Choked Flow Critical Area Question 1

[anubis512]

Hi All,

I'm a pretty new engineer a couple years out of college. The topic of choked flow keeps coming up at my job so I've bought a couple compressible flow books and am attempting to educate myself. Currently browsing through Compressible Fluid Flow by Saad (if you have any other recommendations, please let me know). In the section discussing mass flow, Saad shows that the maximum mass flow occurs when Mach Number = 1 and minimum flow area. The minimum flow area is A* the cross sectional area at M = 1. He goes on to define a ratio A(actual)/A* and states that this value can never be less than 1.

What does this mean? If A* at M = 1 is defined as 5 in2, what would happen if A = 3 in2? It seems to imply this can't physically happen so I'm confused.


Thanks for your time

 

[Latexman]

Since A* is defined as the minimum flow area in a given isentropic flow problem/illustration, how can one have a point with less flow area? This would invalidate the definition.

If you mean something else, you'll have to be more clear. Attach a sketch.

Good luck,
Latexman

Technically, the glass is always full - 1/2 air and 1/2 water.

 

[anubis512]

Attached is a sketch of a pipe line showing my question, I hope. If A can never be less than A*, can you explain why that is in a physical or practical sense?

I'm having trouble understanding choked flow in general, so any guides or references you could point me to would be appreciated.


Thanks

 

[Latexman]

What is being held constant among these three cases? Mass flow rate, P[sub]upstream[/sub], P[sub]downstream[/sub]?

Good luck,
Latexman

Technically, the glass is always full - 1/2 air and 1/2 water.

 

[anubis512]

P[sub]upstream[/sub] remains constant. As you lower P[sub]downstream[/sub], for a given area A, mass flow (G) increases until it's choked. I'm having a hard time visualizing/understanding:
[ol 1]
[li]Why A cannot be less than the defined A*. In the sketch I attached, what happens? Does it just mean flow isn't isentropic?[/li]
[li]Why exactly the gas cannot exceed M = 1. What's physically restraining it? I understand de Laval nozzles can create supersonic flow, but I don't understand why it becomes choked in a reducer/pipe exist.[/li]
[/ol]

 

[Latexman]

I think you are confused on the flow models and what is being held constant. IMO, the only way the 3 flow cases make sense in your sketch is if mass flow rate remains constant. Then, if the reducer in the bottom case is physically smaller than the middle case, the same mass flow cannot be obtained because flow is choked at the same sonic velocity/different mass flow rate in the smaller nozzle area.

When choked (M = 1), downstream pressures cannot be transmitted upstream to affect mass flow. Pressure wave travel at sonic speed. Since sonic speed exists at A*, pressure waves cannot get past A*.

Good luck,
Latexman

Technically, the glass is always full - 1/2 air and 1/2 water.

 

[anubis512]

Oh! That makes a lot of sense and I think that's what I was missing.

One more question, if you don't mind. With regards to pressure drop in a pipe after it becomes choked flow, what would be the best approach to calculate it in your opinion? I've come across some things saying the pressure in pipe after it becomes choked can't fall below P* while others seem to imply only the region directly downstream of the shock is P* and frictional losses are calculated as they usually are.

 

[Latexman]

Directly upstream of the shock is P*.

Directly downstream of the shock is the backpressure created from that point to the exit. A lot of times I've heard folks say to calculate this pressure drop backwards - from exit to choke point.

Good luck,
Latexman

Technically, the glass is always full - 1/2 air and 1/2 water.

 

[zdas04]

You really have to work backwards. The continuity equation says that in a closed control volume, mass flow rate must be the same at any point. So in I know velocity at the choke point is 1.0 Mach, and I know the size of the restriction orifice then I can calculate a volume flow rate at actual conditions. Now if I can determine a density I can get that fixed mass flow rate for the rest of the system. Picking that pressure is not simple or straightforward. With a passage directly from source pressure to atmosphere (e.g., a ball valve with no tail pipe) it is clear that system pressure is appropriate and the math doe match observed conditions).

With a tail pipe longer than about 8 times the tailpipe diameter it gets a lot messier. Let's say that we have a SG 0.6 gas at 10,000 psia upstream of an atmospheric vent with a tail pipe. Critical pressure is 5457 psia (using k=1.3) which is a fair bit above atmospheric pressure, so you would expect a second standing wave in the pipe, and a third, etc. The only way I've ever been able to match physical system performance with a tail pipe on my blowdown has been to use the critical pressure relative to atmospheric pressure (e.g., 27 psia at sea level). This is a much smaller mass flow rate than you would get using the system flow rate, but when I've used the system pressure I've predicted blowdown times that were far shorter than we saw in the field.

Once you have a mass flow rate that you believe, you can calculate flowrates a pressure drops throughout the system.

David Simpson, PE
MuleShoe Engineering

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

 

[anubis512]

So you start at atmospheric pressure if it's discharging to atmosphere. With k = 1.3 and P[sub]critical[/sub]= 14.7 psia, P = 27 psia. Therefore if the pressure in your tailpipe is equal to or above 27 psia, you have a shock wave at this point? Then, in your scenario, you'd have another shock wave where P[sub]critical[/sub] = 27 psia, P = 50 psia. And you'd keep working backwards until you build up to the 5457 psia just downstream of your orifice?

