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Choked and sonic flow
5

Choked and sonic flow

Choked and sonic flow

(OP)
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

RE: Choked and sonic flow

Yes, "choked flow" implies that the exhaust stream is initially moving at Mach 1.0.  That velocity never lasts very long because of fluid friction and changing the momentum of surrounding gas molecules immediately begin slowing it down, but the velocity at the exit plane is Mach 1.0.

David

David Simpson, PE
MuleShoe Engineering
www.muleshoe-eng.com
Please see FAQ731-376: Eng-Tips.com Forum Policies for tips on how to make the best use of Eng-Tips Fora.

The harder I work, the luckier I seem
 

RE: Choked and sonic flow

Choked flow may occur with less than  1.0 mach number- if there are sharp angles then there can occur "oblique shock waves" when the average velocity thru the hole is less than the speed of sound.

For example, flow thru some control valves , such as a globe valve, implies sharp angles and changes of direction.  This is described by the "ISA handbook of control valves" as when the Xt is greater than 0.1  . For a streamlined ball valve Xt=0.1, and no oblique shock waves form. For a typical globe valve, Xt=0.8 and oblique shock waves do form and choked flow will occur at a pressure ration less than critical. For a CCI drag valve , Xt=1.0, and no shock waves form and the flow is frictionally choked, not acoustically choked.

In the case of choked flow thru a hole in the pipe, the discharge coeficient is about Cd=0.82, but will vary according to the ratio of the hole diameter to the wall thickness of the pipe. So, based on 100% hoel area, it never reaches sonic velocity , but based on an apparent vena contracta or due to oblique shock waves it becomes choked.

RE: Choked and sonic flow

If the process is other than adiabatic, say, isothermal the flow will not choke at Mach=1.

 

RE: Choked and sonic flow

Wait a doggone minute here.  

sailoday, can you provide an example of an isothermal flow in conditions described by choked flow (i.e., downstream pressure < upstream pressure * (2/(k+1))^(k/(k-1)))?

Davefitz,  I've found that in the conditions you describe (i.e., standing shock waves downstream of the restricting element) that the pressure at the outlet of the restricting element is higher than the critical pressure from the equation above and the flow is not choked.

My experience says that if you satisfy the equation above, then the velocity will be 1.0 Mach at that exact point.  If something downstream restricts the mass flow rate then pressure will increase at the restrictive element and the conditions for choked flow will no longer be satisfied.

David

RE: Choked and sonic flow

sailoday28 and zdas04, isothermal choke flow may occur in longer pipe lines, that choke at some expansion.

But my understanding was always that Mach=1 in both cases, it is just that sonic velocity for isothermal flow is different from adiabatic flow.

 

RE: Choked and sonic flow

After reviewing my copy of "The Dynamics and Thermodynamics of Compressible Fluid Flow" by Shapiro, the isothermal flow model falls apart at M = 1/k1/2, where an infinite heat transfer per unit length is needed to maintain a constant temperature.  In reality when a subsonic isothermal flow approaches this limiting Mach Number, all fluid properties change rapidly with distance.  Unless heat is transferred purposely, flow is likely to be more adiabatic than isothermal.

Good luck,
Latexman

RE: Choked and sonic flow

Latexman (Chemical)
The original question was
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?
You clearly have chosen an excellent reference(Shapiro), however, choking can occur at other than M=1.
Regards

 

RE: Choked and sonic flow

Quote (ronjoenz):

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

The velocity of sound is thermodynamically defined as a small compression wave moving adiabatically and frictionlessly through the medium.  This is an isentropic process.  If the actual flow process deviates from this, for example the flow is isothermal or there is significant friction or the flow is not adiabatic, it should not be surprising that choked flow can occur at velocities other than sonic.

If a = acoustic velocity (velocity of sound) and a' = the limiting velocity of isothermal flow, then a = a' x k1/2.

Good luck,
Latexman

RE: Choked and sonic flow

In isentropic flow (adiabatic + small friction loss), just as isothermal flow, the mass velocity reaches a maximum when the downstream pressure drops to the point where the velocity becomes sonic at the end of the pipe (e.g., the flow is choked).

