Sonic Fluid Flow 2

[venividivici]

Why is sonic velocity the maximum velocity a fluid will flow under typical conditions (fluid flowing from higher pressure to lower pressure). I know that flow becomes choked at Mach=1, but what physically causes this to be the case?

 

[BigInch]

pressure waves moving upstream never get upstream because the fluid moves downstream equally as fast.

 

[bimr]

When the pressure on the downstream of the gas flow reaches a certain value, further pressure decrease at the end will not cause higher weight rate of flow [kg/s]. It means that the gas flow in the pipe has its maximum depending on the available energy at the beginning. If the pressure drop is sufficiently high, the exit gas velocity will reach the speed of sound, and maximum weight gas flow rate will occur. Further pressure drop and pressure decrease on the outlet will not be felt upstream because the pressure wave can only travel as sonic velocity and the "information" about pressure drop will never translate upstream. Shock waves will occur but actual gas weight flow rate will not increase. The maximum possible gas velocity in the pipe is sonic velocity,

 

[zdas04]

bimr,
That is simply wrong. Mass flow rate is NOT choked. Velocity is Choked. Think of depressurizing a pipeline full of methane and pressurized to 1000 psig at 60F through a 2-inch pipe at sea level.
[ul]
[li]Sonic velocity is 1380 ft/sec[/li]
[li]Flow area of pipe is 0.022 ft^2[/li]
[li]Therefore volume flow rate at actual conditions as long as the flow is choked is 2600 MACF/day[/li]
[li]Initial density is 3.732 lbm/ft^3, so mass flow rate is 111.2 lbm/s[/li]
[li]When the line has blown down to 500 psig, density is down to 1.748 lbm/ft^3 which cuts mass flow rate to 52.4 lbm/s.[/li]
[li]At the end of choked flow (about 27 psia), density is now 0.131 lbm/s, velocity is still sonic so mass flow rate is 3.9 lbm/s[/li]
[/ul]

Not a constant "weight rate of flow [kg/s]" at all. It is a constant volume flow rate at actual conditions, but that isn't a terribly useful number.

venividivici,
In gases, BigInch answered your question--to put it in anthropomorphic terms, the flow stream doesn't have the ability to tell the source that downstream pressure dropped further. This "communication" happens at the speed of sound so the flow of the gas is limited to Mach 1.0. In fluid mechanics terms, the flow creates standing shock waves in the flow that have a very large dynamic pressure discontinuity.

You were careful to use the word "fluid" instead of "gas" or "liquid". I'm not sure why. This phenomena is only seen in gases. In liquids the momentum effects disturb the shock waves and a standing wave is not possible (think of pictures of bullets shot into a pool of water to visualize the process). Very high velocity is liquid flow is possible, but the analysis of these flows is way too complex for a sound byte.

David Simpson, PE
MuleShoe Engineering

In questions of science, the authority of a thousand is not worth the humble reasoning of a single individual. Galileo Galilei, Italian Physicist

 

[bimr]

Flow rate for gases is usually give in terms of mass flow rate, w, or volumetric flow rate, Q. The actual fluid velocity is:
U = Wv/A = Q/A

 

[BigInch]

Higher weight rate of flow, mass flow rate, can be obtained at sonic velocity only by increasing gas density upstream of the choke point, which occurs without any additional increase in velocity. Higher density at same velocity = higher mass flow.
If density remains constant, there is no further increase in either volumetric or mass flow rates.

 

[zdas04]

BigInch,
As long as you say it that way it is true. Kind of like saying, "if you don't get older, you won't age". I just don't see the benefit. I can say (without qualifiers) that "velocity is constant in choked flow". I can also say (with minimal qualifiers) that "Volume flow rate (at actual conditions through a constant flow area) is constant in choked flow".

But if I say "Mass flow rate (at constant upstream density) is constant (into a fixed effective downstream pressure)" I don't even need to mention choked flow because flow from any given pressure and temperature into a sink of fixed pressure will be constant. In choked flow, the sink "effective pressure" is the critical pressure at the standing wave. The whole "mass flow rate is choked" paradigm just adds a world of confusion with zero benefit.

I see the "constant mass flow rate" description a lot (I even submitted a change to the Wikipedia article, my language has since been removed, but the replacement makes it kind of clear that mass flow rate is only constant with constant upstream pressure and temperature), and I have never seen a case where the (effectively) infinite source provided constant upstream pressure. The closest that I see is AOF on a wellhead with the hydrocarbon reservoir being effectively infinite for days or weeks. Even with that, when you first open the wellhead, the pressure at the valve is near reservoir pressure, some time later (usually minutes, occasionally hours), the sum of the parasitic losses in the flow conduits upstream of the wellhead have lowered the wellhead pressure markedly to a new stable point. That new stable point may then be constant for a very long time, but you have a very different mass flow rate when you first open the valve than when you reach the stable point.

