Clarification on flow/force in bends...

[dgsbsme]

I've read through threads I could find on the subject and get, more or less, the same take I had already...

Sum:Fx= P1A1 - P2A2COSa - Rx = M(V2COSa - V1)
Sum:Fy= - P2A2SINa + Ry = M(V2SINa)

And these forces are steady state.

Anecdotaly, I've heard of slugs of condensate in a steam line, breaking off a valve at the end of a run of pipe, resulting in injury (or even death). I'm thinking this is a special case transient event involving an expanding compressible fluid accelerating a slug of incompressible fluid.

The design in question though, is regarding start up of a pump, with ~11 seconds ramp up time. All flow is incompressible. So, maximum velocity will be no more than the pump volumetric output divided by the cross sectional area of the section of pipe in question.

Wouldn't that make the equations above encompassing fluid momentum the highest force at bends?

 

[BigInch]

Assuming that the pipe is "non-expandable" too.

There are scenarios where vapor can be generated in liquid lines, as when the pressure at a point goes below its vapor pressure, in which case the higher accelerations to which you refer can occur on the surrounding slugs of liquid.

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[BigInch]

And, since you mention a pump start case, as if the pump was attempting to start and reached an overload current at hi rpm, initiating shutdown and closure of a discharge block valve, in which case the fluid downstream of the block valve might break and run away. Depends on the system configuration and elevation profiles etc.

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[dgsbsme]

In this particular case, the water temperature is ~80°F so no chance of dropping below vapor pressure (shade under -14psig).

This pump is pumping from a river up to a holding pond. The only valve downstream of the pump is a butterfly near the pond and it has a hole in the disc to prevent dead head of the pump. All manual operation.

On intitial filling of the line, I am assuming that fully developed flow (full pipe) won't occur in horizontal sections untill a vertical climb allows/causes the preceding horizontal leg to fill. So initial transient thrust on those elbows would be "cushioned" a bit by not having full fluid momentum arriving all at once.

Elbows at the top of rises would have full flow reaching them all at once.

In any case, the force on the bends would be dependant on (and only on) Pressure and mass flow rate (momentum), and the momentum could not rise above the velocity of the full pumping flow rate in that size pipe. Correct?

There would be nothing to accelerate the liquid beyond that velocity, unless there were a long enough vertical drop, and that would raise other questions about whether the pipe was flowing full or not.

So, the question I suppose is, are the forces/reactions the same for steady state (fully developed) flow as for initial flow (during first pipe filling)?

 

[BigInch]

No, not exactly. Forces are not the same. When you are filling, you have no liquid in the downstream segments and hence there is no friction or static pressures from any downstream liquid that can create any resistance to flow into the segment. You are creating the eventual steady state system curve. Hence accelerations during initial fill can be much higher as there is no resistance and the filling pump will tend to pass the BEP and run away out to the end of the curve to full flow capacity at zero discharge head, until you begin to build some backpressures, when the pump will start moving back towards BEP and possibly shutoff head. Consequently, you could experience surges up to velocities consistant with the maximum pump flow, not steady state (probably BEP rated flow). You can reduce those possibilities by limiting flow from the initial fill to BEP velocity by restricting the initial fill pump discharge flow.

Depending on velocity of filling flow, you may push the air ahead of the liquid and obtain full pipe liquid flow behind the air, even before reaching a climbing segment.

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[BigInch]

Brings up another point. During initial filling, generally you want to have a velocity fast enough to push the air ahead and not allow liquid to run down the next downslope in a partially full state and pool ahead of arrival of any "full flows". In fact, spheres are often used for that purpose; to maintain full flow behind the spheres and the air in front.

If velocity is slow, such that you have partial flow over an overbend and cascading in the down slope, filling at the underbend beyond and pushing the liquid up the next slope will tend to increase the initial filling pressure at the pump, as it is the sum of the static heads created by the two (or more) upslopes separated by a partially filled segment that must then be maintained to continue filling. So its a good idea not to let initial fill start running ahead of the full pipe flows.

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[dgsbsme]

Thanks for the dialog BigInch.

So the worst case forces on the bends (which will be carried by pipe supports) will be from the same pressure-momentum equations in my original post, but at the velocity of the highest (end of curve) flow that the pump might generate?
Neglecting any partial system resistance as the flow reaches a particular bend.

I interpret your explanation as the system curve is truncated. Or in other words, the system curve is building up as the flow reaches further along the line. Soon after initial pump start, the system resistance is only the first couple of bends and/or valves and a few feet of pipe, therefore the pump will run far out on its curve and begin dropping back as more of the system resistance comes into play. In this particular instance, I have that 11 second ramp-up time in the motor that will mitigate this effect, but for the sake of conservatism, that might be worth ignoring.

I'm not sure I follow your last paragraph correctly:

"...filling at the underbend beyond and pushing the liquid up the next slope will tend to increase the initial filling pressure at the pump, as it is the sum of the static heads created by the two (or more) upslopes separated by a partially filled segment that must then be maintained to continue filling."

If there are two rises separtated by a drop, I would expect the drop to "waterfall" to the lower bend and begin filling the downstream rise. The "waterfall" segment would backfill at the same rate as the downstream segment, matching static head as it goes. With a discontinuity between the two rising legs, the pump would see the static of the first leg plus whatever pressure the air bubble rises to. With a high point vent at the top of the "waterfall", perhaps this bubble pressure would be slightly less than the downstream static height (else the "waterfall" leg could never fill), but still cause the pump to see higher static head. Is this correct?

 

[BigInch]

No problem.

Depends on the velocity and flow regimes at the overbend..

If the liquid is slowly cascading at the overbend, what you are saying about both the downslope and the upslope filling at the same rate is true, but the air is not being forced to move to the outlet during that process. The air there is moving up and back to the overbend as the liquid from the top of the overbend replaces it at the bottom of the following underbend. What happens to the air that eventually gets trapped at the overbend? It gets stuck there until (if) velocity increases and forces it to move along to the outlet. The only way for the velocity to increase is for the pump to increase the pressure by backing up on its curve. If it gets all the way back, and the velocity is still not enough to force the air along you wind up with "vapor lock", which may even be possible even if the pump backs up to shutoff head; depends on the pump dH and the static heads in the system. Under the air trapped at the overbend the fluid might move very fast but only displace the bubble slightly down the overbend into the next valley, thus maintaining a high velocity at the bottom of the pipe, a hydraulic jump at the bottom recirculating back into the air caught in the downslope. It may be that the velocity is high enough to force the bubble down into the underbend, but then what do you have. The same situation again, waiting for the cascade to fill and displace the air back up into the overbend again. If the air begins to be forced up the next rise, you have a lot of weight on the downslope liquid air interface that could compress the bubble and accelerate that column moving towards the underbend very fast. Of course this is all "what if" stuff and maybe it ain't gonna' happen in your little system... but... then again maybe ... it could. Anyway, rather than work out all those "IFs", its better not to let it happen in the first place. My only point is that other things are possible rather than just steady state forces, or even forces at the highest pump velocity, if the design is not thought out through construction and initial filling, startup and finally operation. I can see you are thinking about most of this with the startup case analysis already, which is very good. A lot of people miss the filling and startup bits entirely.

I did this little engine fuel flow schematic a while back showing how vapor lock can increase pressure above that required for steady state flow in a vapor free system, even though the outlet pressure doesn't change.

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