But given:
c = speed of sound
M = molecular weight
R = gas constant
v = spatial averaged velocity
T = constant temperature
1 = reference point 1
2 = reference point 2
* = sonic state

Under isothermal conditions, choked flow occurs when:
v2  = c = v2* = (R*T/M)^0.5

Under Adiabatic conditions (or locally isentropic), choked flow occurs when:
v2 = c = v2* = (k*R*T2/M)^0.5

Reference: Darby, R., Chemical Engineering Fluid Mechanics, marcel Dekker, 2001  

"We don't believe things because they are true, things are true because we believe them."
"Small people talk about others, average people talk about things, smart people talk about ideas and legends never talk."
 

RE: Choked and sonic flow

sheiko,

I have Ron Darby's book and I really don't agree with saying c = v2* = (R*T/M)^0.5 for isothermal flow.  The speed of sound has a very precise thermodynamic definition:

c = (gc(dP/dρ)S)1/2

One cannot calculate with 100% rigor the speed of sound of a medium in isothermal flow.  This is due to the constant entropy constraint in the definition of the speed of sound.  Isothermal and isentropic conditions are definitely not the same thing.

It's a good book, but being too loose with precise thermodynamic definitions makes this section a bit sloppy in my opinion.

Good luck,
Latexman

RE: Choked and sonic flow

I would argue that the mathematics say that isothermal flow chokes at M<1 and the equations are consistent.

However, I would also argue that at the moment that an isothermal flow chokes, an infinite amount of heat is needed to be transferred into the gas in order to maintain the temperature of the gas.  Since this is physically impossible, the temperature must change, and isothermal flow cannot exist in a choked state. I'd say the equations are physically meaningless at that point and so choked isothermal flow cannot happen in reality.  Before you reach the choke point, any real system must always diverge from isothermal conditions.

Exactly how long the isothermal flow equations adequately describe a flow as its velocity builds up depends on the gas, but I'd say that it's pretty conclusive that they do not realistically describe choked flows.

RE: Choked and sonic flow

Doh.  How long the isothermal flow equations hold up depend on the state of the gas and the amount of heat that can get into the gas (i.e. surroundings), not just the state of the gas.  

My apologies.

RE: Choked and sonic flow

The isothermal equations work really well for a body moving in an infinate reservoir (fighter planes at Mach 3 may heat themselves up, but they don't change the temperature of the atmosphere a measurable amount).  There are really no situations where the equations work when the fluid must reach this speed (instead of the body within the fluid).

David

RE: Choked and sonic flow

Latexman (Chemical)

Your response of
"One cannot calculate with 100% rigor the speed of sound of a medium in isothermal flow."
Sound speed in isothermal flow is per the definition in your response
c = (gc(dP/dρ)S)1/2  or
- with or without isothermal flow
c^2=gamma (dp/droh)isothermal

RE: Choked and sonic flow

sailoday28,

Maybe I'm making too much of a big deal of the definition of sound speed being for isentropic conditions.  What got me started was the equations in Darby's book:

c = (kRT2/M)1/2 for Adiabatic flow

and

c = (RT/M)1/2 for Isothermal flow

but

(kRT2/M)1/2 is not = (RT/M)1/2

A different symbol should have been used for the isothermal case or subscripts or something.  It's confusing.  Anyway, whether one uses c2 = (dP/drho)S or k(dP/drho)T they are calculating sound speed for isentropic conditions in both cases.  I don't see how this has meaning to the limiting velocity for isothermal conditions which = c/k1/2 or (dP/drho)T (with no k in the equation).

Good luck,
Latexman

RE: Choked and sonic flow

"Mass velocity" simply does not mean anything.  The quantity you described is actually "mass flow rate".  "Velocity" is a distance over a time in a direction ("speed" is a distance over a time).  Mass per second does not meet the definition.