I think that people talking in "constant mass flow rate" terms just confuses the issue. I know I was very confused about this the first time I saw the discussion on eng-tips.com. Before that time I was sure that "constant mass flow rate" was a law of nature and was (surprisingly) belligerent about people that said mass flow rate changed with changing upstream pressure. Once I got over my obstinacy (and later embarrassment), I decided I would make it a point to challenge the concept of constant mass flow rate in choked flow every time I saw it.

bimr,
I'm not sure what your point even is.

David Simpson, PE
MuleShoe Engineering

In questions of science, the authority of a thousand is not worth the humble reasoning of a single individual. Galileo Galilei, Italian Physicist

 

[BigInch]

zdas, no argument. I just tried to say something that would bring together both of your and bimr statements. I think bimr was simply assuming that it was a steady state case and density upstream would not be increasing, in which case I believe he is also correct.

 

[bimr]

Right, you lost me with the incorrect usage of anthropomorphic and phenomena.

 

[katmar]

I think you guys are arguing at cross purposes. All the fluids text books will give the "constant mass flow" result because they can easily assume constant upstream pressure. Just like they assume frictionless pipe when proving Bernoulli. In the real world, like when a gas pipe is being depressurized, it is likely that the pressure will steadily fall.

But there will also be situations where the upstream condition is controlled to be constant. We probably all did these experiments in college where a sufficiently large source of air, nitrogen or steam was available for us to take a series of flow readings while the downstream pressure was progressively decreased (and the upstream pressure was held constant by a regulator of some sort).

It will be a rare text book that gives a workable solution to the pipe depressurization problem. I bet David had to develop those solutions for himself and could not simply lift them from a college level text book.

Katmar Software - AioFlo Pipe Hydraulics

http://katmarsoftware.com

"An undefined problem has an infinite number of solutions"

 

[zdas04]

an·thro·po·mor·phic (anTHrəpəˈmôrfik) adjective relating to or characterized by anthropomorphism. Having human characteristics.

to put it in anthropomorphic terms, the flow stream doesn't have the ability to tell the source that downstream pressure dropped further

Seems like attributing "the ability to tell" to a flow stream fits the definition of "anthropomorphic"

The singular of phenomena feels pretentious to me, so I often use it incorrectly. Sorry.

David Simpson, PE
MuleShoe Engineering

In questions of science, the authority of a thousand is not worth the humble reasoning of a single individual. Galileo Galilei, Italian Physicist

 

[venividivici]

Thank you everyone for the replies.

It's funny, in a cursory search for the same question prior to posting, I found the same debates surrounding "choked mass flow" and "choked velocity". In my application, we have an 150PSIG natural gas supply from a gas company. This is regulated down to 50PSIG, then down to 15 PSIG, and the flow eventually exits to essentially atmosphere.
For this application, could you then say that at a given pipe size and flow (let's just say 3" at 800 SCFM) that the upstream pressure is constant?

zdas04, I was careful because I was not sure whether or not the same phenomena occurred in liquids. I deal with gases in my line of work, but I wanted to word the question as broadly as possible to gain the best understanding. Does this not apply to liquids because liquids can be assumed to not be compressible?

"pressure waves moving upstream never get upstream because the fluid moves downstream equally as fast. "
"the flow creates standing shock waves in the flow that have a very large dynamic pressure discontinuity."

I'm sure these statements answer my question, but I am not sure why. Does the first statement imply that with subsonic flow of a gas, the amplitude of the pressure waves moving upstream dictate the magnitude of the flow? I also assume these mean that the maximum velocity that these waves can travel at is sonic velocity - which I guess is my real question - what restricts both the pressure waves and gas flow to sonic speed. I am thinking about this in terms of the speed of sound. With a given medium, you have a given sonic velocity. Is this the highest velocity of which a mechanical wave can pass through the same medium before the molecules of that medium can no longer impact any faster?

 

[zdas04]

I would say that your supply from the gas company is essentially an infinite reservoir.

What I'm not certain of is whether any of this discussion applies. Choked flow requires an abrupt transition that is basically frictionless. Most pressure regulators take their pressure drop over a considerable distance to prevent getting to a significant portion of sonic velocity (transition from incompressible flow to compressible flow is usually taken to be 0.6 Mach). In other words, your regulator gives up its pressure to friction, noise, J-T cooling, etc instead of to compressible flow and dynamic pressure loss.

Let's say that your sonic velocity is 1380 ft/sec and area of the pipe is 0.049 ft^2. That gives a flow rate at standard conditions of 65 MMSCF/day. Your 800 SCFM is 1.15 MMSCF/day. Obviously something other than choked flow is happening here.

