Transonic Flow without Wings

[zdas04]

I've tried to calculate blowdown times for pipelines for 15 years with pretty miserable success. I can predict the end of choked flow with pretty good success, but the flow after the line reaches the critical pressure has always been way off.

I recently found an equation for the sub-critical flow, but found that the mass flow rate at Mach = 0.99999 was 66% of the mass flow rate at Mach=1.0. I've spent the day today digging through all the compressible flow material that I can find and every discussion I can find suggests the sub-critical flow equation (incompressible) probably eases into applicability between Mach = 0.6 and Mach = 0.8. In the missing region, the flow is "transonic" and is sort of sonic-ish and sort of incompressible-ish.

Everything I can find about transonic flow is focused on transonic behavior on an air foil. I don't have an air foil, just a line blowing into the atmosphere. Anyone have any suggestions? [Host hands over a stack of bar napkins and a grease pencil]

David

 

[zdas04]

Well, I finally decided to stop searching for something that doesn't seem to exist and started playing with arithmetic.

I took the first and second derivatives of the sub-critical flow equation (with regard to upstream pressure) and found that the first derivative (which looks kind of like a hyperbolic decline, but it isn't log-log straight) is approximately a straight line (second derivative = constant) for the first 14% below the critical pressure. I ran it for a half dozen atmospheric pressures and the number kept being about 86% of critical pressure. I'm taking that as a physical property and drawing a line from the critical flow end to the sub-critical at 86% (it should be an "S" shaped curve, but that would imply more science that I actually put into this).

David

 

[Latexman]

The gas mixture would have to be exposed to about -40^oC at your max. pressure to condense out any liquid CO₂, so that doesn't seem likely unless it is near one of the poles.

I've never been let down by the age old advice:
1. 10% > dP use incompressible with inlet or outlet conditions
2. 10% > dP > 40% use incompressible with an average of inlet or outlet conditions
3. dP > 40% use compressible methods

But being off by 9X makes me think something is going on with an accumulation and a phase change or something.

I wonder if CH₄ or CO₂ has an affinity to adsorb onto CS and/or rust? A Google search on each had lots of hits. Too many to get a feel if it's real or not. I suspect CO₂ more than CH₄. A monolayer of molecules slowly desorbing may drag things out quite a bit, and you got lots of surface area there!

My mind immediately jumped into problem solving mode, wanting a lab, steel wool, a balance, a containment vessel, and 150 # CH₄ and CO₂, but . . .

Good luck,
Latexman

 

[zdas04]

Being off my 9X actually meant I had some really lame math that I got from someone else (who got it from someone else, etc. back 30-40 generations before someone stopped to check it out) and didn't check out. When I look at it today, it still embarrasses me.

David