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

 

[Latexman]

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

 

[mbeychok]

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)
.

 

[sailoday28]

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.

 

[mbeychok]

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)
.

 

[zdas04]

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

 

[mbeychok]

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)
.

 

[25362]

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.

 

[sailoday28]

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

 

[sailoday28]

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

 

[Latexman]

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/k^1/2. 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

 

[sailoday28]

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

 

[sailoday28]

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

 

[zdas04]

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

 

[sailoday28]

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

Regards

 

[zdas04]

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

http://www.muleshoe-eng.com

Please see FAQ731-376 for tips on how to make the best use of Eng-Tips Fora.

The harder I work, the luckier I seem

 

[sailoday28]

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

 

[rbcoulter]

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?

 

[sailoday28]

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

 

[rbcoulter]

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?

 

[sailoday28]

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