 

[zdas04]

I used 14.5 psia instead of 14.7, but kind of yes. The multiple shock waves are conjecture, but actual data matches that model pretty closely. What seems to be happening is that at the location of the next to last shock wave, the velocity is a bit less than Mach 1.0 because the system cannot move the mass of the higher pressure gas at sonic velocity (it would be too much mass flow rate for sonic velocity out the end). Mass flow rate at every point in the tailpipe must be equal to every other point in the tailpipe. Pressure drop in the tail pipe is far too high for the incompressible assumptions of the standard gas flow equations to be valid so the best I've been able to develop is a conceptual model that is a very long ways from a theoretical model or a mathematical model.

Field measurements have matched my conceptual model in dozens of system blowdowns. If (with a tailpipe) I use a critical pressure bulked up from local atmospheric pressure I generally can project a blowdown time within a few minutes of actual. If I use system pressure and recalculate upstream pressure every second I predict blowdown times that are 5-10% of actual.

David Simpson, PE
MuleShoe Engineering

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

 

[anubis512]

Very interesting. How do pipe fittings factor into the pressure drop? Do you calculate their losses via traditional equivalent length methods, at the given velocity "between" shock waves?

Attached is a quick sketch of the tailpipe of the scenario you described, with some fittings added. I'm imagining you start just inside the pipe end at 27 psia (last wave). Then you march up the pipe, adding back any pipe friction/fitting losses until you build up to the next shock wave?

I believe in this situation mass flow must be same all throughout the tailpipe, but the velocity changes significantly after each of those shock waves which drops the pressure. Or am I going about it the wrong way?

 

[Latexman]

Handle fittings the normal way. K = fL[sub]eq[/sub]/D. Yes.

Yes.

No.

Good luck,
Latexman

Technically, the glass is always full - 1/2 air and 1/2 water.

 

[anubis512]

One last question I think.

27 psia is the pressure upstream the last shock wave, just inside the pipe. How do you determine where the other shock waves are? In other words, how do you know it doesn't just go from 14.7-> 27-> 50-> 92-> 169-> 310 psia, etc, up to 5,457, in consecutive shock waves? What determines the spacing?

Thanks for all your help

 

[zdas04]

You are talking COMPRESSIBLE flow here. There are ZERO neat and pretty techniques, correlations, or equivalencies. None. There is also no known way to calculate a friction factor in compressible flow. If you ever know anything about a compressible flow it is from experiment on an exact configuration. If you could calculate where the standing waves might be (you can't), it would be meaningless because every point that you could nail down would only be valid for a few miliseconds.

Let's say that your blowdown was in the middle of a multi-mile line. Among a million other things that you don't know, you have no valid way to apportion the flow from the left and the flow from the right. And if you were able to fix it at a point in time, it would change in the next milisecond.

My technique is to:
[ol 1]
[li]Determine mass flow rate out the end of the pipe with critical pressure on the upstream and of the tailpipe and atmospheric pressure immediately after the shock wave.[/li]
[li]Assume pressure at the trunk (on the tailpipe side) is at critical for the system pressure (gives you a dP down the tail pipe, but be really careful trying to pretend that velocities in this transonic region mean anything with regard to pressure drops or friction).[/li]
[li]Jump to the head(s) of the pipe and measure the pressure(s)[/li]
[li]Use some method to apportion the mass flow rate that is leaving the system to the various flow paths (I use percent of total pipe volume for my first iteration, you have to use something)[/li]
[li]Using the mass flow rate (converted to volume flow rate at standard conditions) to convert the upstream pressure to a system pressure at the hole.[/li]
[li]Do that for each leg and when you don't get the same value for pressure at the outlet from the various legs, tweak the relative flow rates until you do.[/li]
[li]Then move the clock ahead a few seconds and do it again.[/li]
[li]Repeat until you reach your target conditions (I do this to estimate blowdown times, I can't think of another reason to do it).[/li]
[/ol]
It is really ugly, non-theoretical, empirical, and cumbersome. You really have to have a good reason to put yourself through it. I wrote a MathCAD program to do all the picky iterations and friction factor calculations so it doesn't hurt quite as bad as it used to, but it still is a pain and it only gets you to the start of the transonic exit region. I've never found a good way to do any calcs in the transonic region at all, so I typically guess a duration to 0.6 Mach. Once exit velocity drops below 0.6 Mach, you can go back to incompressible math and get closer to theoretical activities.

I have never found many people who were all that interested in working with compressible flow within a pipe. Everything I've ever done with it has started with aerospace calcs and tweaked them to work in a pipe. I never published my tweaks because I never found anyone who cared enough for it to be worth the effort.

David Simpson, PE
MuleShoe Engineering

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

 

[hydtools]

anubis512, you might want to find an download Shapiro's Dynamics and Thermodynamics of Compressible Flow, vol 1. It is a PDF file.

Ted

 

[anubis512]

zdas,

Wow. I knew compressible flow was infinitely uglier than incompressible flow but I didn't know we couldn't analyze so much of it in a pipe. This came about from not fully understanding the compressible flow software we have and trying to educate myself to hopefully explain what it was doing. But that sounds unlikely now haha.

Hydtools,

Thanks! I'll definitely go through that.

 

[zdas04]

The good news (and there is good news) is that the only time I ever have to deal with compressible flow (i.e., I can't find a way to break a pipe up into small enough chunks that density change is less than about 10% over the length of the segment) is blowdowns and PSV tailpipes. I've tried a couple of critical flow chokes, but the results were pretty poor both from a precision and a maintenance standpoint.

David Simpson, PE
MuleShoe Engineering

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

 

[SNORGY]

anubis512,

See attached article. Quite interesting although a bit of a departure from your original question. It does illustrate a lot of the phenomena explained by zdas04.

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

 

[anubis512]

SNORGY,

Thanks for the article. I've actually come across that one before and it's very helpful.