Choked and sonic flow 5

[robjoenz]

The definition of choked flow is when a reduction in downstream pressure does not result in an increase in flow through an orifice. In the case of a tiny hole in a pipe if flow becomes sonic it will also be choked.

My question is... if you have choked flow does it necessarily mean you will have sonic flow? i.e. can choked flow exist for sub-sonic velocities?

Thanks,
Rob

 

[rbcoulter]

Sailoday28:

Don't understand the use of A / Ao. The case I am referring to is an isentropic nozzle with a fixed throat nozzle area.
For an ideal gas / isentropic flow, the exit velocity (choked flow) is fixed by the discharge temperature and the heat capacity ratio (k) and is independent of the throat area. The throat area does affect the mass flow rate but not the discharge velocity (choked flow only).
So I still don't see how the velocity can change if the inlet temperature stays the same (choked conditions).

 

[sailoday28]

rbcoulter (Chemical)
A/Ao=sqrt(T/To)
A=sound speed The subscript, o , for stagnation accounts for KE effect
The stagnation temp To is related to stagnation pressure by

(P/Po)=(T/To)^[k/(k-1)]
If static temp, T remains constant and either P or Po change, then To and Ao change.
From enery equation at choked conditions (Ao/A)^2=(k+1)/2
With change in Ao, then A the throat velocity will change.

Regards

 

[rbcoulter]

Sailoday28:

OK. An ideal gas is flowing through an isentropic nozzle at choked conditions. We now increase the inlet pressure without changing the inlet temperature. Does the throat velocity increase?

At choked conditions the pressure ratio (P/Po) does not change. Also, the temperature ratio does not change. If only the inlet pressure is increased this only changes the throat pressure but not the throat velocity which is determined by the throat temperature. By your equation:

A = Ao * sqrt(T/To)

indicates that the throat velocity (choked) doesn't change if the temperature ratio doesn't change.

For an ideal gas, the sonic velocity = sqrt(kgRT). The speed of sound at the inlet doesn't change if the temperature doesn't change.

 

[sailoday28]

rbcoulter (Chemical)
You have increased the inlet pressure and therefore, while you have maintained, T as constant, To, the stagnation temperature must increase. Ao must increase and the resulting energy equation yields an increase in the throat sound velocity.
Regards

 

[Latexman]

For ideal gas:

T.P^-(1-1/k) = constant

If inlet pressure is increased and temperature remains the same, we have a different constant than before with lower inlet pressure. Therefore, T and P at the throat will be different too. Likewise, the speed of sound

c = sqrt(kgRT) = sqrt(kgP/[ρ])

will change.

Good luck,
Latexman

 

[rbcoulter]

Latexman:

Temperature at throat (T*) (choked flow, ideal gas, isentropic) is:

T* = To*(2/(k+1)) ; To = inlet stagnation temperature

Sonic Velocity is:

c* = sqrt(kRT*/Mw) ;

T* is only dependent on k and inlet temperature. So sonic velocity doesn't change.

The above from Perry's 7th edition page 6-23.

 

[sailoday28]

rbcoulter (Chemical)
Either Perry's is wrong or you are misintrerpreting it.
I have obtained and written in this forum how Mechanical Engineers Hanbook (McGraw-Hill) was to be corrected to change upstream p and T to be Pstagnation and T stagnation.

apparently some readers on Eng-tips have chosen to ignore my previous (over the years) comments.

When the upstream pressure is increased, the Stagnation temperature increases. For an adiabatic steady flow of perfect gas, the stagnation temp remains constant with the flow.
Use the energy equation with increased "stagnation" temperature and the resulting sonic velocity will increase.

or Ho=H +u^2/2 Cp(To-T)=u^2/2=a^2/2=kRT/2
Since To increases, solve for T
Regards

 

[rbcoulter]

I am not convinced. I would be interested in reading the
Mechanical Engineer's handbook version of the problem.

 

[sailoday28]

rbcoulter (Chemical)
Reading is not enough. The equations were similar to those posted by mbeychok (Chemical; His equations should specify that the pressures and temperatures are at stagnation.

I'm sorry, but you should be reading a good text, such as "Volume 1 of Shapiro" in compressible fluid flow. That type of text will clearly spell out that the upstream conditions are at stagnation.

