Small pipe discharge: open channel or siphon 3

[bulkhandling]

For a 3" sch 80 water pipe to a high point and down hill to a low point discharge, is the downhill part pipe full or open channel? The pipe elevation is 6 ft lower than the high point and total length is about 10 ft.
As we know, for a small pipe (or tube), the pipe will always be full flow no matter what is the flow velocity or elevation difference. That is because of the siphon effect or say surface energy (pipe inner surface and water). When pipe diameter becomes bigger and bigger, the pipe will be open channel instead of siphon (full flow)if the slope is big enough (Manning Formula). My question is: how big is the pipe size for this transfer and is it independant of the velocity?

 

[katmar]

For a 3" pipe you can disregard surface tension effects.

Consider the down hill part of the pipe. You need to consider the relationship between the frictional effects and the change in elevation. Assume that the pipe is full and use Darcy-Weisbach analysis. For every foot of elevation you drop, one foot of head can be consumed as frictional drop. If the friction drop caused by the flowrate is less than the head contributed by dropping that foot in elevation it means that more energy is available to transport the water than is actually needed, so the water will accelerate. If it accelerates then it will not fill the pipe and you will switch to Manning flow. On the other hand, if the friction drop is greater than the static head contribution then the pipe will run full.

This is probably best explained with an example.

If your down hill section is at an angle of 45 degrees, there will be 1.41 ft of linear pipe for every ft you drop in elevation. Taking the pipe ID as 2.9" and calculating the flowrate that would give a friction pressure drop of 1 ft of water over a length of 1.41 ft gives 500 gpm. If your actual flow is less than this the downhill pipe will not be full. If it is more than this it will be full.

In fact there is not a sharp switch over from Manning to Darcy-Weisbach flow and you will experience a range of slugging as you get into the syphon range, but as 500 gpm is equivalent to a velocity of 24 ft/second in a 2.9" pipe the chances are that you will actually be well below this cut-off and will have a partly full pipe.

There was a long and (to me) very interesting discussion on a similar topic way back in thread378-81608 This concerned diesel flowing down hill all the way to the bottom of a mine. There were some very useful references in that thread given by Art Montemayor. Another useful reference on syphon flow that is available on the internet is

http://www.agr.gc.ca/pfra/water/syphon.pdf

 

[bulkhandling]

Thank you, katmar.

Another scenario: Assume a centrifugal pump discharge. after a distance of run, turns vertically down for 10 ft and turn horizontally for 2 ft (or any other distance less then 10 ft friction loss). Will the verical down pipe be full or partially full? or eventually expel all the air in the pipe and reduce the pump total head required?

 

[katmar]

You need to consider the end horizontal section (2 ft in your example) and the vertical downflow section separately.

Start first with the horizontal section and calculate what pressure is needed to drive the liquid through it at the required flow. This will tell you if the liquid will back up into the vertical section. Using your 2 ft of horizontal pipe, and assuming again an ID 2.9", a flow of 60 gpm would give a frictional pressure drop of 0.3" WG. If you take this head above the centre line of the horizontal pipe you are still inside the pipe and you will be running part-full in this section. In this case there is no back up into the vertical section and with this flowrate the vertical and end horizontal sections will be part-full and there will be no syphon.

Using the flowrate of 500 gpm from my previous post the pressure required to drive the water through the 2 ft of horizontal pipe is 17" WG, so there will definitely be a back up of water into the vertical section. By the same calculation, the frictional drop down the 10 ft vertical section (assuming the pipe is full) would be 5x17"=85" WG. Add to this the 17" required for the horizontal section and you can see that you need to provide 102" of WG to give this flow through the combined vertical and horizontal sections. The vertical change in height provides 10x12=120" WG. This would give a pressure at the top of the vertical section of 102-120 = -18" WG.

The risk is if this pressure is below the vapor pressure of the liquid it will start boiling and generate vapors. In this example, assuming water as the fluid, unless the temperature is over 98 deg C the water will not boil at this pressure. So, with 500 gpm the pipe would be full and your pump would get the benefit of 18" WG "pulling" the water through.

