Altering exit and inlet pressures for choked flow

[Tsiolkovsky]

Hello All.

This is yet another choked flow thread but worry not for I have searched previous threads to make sure my questions are not redundant.

Before my question(s) is put forward, a brief theoretical background:

A scenario in which two pressure reservoirs are connected with a pipe in which a throat lies.

Say the inlet to the pipe is at a higher pressure and flows to the exit lower pressure. This inlet is now increased in pressure so that the fluid flows faster and compressibility effects start, this increase at the inlet is continued until we reach choked velocity flow at the throat.

Any further increase in inlet pressure will still limit the throat/orifice to Mach 1 however, the mass flow rate will continue to rise linearly due to increased density.

The problem however arises when the literature says the following, and I quote:

In the case of vacuum conditions on the outlet
of an orifice and where the inlet is at ambient
atmospheric pressure, both the air velocity
and the mass flow rate become choked
(limited) when sonic velocity is achieved
through the orifice.The reason for the mass flow rate limitation is the fixed inlet density combined with the fixed velocity.
As in the case of the above vacuum conditions
there are certain situations in which choked
flow does occur for positive (above atmospheric)
pressure. By maintaining a fixed inlet
pressure to the orifice and allowing the
outlet pressure (back pressure) to vary, there
is a range of outlet pressures over which the
mass flow rate is fixed.

It is indeed true that if the outlet pressure is decreased significantly it will eventually result in choked velocity flow. It is also true that if the inlet density is fixed, this, coupled with fixed choked velocity, will result in constant mass flow rate. What I do not understand however is why on earth we would consider a scenario in which there be a fixed density at the inlet. Naturally, when you decrease the outlet pressure, higher flow velocities result, these higher flow velocities result in compressibility effects at the inlet which continually increase inlet density; so a "fixed" inlet density makes absolutely no sense whatsoever. (I dont see how fixing the inlet density would be achievable anyway.)

The exact same is true for increasing the inlet pressure. Eventually the inlet density will continually rise after the flow is choked and it is precisely this continual rise of inlet densities that accounts for an increase in mass flow. I therefore see no reason for considering a scenario in which there is "fixed density" at inlet. How is this scenario unique. Why do we pay special attention to it? Further, what difference does it make if the exit pressure is decreased (vacuum condition) or if the inlet pressure is increased? Pressure, like its voltage analogy, holds no preference, rather its the ratio of pressure that matters.

Just to further elaborate, if the quote above mentioned what it did, why couldn't it also have added:
"An increase in inlet pressure after sonic velocity condition at throat will yield no increase in velocity and no increase in mass flow CONSIDERING INLET DENSITY IS FIXED"
I see this as useless ofcoarse because the inlet density wont be fixed, it will rise from compressiblity effects.

 

[katmar]

The inlet density is a function of inlet pressure. If you vary (decrease) the outlet pressure to induce sonic flow you have done nothing to vary the inlet pressure, and therefore the inlet density remains fixed. If inlet pressure and density are fixed you hit a maximum mass flow rate at some point.

On the other hand if you induce sonic flow by increasing the inlet pressure then you also increase the inlet density, thus increasing the mass flow rate.

This seems so obvious that I am concerned that I have misunderstood your question. I think the crux of the misunderstanding is in your statement "Naturally, when you decrease the outlet pressure, higher flow velocities result, these higher flow velocities result in compressibility effects at the inlet which continually increase inlet density". I do not think that decreasing the outlet pressure can increase the inlet density.

Katmar Software
Engineering & Risk Analysis Software

http://katmarsoftware.com