Regards

 

[Latexman]

rbcoulter,

Inlet temperature [≠] inlet stagnation temperature.

Stagnation temperature is the temperature the fluid would attain were it brought to rest adiabatically without the development of shaft work.

The more energy a stream has (i.e. higher pressure) the higher it's stagnation temperature.

Good luck,
Latexman

 

[rbcoulter]

Are not the stagflation and inlet temperatures essentially the same when there is near zero approach velocity (in the case of a large tank with a hole)? If you are arguing that this is never the case in real life then I see your point. However, must release models/equations assume these conditions. Of course, there is no ideal gas or isentropic flow in real life neither.

 

[Latexman]

The stagnation and inlet temperature can be essentially the same when there is near zero approach velocity in a single pressure scenario. However, your question was

An ideal gas is flowing through an isentropic nozzle at choked conditions. We now increase the inlet pressure without changing the inlet temperature. Does the throat velocity increase?

There were no qualifiers or specifications on how much the pressure was increased. It's a question that needs a yes or no answer. The correct answer is yes, the throat velocity increases.


Good luck,
Latexman

 

[sailoday28]

rbcoulter (Chemical)
'However, must release models/equations assume these conditions. "
While Latexman has answered your original question", please note, that even with a perfect gas, those models are "quasi-steady".
My response are based on steady state flow models.

I hope that these last responses have put to bed the incorrect perceptions relating to throat velocity not changing with the increased source pressure.

 

[rbcoulter]

All my posts have been in reference to Milton's posting of the discharge equations above. Sorry if I didn't make myself clear. If one assumes the following:

1. Isentropic flow through a nozzle (before and after)
2. The same ideal gas (before and after)
3. Heat capacity ratio doesn't change with temp / press
4. Large entrance area to nozzle area (stagnation temp = inlet temp) AND (stagnation pressure = inlet pressure) (before and after). (No entrance pipe between stagnation zone and nozzle.)

THEN:

Throat velocity does not change with only an increase in the stagnation (also inlet pressure in this case) pressure. The mass flow rate does increase.

If you are saying that there is a pipe between the stagnation zone and the nozzle where the pipe diameter is significant in comparison to the nozzle throat diameter, then that would be a different matter.

 

[sailoday28]

rbcoulter (Chemical)
"Throat velocity does not change with only an increase in the stagnation (also inlet pressure in this case) pressure.?????? The mass flow rate does increase."

If stagnation pressure=inlet pressure,the stag temp=static temp, then throat velocity, mass flow remain fixed.

I don't know what you mean.

 

[rbcoulter]

Assume conditions as previous post:

Case #1:

1. Po = Stagnation Pressure = Inlet Pressure
(inlet nozzle area is very large compared to throat)
2. To = Inlet temperature
3. T* (throat temperature) = To*(2/(k+1))
4. Throat velocity (choked) = c* = sqrt(kRT*/Mw)

Case #2:

1. 2Po = Stagnation Pressure = Inlet Pressure
(inlet nozzle area is very large compared to throat)
(Pressure double of case #1)
2. To = Inlet temperature (same as Case #1)
3. T* (throat temperature) = To*(2/(k+1)) (same as Case #1)
4. Throat velocity (choked) = c* = sqrt(kRT*/Mw) (same Case #1)

Conclusion:

Case #1 and Case #2 have the same throat velocity (choked).


What could I possibly be missing here?

 

[Latexman]

Is inlet nozzle area _{Case #1} = inlet nozzle area _{Case #2}?

Good luck,
Latexman

 

[rbcoulter]

Oh. So now I see your trick.

The inlet area is considered very large compared to the throat in both cases. Trying to define a fixed inlet area does not work for this equation. May not sound realistic but it is my understanding that is the assumption behind the derivation of the equation.

The inlet area is infinite in both cases. The approach velocity is zero in both cases.

 

[Latexman]

There's no trick. Just defining the boundaries.

Inlet nozzle area _{Case #1} = [∞] = inlet nozzle area _{Case #2}

Right?

Good luck,
Latexman

 

[rbcoulter]

Yes. But neither is finite for this model. If you try to get me to accept that then the model in invalidated and you can make your case.