If you start with a flow of 500 gpm and gradually throttle it down the pressure at the top of the vertical section will decrease and the question becomes whether the water will start boiling before you get to the part-full regime. It depends on your actual installation, particularly the vertical length. If the vertical length is more than 34 ft, and the horizontal section is just flooded, there will always be boiling and potential slugging and vibration.

 

[bulkhandling]

katmar,

I’ve noticed you "assuming the pipe is full" at the 500 gpm flow. That implies that the pipe can be possibly not full(?). This is actually my question.
Let's take a normal flow for this example, 100 gpm (4.85 fps). This flow will only build 0.75" WG back pressure at the 2 ft horizontal pipe and at the beginning when the pump is started, most part of the vertical column will not be full flow and the horizontal part will be full, so no air can back enter the pipe. While fluid from the pump will bring the air out from the vertical pipe little by little and eventually the vertical pipe will be full. So the vertical column will reduce the system static head and make the pump run at a greater flowrate.
Is this true? If not, how about 200 gpm (9.7 fps) or 300 gpm (14.5 fps)? Or only when the flow is big enough to make friction loss in the pipe equal or greater than the static head, the vertical pipe will be full?
In other words, if the end of the pipe discharge is always full of fluid, then the downward pipe will be running in full flow (within vapor pressure) if only the flow (velocity) is bigger then some datum. Is this true? What is this datum?
Appreciate for any comments.

 

[katmar]

The reason I stated "assuming the pipe is full" is that I wanted to use Darcy-Weisbach to calculate the friction head, and this method applies to full pipes. Once you have the result, you need to go back and check if the assumption was true.

Getting on to your question of whether the pipe is full or not. As you stated, as the flowrate of the water is increased it will reach a stage where the water carries the air out of the pipe. According to the articles by Simpson and Hills referenced in thread378-81608 the air will start to be flushed out once the Froude number exceeds 0.31. With a 2.9" pipe this is at approximately 18 gpm.

At the other end of the scale, Simpson says the Froude number needs to reach 2.0 before the pipe is totally flooded - if the discharge point is not submerged. This would be about 115 gpm. For flowrates between 18 gpm and 115 gpm you would get pulsating flow until the air was all removed. But if the discharge is not submerged, or if the final horizontal section is not full, air can flow back up the pipe and you will get unstable conditions. Between these two extremes, Simpson gives the value of the Froude number as 1.0 at which syphons form readily, although he notes that there may be pulsating flow.

If the end of the pipe is not submerged it becomes important to be able to predict at what flow the horizontal section becomes flooded. I would say the friction head through the final horizontal section would have to exceed half the diameter of the pipe for this to guaranteed. To get a friction head of 1.45" WG in 2 ft of 2.9" ID pipe you would need a flow of about 140 gpm. Simpson's value of Fr=2.0 at flooding would suggest that my estimate is a bit high.

If the pipe is flooded, it becomes an easy task to calculate the actual pressure at the top of the vertical section, by off-setting the friction losses against the static height. This needs to be done to check if the liquid will boil.

At some higher flowrate the point will be reached where the friction head matches the static head exactly, and there is no more risk of boiling. For 10 ft of vertical pipe and 2 ft of horizontal pipe this would be at about 540 gpm.

In summary I see the following 4 flow regimes
1. Below 18 gpm the water runs down the walls of the pipe and the air remains in the pipe.
2. Between 18 gpm and 115 gpm the air will be swept out of the pipe, but it may flow back up and pulsating flow can be expected.
3. Between 115 and 540 gpm the pipe will be full of water and the flow will be stable - provided there is no boiling.
4. Above 540 gpm there will definitely be no boiling, but of course the cut off point for boiling will be a bit below 540.

If the pipe is full then then you can confidently take the change in height as an aid to the pump head (subject to the boiling check).