David Simpson, PE
MuleShoe Engineering

In questions of science, the authority of a thousand is not worth the humble reasoning of a single individual. Galileo Galilei, Italian Physicist

 

[venividivici]

I do not have an actual application. Just trying to understand why the maximum velocity for choked velocity is sonic velocity.

 

[bimr]

Yes, but that is not quite what you posted above. If the flow stream had a beret, a bow tie, a smiley face, and "the ability to tell", it would be anthropomorphic.

Instead, you posted that the flow stream does not have the human characteristic of speech. "the flow stream doesn't have the ability to tell the source"

an•thro•po•mor•phic /ˌænθrəpəˈmɔrfɪk/also ˌan•thro•poˈmor•phous,
adj.
thought of as being human in form or attributes:
ancient anthropomorphic gods.

attributing human characteristics to nonhuman objects.

Attributing human characteristics to nonhuman objects is the definition of "anthropomorphic". Writing that the object does not have human characteristics is not the definition of "anthropomorphic".

 

[zdas04]

Really? That's what you got? Good on you. Have yourself a very nice day.

David Simpson, PE
MuleShoe Engineering

In questions of science, the authority of a thousand is not worth the humble reasoning of a single individual. Galileo Galilei, Italian Physicist

 

[bimr]

I also do not understand the concern.

To determine the size of piping used for gas piping, the factors that are considered:

1. Allowable loss in pressure from point of delivery to point of use.
2. Maximum demand
3. Length of piping and number of fittings
4. Specific gravity of gas.

Gas piping (not pipelines) is typically designed with a maximum dry gas velocity of 100 ft/s, while the natural gas sonic velocity is approximately 1,470 ft//s.

http://www.aft.com/documents/AFT-CE-Gasflow-Reprint.pdf

 

[bimr]

The choked flow condition is a common occurrence in PSV's, vent headers and control valves.

In a conduit, the mass flow rate increases with decreasing downstream pressure. The increase in mass flow has a limit. There will come a point, when the downstream pressure is about half the upstream pressure, when the mass flow rate ceases to increase with decreasing downstream pressure. The mass flow rate will remain fixed no matter how much lower the downstream pressure becomes. The flow is said to be choked. The practical importance of the choked flow phenomenon is that, for a given mass flow rate, the pressure within the exit of the conduit at the choke point will be higher than immediately downstream. The choked mass flow rate can be made to increase by increasing the upstream pressure, but not by decreasing the downstream pressure.

A better explanation than animism can be found from analyzing an ideal nozzle. It can be demonstrated that a velocity equal to sound in a gas can only be reached within the throat area of an ideal nozzle, when the change in area along the length of the nozzle, dA, equals zero.

In a conduit of constant cross-section, the change in area along the length, dA is zero. Since increased velocity is obtained by decreased pressure, and since pressure increases in the upstream direction, the velocity must decrease upstream. The maximum attainable velocity must be obtained at the discharge of the conduit. If the maximum velocity in the throat of an ideal nozzle is Mach 1, it cannot be greater than this in a conduit or in any orifice. If a converging-diverging nozzle is connected to the entrance of a conduit, supersonic velocity can exist within straight conduit.

 

[georgeverghese]

The reason for choked flow in gases is based on entropy changes in the gas as it expands out, and how it matches up with frictional pressure drop. I've yet to get to understand how this relates to Fanno lines and Rayleigh lines and (d - rho / dp) at constant S, but I'm working on it.

 

[zdas04]

bimr,
There is a point where the horse is actually dead. You are trying to explain these concepts to yourself and you are getting them wrong.

There is something called the "Continuity Equation" that says that mass flow rate everywhere in a conduit is the same as everywhere else in the conduit (regardless of cross-sectional area by the way) as long as no mass can be added or removed between the start and the end. So to say

In a conduit, the mass flow rate increases with decreasing downstream pressure.

violates the continuity equation and is therefore incorrect. If you replace "mass flow rate" with "velocity" it is correct, but you are not going to do that.

Katmar,
You are right about not finding the blowdown behavior in a text. I estimated blowdown time (so that the field guys would know when to start looking for leaking valves) and got it way wrong. Refined my process and got is wrong again. Over 50 gathering system depressurizations, I finally got to the point where I could predict the end of choked flow within a few seconds. It was after I retired that I was able to spend the time analyzing my old data and found an equation that matched it. In upstream Oil & Gas, declining upstream pressure in blow-downs or PSV is the common problem, not constant upstream pressure.

David Simpson, PE
MuleShoe Engineering

In questions of science, the authority of a thousand is not worth the humble reasoning of a single individual. Galileo Galilei, Italian Physicist