Analytical evaluation of viscous fluid draining from a vertical pipe.

[electricpete]

Let's say you have a vertical pipe of radius R, height h.
It is assumed (for simplicity) open at the top and has a frictionless gate valve at the bottom.
Prior to time t0 the pipe is full and valve is closed.
At time t0 the valve is opened and oil begins to drop out of the bottom (resisted by viscous friction)
Let's make a simplifying assumption that (since the oil tends to stick to the pipe), at a snapshot in time, at any elevation within the pipe the oil forms a "washer" shape which extends from the outer radius of the pipe to some intermediate radius r (0<r<R).
By virtue of the assumption, the problem is axisymmetric and we could characterize the oil distribution in the pipe as
r(y,t) = f(R,rho, mu,h)
where r is the the innner radius of that washer shape volume of oil at that elevation and time (t).
y is independent variable from 0 to h
t is independent variable from 0 on
The forces acting on the fluid are known (gravity and viscous forces).
I'm thinking maybe this could be analytically evaluated as a partial differential equation, but it's beyond my capability at the moment.
So my questions are:
Does anyone want to take a challenge to try to solve the above problem?
Or maybe familiar with a worked-solution to this problem?
Or aware of any empirical information about expected drain times of oil from pipe?

By the way, here's my motivation. We drained oil out of a piping section drain in order to check for oil inleakage at a remote/ inaccessible location. Oil drained very slowly and rate of drain decreased very slowly. Something like pencil stream initially, then one drip per second after 15 minutes, one drip per 5 seconds after 30 minutes, one drip per minute after 60 minutes. We didn't have the luxury of letting it drip any longer since it must be attended while dripping. It's hard to distinguish what is residual oil and what might be inleakage because I don't know how fast the residual oil should come out. The particular section of piping drained is two vertical sections and one horizontal section (so not identical to the analytical problem, but I figure the simpler analytical problem would be a good starting point to understand this type of problem).


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(2B)+(2B)' ?

 

[MacGyverS2000]

Perhaps I misunderstand the problem, but...

If you're concerned about residual oil (sticking to the walls of the pipe after the initial drain and slooooowly working its way down), I would drain as much as reasonable, then plug it back up. Let it sit for a day, then drain it again. If there's no leakage from somewhere else, a few days of this should give plenty of time for any residual oil clinging to the walls to drain to the plug, and since it's plugged most fo the time, there's no need to babysit.

If you unplug after doing this for a week and continue to get drips, that would suggest to me new/leaked oil is being introduced into the system. But it doesn't allow for any fun math, soooooo....

Dan - Owner

http://www.Hi-TecDesigns.com

 

[LittleInch]

Ummmm. What you describe sounds like laminar flow to me, but complicated by the fact that as time goes on what you have is actually annular flow with a central empty core, essentially a film of oil around the outside of the pipe so you're bringing in surface tension as well as thin film flow.

Try a search for "vertical thin film annular flow" Lots of things that look similar to what you describe.

Things like this

http://www.thermopedia.com/content/19/

Remember - More details = better answers
Also: If you get a response it's polite to respond to it.

 

[rb1957]

can you vent above the oil ? (ie is the draining oil creating a vacuum ?)

I guess it's possible that the oil is sticking to the edge of the drain, and then more oil sticks to that oil since the flow rate is low, and so on, eventually closing the drain.

another day in paradise, or is paradise one day closer ?

 

[Compositepro]

You will get nowhere with calculations unless you have lots of empirical data on your fluid and piping. If you had that you wouldn't need a calculation. Oil is designed to be non-Newtonian. If you leave your car in the garage for a year you can start it right up because the engine parts are still coated with oil.