And, if the upstream pressure increases, then the velocity will increase AND the mass flow rate will increase.  Sonic velocity is dependent on the media in which the sound is traveling--if the media becomes more dense then the speed of sound increases.  In the equations you re-quoted above, the mass flow rate (that I've never seen given the designation "Q" before, but that is a quibble) will increase as the pressure increases--SO WILL THE VELOCITY.  By the way the limiting equation that you included is the same one I posted in this thread three days ago.  

The only thing special about choked flow is that as long as the pressures satisfy the limiting inequality the downstream pressure does not matter.

David

RE: Choked and sonic flow

In the reference i mentionned in post dated 24 Apr 08 11:11 (Darby 2001) the mass velocity is the mass flux: G = m / A = rho*v, with:
m: mass flow rate
A: cross sectionnal area
rho: density
v: spatial averaged velocity

"We don't believe things because they are true, things are true because we believe them."
"Small people talk about others, average people talk about things, smart people talk about ideas and legends never talk."
 

RE: Choked and sonic flow

zdas04,
       I agree with you that there is no such thing as mass velocity. As you say you can have "mass flowrate" or velocity but not mass velcity. even the units quoted of kg/s are not "velocity" related.

RE: Choked and sonic flow

In the Chemical Engineering field, mass velocity is a term used commonly.  Mass velocity = G = mass flow rate divided by flow area perpendicular to flow.  It is more properly called mass flux, but mass velocity is used by many references.

Good luck,
Latexman

RE: Choked and sonic flow

2
zdas:

I don't believe we should get bogged down by semantics. Each science or engineering discipline has its own jargon. As Latexman said, chemical engineers often use the term mass velocity. But whether we call it mass velocity, mass flow rate or mass flux, the equations I gave above speak for themselves in the universal language of mathematics and the references that I gave also speak for themselves. In fact, you will note that when I presented the choked flow equations, I also used the term mass flow rate.

Nor does it matter whether the mass flow rate (mass velocity) is expressed as Q or G or m or whatever, as long one spells out what the symbols are.
 
What I like about the equations that I gave is that they don't get involved with Mach numbers or the speed of sound, which in my opinion is much simpler

As I said before, the equations that I gave have been used for the last 15-20 years in quantifying accidental releases of gases from piping or vessel holes or similar release sources for the consequence analyses required by law ... not only in the USA but in some other countries as well.

Milton Beychok
(Visit me at www.air-dispersion.com)
.

 

RE: Choked and sonic flow

To clear up a simple point, a flow is choked when for given upstream conditions, the mass flux; W/A, will not incrrease when the back pressure is lowered.

 

RE: Choked and sonic flow

sailoday28:

Just add the one sentence: "However, the mass flux will increase if the upstream pressure is increased."

I don't mean to be too pedantic ... but too many engineers don't realize that the mass flux can be increased even though the linear velocity can not be increased.

Milton Beychok
(Visit me at www.air-dispersion.com)
.

 

RE: Choked and sonic flow

Milton,
What you are saying is incorrect.  If the upstream pressure increases then BOTH the mass flow rate and the velocity increase.  If upstream density decreases then BOTH mass flow rate and velocity decrease.

The only time mass flow rate and velocity are constant is when upstream pressure and temperture are constant and a change in downstream pressure does not take you out of choked flow.

David

RE: Choked and sonic flow

zdas:

I guess we will just have to agree to disagree. If you look at these "choked flow" equations:



or this equivalent form:



It is obvious from the above equations that the mass flow rate (Q) increases if the upstream pressure (P) is increased ... but the gas velocity is still choked meaning that the linear velocity is still the maximum velocity (i.e., sonic velocity).

 

Milton Beychok
(Visit me at www.air-dispersion.com)
.

 

RE: Choked and sonic flow


That's correct, but let's not forget that the sound speed in gases measured by (k×P÷ρ)1/2 changes slightly in line with the correction by zdas04.

