governing equestions of unsteady flow in hose?

[salmon2]

Experts,

Got a real problem, can't figure out what are the governing equations to solve, which is different from school

I have an umbilical or hose running from sea surface down to 10,000ft deep. The hose is used as a hydraulic control line for subsea equipment. The top end of hose is connected to a actuator with a valve and bottom end of hose is a piston. When the valve is opened, the pressure from actuator will pressurize the hose and all the way down to piston - pushing the piston eventually. Assume the fluid in the hose is incompressible and ideal and the pressure in actuator is a constant pressure, say 10ksi. I need to know the average fluid velocity and pressure at any spot from opening the valve til flow becoming steady. Additionaly, the hose will expand when pressure increase. I can ignore the piston for now.

My biggest question is what are the governing equations I suppose to crack for this case. I know how to numerically solve differential equations but just don't what to solve.

Additional specific questions:
1) This is a initial condition problem, right? Do I have enough initial conditions?
2) Can I treat this one as one-dimensional problem given expanding from pressure?
3) Does the famous one-dimension incompressible flow equation, A*V = constant, still hold? Where A is hose cross section and V is average velocity.

Thanks a lot in advance for your any comments and inputs.

 

[quark]

Never came across such things but the problem is definitely food for thought for me.

1) Didn't get you.
2) Seems to be one dimensional, but will revert back
3) Yes, unless there is leakage (V is localised velocity, though)

Average fluid velocity can be piston traverse/time of traverse at the CS area near piston. If the CS area varies, use the continuity equation.

Nevertheless, I have a vague feeling that I am oversimplifying things. Let us see what others and my second thoughts say.

 

[BigInch]

"Assume the fluid in the hose is incompressible and ideal and the pressure in actuator is a constant pressure", if you assume those things then you have no transient pressures at all. The flexible hose won't expand much with all that external seawater pressure on it, unless it is in a collapsed state before you open the valve, but then why would you.

Let your acquaintances be many, but your advisors one in a thousand’ ... Book of Ecclesiasticus

 

[salmon2]

Thanks a lot for your inputs.

To Quark:

1) Maybe I should have called it as "initial value" problem? I received my education outside US, so not sure about the terminology. For fluid mechanics, there are two types of problem: initial condition and boundary condition. That is what I meant.

2) & 3) the reason I want to use average velocity because I figure then I can possibly use one dimensional approach.

To BigInch:

I am trying to simply the problem, hope not too much. Assumption of "incompressible" can stay, right? I don't mind to include viscous effect. The pressure in actuator can be related to outgoing fluid volume, can be included as well. But if the actuator is big compared with fluid volume being moved, I feel it is a valid assumption.

Before opening valve, pressures inside and outside the hose is balanced since the inside is not empty. The data from manufacturer told us expansion at 10ksi differential pressure will be 20% with a reducing trend, meaning hose expand pretty quickly at low pressure.

Thanks a lot to both of you again. Welcome any discussion.

 

[salmon2]

To Quark again,

Forgot to say: CS of piston is fixed, but the CS of hose at different spots is different, a function of location and time within unsteady stage because of expansion.

 

[BigInch]

Simplify, simplify ...

OK. Liquid is incompressible, hose not: Means, As the hose expands, then flowrate into hose must increase while hose is expanding.

As flowrate increases, velocity increases and pressure drops. That happens until hose is fully expanded, presumedly at a steady state pressure that is held for some time. When (if) flow is stopped, pressure increases again as velocity slows once more, reaching a high as fluid finally comes to a halt, which probably expands the hose some more. When hose is finished expanding, the hose, now expanded at the highest pressure, tries to squeeze some of the fluid out of the hose, in any direction it can, until equilibrium is reached. If the fluid was stopped by a closed valve, or other blockage downstream, the new flow direction will most likely be in the prior upstream direction. That wave travels back until the pressure difference driving that flow is consumed by friction of the flow against the walls and by pressure reductions due to the once again slightly expanding wall of the conduit. If the fluid was compressible, the results would go a little faster as that energy in the compressed fluid was converted into velocity when pressure reduced, or into a slowing wave front as increasing pressure expanded the conduit at the same time, making room for some volumetric storage in the larger diameter, thus slowing the pressure wave's advance. OK, ya I think that's pretty close to what happens.

Let your acquaintances be many, but your advisors one in a thousand’ ... Book of Ecclesiasticus

 

[stanier]

Prof Thorley covers this very application in his Fluid Transients in Pipeline Systems. Refer chapter 2.2.5. The section is too large to replicate here and I suggest you buy the book.

He does warn that the pressure could exceed the Joukowski prediction by many times at the closing valve.

ISBN 1 86058 405 5

"Sharing knowledge is the way to immortality"
His Holiness the Dalai Lama.

http://waterhammer.hopout.com.au/

 

[quark]

Salmon2,

Assuming balance of inside and outside pressures before filling the hose is not a valid assumption as pressure exerted by a fluid column differs for different fluids. Outside fluid is sea water and pressure balance happens if the inside fluid is also sea water.

Considering uniform presure inside the hose, after pressurisation, the expansion of hose is a function of depth (i.e external pressure). You can calculate localised velocities as a percentage of initial velocity (w.r.to depth) by considering expansion of hose is linearly dependant upon the pressure difference. Just showed a sample calculation in the attached spreadsheet.