "trying" to accelerate it further. This lack of acceleration is due to the force provided by the airs resistance to its own flow - and that force is equal, but opposite, to the force provided by the tank.

In other words, once flow is choked your tank is pushing it with 6000N of force, but that force is being countered by 6000N of backpressure.

And keep in mind, just as the tank imparts momentum onto the air due to the 6000N of force available to it, the choked air is in turn pushing back on the tube/tank with 6000N of force. It is this equal, but opposite, forces that prevent the further acceleration of the gas, and it is that opposing force that causes m(dot)Ve to be zero in a tube.

What this means is that your tube-based system experiences a force, and thus gains momentum, from that initial acceleration of the air from rest to the speed of sound - a force determined by [Pt-Pe]Ae. However, the force generated by mass flow - and thus that momentum - is countered by an equal but opposite "choking" force. Ergo m(dot)Ve is zero with a tube.

All of that could be derived by looking at the formulas provided, in a lot fewer words, but the above is the plain-ol-english version.