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I've been thinking about a hypothetical problem concerning fluid flow across two distant countries. Assume that country A wants to share water from a river with country B. Country A builds a dam to collect water in a large reservoir. A horizontal pipe is connected from the bottom of the reservoir to country B in an attempt to transport water from the reservoir to country B (We are talking a 1000km~2000km long pipe). No pumps are used in this assembly. Since the reservoir is constantly collecting water from the river, we can assume an infinite supply of water (water level in the reservoir does not decrease) My question is: Can the friction forces in the pipe be so great that it hinders the flow to impractical levels or stop the flow all together?

My first guess is that, using energy conservation, the potential energy in the reservoir must be greater than the friction work done by the pipe. If the pipe is very long, then at some point the friction work done by the pipe will overcome the potential energy and the pressure difference energy available in the reservoir, and thus flow will stop. Pumps will most likely be needed for water to reach country B.

However, I've been discussing this with a friend and he theorises that water will reach country B eventually regardless of friction in the pipe. Because the pipe is horizontal, and water supply is unlimited, water will eventually reach the destination even if it takes weeks or months. After flow reaches the other side, country B can extract and use water from the assembly.

What will most likely happen? Will flow stop or will it continue? Will pumps be needed in this assembly? And why? And if my friend's theory is correct, What will happen when country B tries to collect water at a certain mass flow rate?

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  • $\begingroup$ Is your friends logic that the river behind it will continue to push more water through? Because if that's the case, it wont keep forcing water through if the stream doesn't have high enough pressure. The pressure at the river end of the pipe would be the same as the river so it wouldn't keep flowing into it. $\endgroup$ – JMac Apr 17 '17 at 21:43
  • $\begingroup$ If it stops, there's no friction... $\endgroup$ – DJohnM Apr 18 '17 at 1:28
  • $\begingroup$ Why don't you just assume an inlet pressure upstream and an outlet pressure downstream, and, assuming that the pipe runs full, calculate the steady state flow rate through the pipe for a typical diameter and fluid temperature? $\endgroup$ – Chet Miller Apr 20 '17 at 0:28
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There are many parameters you have not considered. Are you modeling this as if the pipe is starting full when you observe it or if the pipe is being filled.

The nature of fluid resistance is such that resistance decreases as flow decreases.

If the pipe is full and you observe what happens, due to insufficient head pressure, the open end of the pipe at the destination lake will begin to drain and air will replace the lost water. If left in this condition long enough, the air will move up the pipe towards the source until the water pressure is sufficient to displace it. Then you will will get a slug of water pushing air and compressing it. The water will surge and compress then slow, this will probably continue in a chaotic manner with slugs of water exiting the pipe propelled by the compressed air followed by periods of slow water draining as air enters the pipe.

Should you start with an empty pipe, you should have a similar air / water slug cycle as the water tries to settle to the bottom of the pipe until it is backed up and seals in a section of air that can be compressed.

If you begin with the pipe flooded and under the surface of the destination lake then flow will likely begin very slow but might increase to a maximum until the friction slows it again and it will settle at a flow rate where the resistance is insufficient to stop the flow. If you lengthen the pipe to increase the resistance, you will approach a point where resistance and flow approach zero.

If you want to move the water over the distance without these problems you would do better with an open channel design where the water can flow along the bottom of a pipe or similar aqueduct. Water flow will not be very fast unless the depth and width is large.

Pumps would not be needed if the pipe is sloped from the elevated lake to the destination.

Drew K

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