RE: Choked and sonic flow

zdas04 (Mechanical)
Thank you for input in helping to correct the misconception of choked flow increasing with upstream pressure increases.

mbeychok (Chemical) The critical pressure ratio, can be easily obtained by fixing upstream conditions and calculating mass flux as back presssure is lowered. The critical pressure ratio is reached when the mass flux no longer increases as back pressure is lowered.--When you increase upstream source pressure after the critical pressure ratio is reached, the source stagnation conditions are changed--as pointed out by zdas04 (Mechanical)
MOre simply-for isentropic flow AND fixed upstream condtions, combine the conservation of mass and energy equations-take the derivative of G,(W/A), with respect to back pressure and the choked flow equations and critical pressure will be obtained when dG/dp=0.

Please point out a fluid text which defines "choked flow" in another manner.

Regards
 

RE: Choked and sonic flow

Latexman (Chemical)
I don't disagree with you with regards to infinite Q at critical flow for isothermal conditions. My Shapiro is in storage and I don't remember the proof of infinite Q. Is it it the text or do you remember the derivation?

Regards

RE: Choked and sonic flow

sailoday28,

I don't believe there is a derivation.  The text says something like, "it's obvious from Equation number ?".  That equation is in the form of T/T* and when you do some simple algebra you can see it is indeterminate at c/k1/2.  You have to infer there must be infinite heat transfer per unit length to counter the infinite T/T* in the equation referenced.

Good luck,
Latexman

RE: Choked and sonic flow

There are two formualas given on this thread for the mass flow rate,Q.   The pressure, temperature an density should be at the upstream STAGNATION conditions.

The second formulation includes a term,k,which one MIGHT presume to be the same specific heat ratio specified in the first equation.  Including compressiilty being a constant,  Cp-Cv=ZR and the k should be adjusted accordingly.  Obtaining a Cp or Cv allows computation of k.
I would be interested in how either the Cp or Cv is obtained.

Further for choked flow, I would expect a different critical pressure ratio, than that used with the first equation.

Regards
 



 

RE: Choked and sonic flow

zdas04 (Mechanical) 23 Apr 08 9:09  
Wait a doggone minute here.  

sailoday, can you provide an example of an isothermal flow in conditions described by choked flow


Consider a horizontal constant ID duct/pipe.
momentum equation
udu/dx  + vdp/dx    +fu*u/(2D)=0       (1)
G=mass flux, rho*u
d(lnu)/dx +  dp/dx/(G*G*v)  +f/(2D)=0    (1a)
isothermal flow

d(lnu)/dx   pdp/dx/(G*G*RT)  +f/(2D)=0    (1a)
conservation of mass
Pu=constant                                (2)
combine (1a) and 2
-d(lnp)/dx  + pdp/dx/(G*G*RT)  +f/(2D)=0     (3)

integrate 3
-ln(p2/p1)+ (p2*p2-p1*p1/(2G*G*RT)+fL/(2D)=0   (4)

To obtain Gmax, for a given L  differentiate G wrt P2

p2*p2=G*G*RT    but    p2=rho2RT  and G=rho2*u2
This will yield u2*u2=RT= a*a/ gamma    
where a = sound speed     u/a= M= sqrt(1/gamma)

Substiture p2 into  (4) and get relation of length to choked flow.

Please check for algebraic errors

For isothermal process, pgas.  dh=0
and
dQ/dx  =udu/dx    substitute into above equations at choked flow  and udu/dx=friction/zero   infinite heat transfer.

Regards


 

RE: Choked and sonic flow

Couple of things.  With the confusion that has existed in this thread about terminology, what is "u", "v" (I'm assuming you are using the Fluid Mechanics convention of "u' being velocity in the "x" direction and "v" being velocity in the "y" direction, but I don't want to start analysing your arithmetic withou knowing for sure).

Also, I'm looking at a p-h diagram on my wall (you caught me when I was working of options for a CO2 sequestration project) and an isothermal process is anything but constant enthalpy and dh never equals zero (i.e., the constant temperature line on a p-h diagram is never vertical).

David

RE: Choked and sonic flow

v is specific volume
 dh=0 for pv=RT    Try the CO2 and low press high temp and look at const enthalpy lines.  

