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We have a river at the bottom of a slope. The only arable land nearby is at the top of the slope. We can build a water wheel and turn a screw inside a pipe, so the pipe can lift water up the slope. The slope is about one hundred feet high. At some point, the weight of the water overcomes the power of the screw to lift it (that's all I can glean from hydraulics).

My question has two parts. One, what is the maximum height of the pipe? Does the angle or diameter of the pipe make a difference? Two, if necessary, would it be feasible to use a second water wheel to turn a second screw at the top of the first pipe?

Of course, if this idea isn't feasible, alternatives are welcome.

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    $\begingroup$ I would suggest that this question be migrated to Physics(Physics.stackexchange.com) as it more considers the mechanics of fluid dynamics and work in/work out. $\endgroup$ – GOATNine Aug 3 '17 at 19:10
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I think your ideas are a bit confused.

  1. The height limitation for pumping is only present if you "suck" water from above. If you pump from below you can pump it as high as you want and as high as the pump allows you.
  2. An archimedean screw has no limitation, too. it just lift water along a tilted plane.
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  • $\begingroup$ In order to pump it as high as you want, you'd need a pump with infinite power. Also, to pump it as high as you want with an archimedean screw, you'd need infinitely strong people operating it with material of infinite tensile strength. So there are definite limitations to both of them. $\endgroup$ – AmagicalFishy Aug 3 '17 at 19:17
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This is a noria in Hama, Syria, lifting water from the river Orontes to a height of about 20 meters (65 feet). It is the largest noria still extant, built in the 12th or 13th century. The technology is much older: we have pictures of norias from the 5th century (found at Apamea in Syria), and descriptions from the 3rd century.

Noria is Hama

(Hama, Syria - a view of 3 norias in front of the Azem palace. Picture by Heretiq, available on Wikimedia under the CC BY-SA 3.0 license.)

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Think about it in terms of pressure

A 1-inch-square column of water weighs 0.433 pounds per vertical foot (at a cool 39 degrees F, somewhat less if warmer). So a 100-foot column of water weighs 43.3 pounds per square inch. You may know there's a unit of measure called "pounds per square inch".

As it works out, size doesn't matter. If you have entrapped a column of water, of any size, that is 100 feet above you, that pressure will be 43.3 psi. If you want to push water up 100 feet, you will need 43.3 PSI to do it.

Your car tires have about 30 PSI in them.

How to do it

Pumping 100 feet would be practical in the early Industrial Revolution. Look at the British canal system - every canal loses water to seepage and lock operation, and not every canal was below an abundant natural water supply. They made up the difference with pumping stations.

If you are more medieval than that, then you do smaller steppes, pun intended. 1-9 intermediate pools with 10-50 feet of lift each. If you have to get crops to market, you may want to also make that a canal/lock system. A flight of multiple locks typically has a pond*** at each intermediate level.

Lastly if you are in the electric age, pumping is easy obviously, but you may want to combine it with backpumping to store electricity.


** Take 1 square inch, i.e. 1" by 1". Now imagine all the air in the atmosphere above that 1 square inch (mind you, that rectangle is more of a wedge, due to the curvature of the Earth.) All that air, all the way out, weighs 14.7 pounds if you live at sea level, somewhat less if you live higher. Hence, atmospheric pressure is 14.7 pounds per square inch -- literally.

*** to be more precise a Pound, short for impoundment.

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This question may be a bit more appropriate for the Physics.se site.

IF you only use energy from the water flow itself to power the pump, then the pump is limited by how much pressure the water in the river exerts on the pump apparatus (physics.se can calculate it for you). If you are hand-wavium'ing an energy source for the pump(slave labor?), then max height is effectively equal to work in (again, Physics.se can help). This is due to the natural incompressibility of water (IANAPhysicist).

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The maximum height of the pipe depends on a lot of things (what the pipe is made of, how strong your pump is, the pipes radius, etc.). To pump water as high as you want, you'd need people with infinite power turning the screw, and material of infinite tensile strength. This site says that the screws were usually used to lift water about 5 ft.

The weight of water is about $60$ lb. per cubic feet. Let's say you had a pipe of radius $r$ (in feet) that was $h$ feet high. The total weight of the water would be $\pi r^2 h$. So a mechanism lifting water through a pipe with radius of $1$ ft. going $100$ ft. high would have to lift about $314 \times 60 = 18, 840$ lb. of water. That's... a lot of weight to put on an Archmedean screw.

That's about the weight of one of these guys. To contrast that, the $5$ ft. Archmedean screw w/ a pipe radius of $1$ ft. would have to lift about $942$ pounds (20 times less). I'm not sure if the site I linked earlier is particularly credible, nor am I sure how wide the pipes they used were—but $18,840$ lbs. seems to be a lot for people to lift in any kind of manually operated contraption.

So, yeah, the diameter of the pipe really affects how difficult it is to raise water a certain height.

The angle of the pipe will affect how hard it is to push the water. If it's straight up-and-down, then you're fighting against gravity and gravity only. The more the pipe is angled, the less you're fighting gravity and the more you're fighting the frictional force—which is usually a lot less than gravity.

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An alternative idea that allows lifting to effectively unlimited height (even taking material strengths into account) is a chain drive bucket system.

The idea is as simple as buckets of water on ropes (the buckets can be as big as barrels).

If the rope would be too long; you can build in a relay system (basically gearing); so (for example) your water wheel is designed to turn an axle, the axle turns a rope with knots in it (to prevent slippage), the rope has permanently attached wooden buckets about the size of barrels. seal them with pitch on the inside, it is waterproof and lasts a long time.

Say our first stage is going to lift the water 50 feet (the height of a modern 4 story building).

These are lifted up, full of water, at the top of the wheel the rope goes around the top axle; which it is also turning. The buckets are guided by simple barriers to dump their water into a chute (also pitch sealed). Either you are now done, or you go on to the next stage: That same top axle, on the other end, has the same setup: Another chain drive and buckets it carries upward to be eventually dumped in another chute.

That is the Stage 2 Lift: Stage 1 collects water from the river (and is turned by the energy of the river). At the top of Stage 1, the chute (and timing) are designed to automatically dump the Stage 1 bucket into a chute, that fills an empty bucket of Stage 2, which is designed to be "going by" at that time.

Stage 2 rises another 50 feet, and dumps its bucket into a chute: Which could lead to a Stage 3 lift.

There is the strain on the rope of turning all the Stages at the same time, so there is a limit. But the rope does not have to be the breaking factor, you can just use 2 or 3 ropes if the strain is too much. The breaking factor is now strong you can make the water wheel before it breaks, or a tree trunk axle breaks, under the strain. Also, fewer buckets can be used if need be; each stage only needs to be carrying one at a time in order for this to work.

This scheme bypasses the need for a super-strong ropes that might break.

The gear that holds the rope can have a "string of beads" design; imagine a rope with regularly spaced knots: It casts a shadow that looks like a string of beads; basically circles connected by a thick line. That same shape should be in the trunk of a large tree or on a wheel, so the rope, with knots in it, has knots that fall into circles, and the rope between the knots falls into thinner channel. It is way to make a chain drive without any chain or gears. (You want the chain drive for its accuracy in a clockwork mechanism like this).

Stages above the first would need platforms built into the side of the cliff; but those do not have to be too extensive; people don't have to live there or anything. You can "dig in" to put some tree trunks about 20 feet into into the side of the cliff; they will take the strain of holding the axles in place and supporting the few chutes you need, along with some flat space for a maintenance worker to make any repairs (or rope replacement) that might be needed.

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