Regards

RE: Choked and sonic flow

That is in the liquid region (and looking at the data, the lines are almost vertical, but not quite).  I thought this was primarily a gas discussion and those lines are not even close to vertical compared to the grid (they leave the saturation bubble horizontal and then turn toward vertical, but don't make it).

David

David Simpson, PE
MuleShoe Engineering
www.muleshoe-eng.com
Please see FAQ731-376: Eng-Tips.com Forum Policies for tips on how to make the best use of Eng-Tips Fora.

The harder I work, the luckier I seem
 

RE: Choked and sonic flow

zdas04 (Mechanical)
That is in the liquid region-Generally, a fluid at low pressure and relatively high temp(with respect to the critical point) will be gaseous.  For gases following pv=RT, the enthalpy is strictly a function of tempeature. Therefore, with no change in enthalpy, there will be no change in temperature.

Regards

RE: Choked and sonic flow

I believe Milton is correct only in the case of an ideal gas expanding through an isentropic nozzle.  The choke pressure is defined by a pressure ratio determined from the heat capacity ratio.  The temperature ratio is determined in an analagous relationship also depending on the heat capacity ratio.  So if only the inlet pressure is increased at sonic flow conditions the throat temperature (which is somewhat cooler than the inlet) would not change.   For an ideal gas, the sonic velocity can be fixed by a constant heat capacity ratio and temperature.  So the velocity does not increase but the mass flow rate increases because of the pressure change.

For a real gas, I guess that the throat temperature would increase (non-isentropic flow creating heat?).  So maybe for a real gas both velocity and mass flow rate would increase?

RE: Choked and sonic flow

rbcoulter (Chemical)
If you change any upstream conditon, you are changing the stagnation conditions and therefore have a new problem,
For the perfect gas, for choking the critical pressure ratio will remain the same, however, the throat pressure, temperature and density will also change.  Since the throat temp will change, the acoustic velocity and hence throat velocity will change.

Regards

RE: Choked and sonic flow

I still don't follow.  Let's assume initially that an ideal gas is flowing through an isentropic nozzle at choked conditions (sonic flow in the throat).   Now, increase only the inlet (stagnation) pressure but keep the inlet temperature the same.  The calculations will indicate, unless I have missed something, that the throat temperature does not change.
The throat pressure does change (it increases).  So if sonic velocity is the same in both cases and choked flow occurs, wouldn't the velocity be the same in both cases?

     

RE: Choked and sonic flow

rbcoulter (Chemical

Adiabatic steady state perfect gas, constant specific heats
Ao^2=kRTo =A^2+U^2(k-1)/2   (1)
A^2=kRT   subscribt o refers to stagnation conditions

(P/Po)=(A/Ao)^(2k/[K-1])  (2)
If local upstream source static temp is held contant and the upstream static pressure is increased, Po and Ao will change.
If Po increases Ao will increase

At throat with choked flow the energy balance (1) yields
Ao^2=A^2 +A^(k-1)/2    =A^2(k+1)/2   
 Increased Ao yields increased A at throat.   Increased A at throat yields increase T static at throat.
  Note stagnation temp is constant but has increased because of increased stag pressure at source.
Regards


 

RE: Choked and sonic flow

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).


 

RE: Choked and sonic flow

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

RE: Choked and sonic flow

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.
 

RE: Choked and sonic flow

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

RE: Choked and sonic flow

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

RE: Choked and sonic flow

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.

 

RE: Choked and sonic flow

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
  

RE: Choked and sonic flow

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

RE: Choked and sonic flow

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


 

RE: Choked and sonic flow

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

RE: Choked and sonic flow

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.

  

  

 

RE: Choked and sonic flow

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

Quote (rbcoulter):

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

RE: Choked and sonic flow

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.




 

RE: Choked and sonic flow

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.


 

RE: Choked and sonic flow


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.



 

RE: Choked and sonic flow

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?


  

RE: Choked and sonic flow

Is inlet nozzle area Case #1 = inlet nozzle area Case #2?

Good luck,
Latexman

RE: Choked and sonic flow

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.    

RE: Choked and sonic flow

There's no trick.  Just defining the boundaries.

Inlet nozzle area Case #1 = ∞ = inlet nozzle area Case #2

Right?  

Good luck,
Latexman

RE: Choked and sonic flow

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.


 

RE: Choked and sonic flow

I don't understand what that means.  I just want to establish that the only parameter or condition that changes from Case #1 to Case #2 is the inlet pressure.  Keep it simple.

Good luck,
Latexman

RE: Choked and sonic flow

Yes. Only the inlet "stagnation" pressure changes with all the assumptions mentioned earlier.  This is "P" in Milton's equation which assumes a zero approach velocity.

It sounds like your trying to remove the constraint of zero approach velocity going from case #1 to case #2.  In this way you can claim that the kinetic energy of approach is greater in case #2 because of the higher pressure then saying that the stagnation temperature is higher and then that the throat temperature is higher then that the choke velocity is higher. This model doesn't allow for this (Milton's equations).  Other more realistic models may (please provide one or a reference).

My only point is that if you use Milton's equations, without deviating from the assumptions in which it was derived, then the choke velocity does not change if only "P" is increased.





 

RE: Choked and sonic flow

rbcoulter (Chemical)
"Miltons equations........"
I believe the basic topic is steady state flow.
The "accidental release" equations are at best an APPROXIMATION used in "quasi steady flow"
Of course you can have a large container with a small break and approxmate zero velocity within the container. Initial conditions would set the starting(initial conditions) pressure and tempearture.  The relation between the stagnation conditions would be related to heat transfer to the vessel and mass removed. For each step of the "quasi steady" analysis, "Miltons first equation can be used. (I question the second with regard to calculation of gamma),  
However, I don't see how you are tying this in with sonic velocity other than the simple relation between the upstream and throat conditions of the flow.
 

RE: Choked and sonic flow

Regarding the many references in this thread to "Milton's equations", I would like to make it clear that the equations I presented in this thread are not "my" equations. Although I have thoroughly checked their derivation, as stated above they are the equations given in:

(1) "Risk Management Program Guidance For Offsite Consequence Analysis", U.S. EPA publication EPA-550-B-99-009, April 1999
(2) "Handbook of Chemical Hazard Analysis Procedures", Appendix B, Federal Emergency Management Agency, U.S. Dept. of Transportation, and U.S. Environmental Protection Agency, 1989
(3) "Methods For The Calculation Of Physical Effects Due To Releases Of Hazardous Substances (Liquids and Gases)", PGS2 CPR 14E, Chapter 2, The Netherlands Organization Of Applied Scientific Research, The Hague, 2005

I could also add:

(4) Equations 5.20 and 5.21 on page 5-14 of the Sixth Edition of Perry's Chemical Engineers' Handbook, 1984. (Perry's equations include the local acceleration constant, g, because they are in the U.S. Customary units rather than SI Metric units).  

(5) For those of you who may be in the United Kingdom, exactly the same results are obtained by using Ramskill's equation. Ramskill, P.K., "Discharge Rate Calculation Methods for Use in Plant Safety Assessments", Safety and Reliability Directory, UK Atomic Energy Authority.

I would also point out that the equations are not merely "accidental release" equations. As far back as the 1950's, when we were designing Exxon's Model IV fluid catalytic crackers in refineries, those equations were used to size the choked flow orifices that injected steam into the catalyst circulation system to fluidize the catalyst ... and if we needed to increase the mass flow rate of steam injection, we simply raised the inlet steam pressure. The point being that the equations I presented have been in use for over 60 years.

In my humble opinion, 67 postings in this thread is beginning to get somewhat ridiculous.  

Milton Beychok
(Visit me at www.air-dispersion.com)
.

 

RE: Choked and sonic flow

mbeychok (Chemical)
Equations being used for 60 years is fine, if the user understands that the upstream Pressure and Temps are stagnation.
With reference to the 2nd equation, I still would like to know why the definition of k as used with compressibility is not spelled out.



Regards

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