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For a civilization that depends heavily on canals for transportation, what limits are there on how steep canals can before they flow too fast to traverse? I know that barges were normally pulled by animals upriver, if winds were unfavorable, so would the maximum slope of the canal depend on how strong the beasts of burden are? It would be neat to be able to calculate the slope in like meters dropped per kilometer in a general sense.

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    $\begingroup$ Going downstream, you could go all the way to 90 degrees if you wanted to. White water rafting with cargo. $\endgroup$ – Tim B II Apr 8 at 23:39
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    $\begingroup$ You can be the first on the pulleys to lift the barge back up the cliff. $\endgroup$ – GiruŠatuku Apr 8 at 23:41
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    $\begingroup$ It isn't having to go uphill that would be the problem. The real problem would be the speed of the water rushing downhill. As soon as that speed gets anywhere near the speed of your boat, your're not going to go anywhere. And it doesn't take much of an angle to make water flow quickly. If you can notice a slope, you're not going to be sailing up it. The Mississippi is long and navigable, and it rises only .12m per kilometre. That's .012%. The purpose of locks is to eliminate the slope. $\endgroup$ – Ray Butterworth Apr 9 at 2:24
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    $\begingroup$ This seems to be what you want : lmnoeng.com/Channels/trapezoid.php $\endgroup$ – Ville Niemi Apr 9 at 4:26
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    $\begingroup$ The problem is never how hard it is to pull barges upstream, the problem is always how to keep the canal filled with water. If the canal has any amount of appreciable slope water will flow downhill. Where does the water come from? $\endgroup$ – AlexP Apr 9 at 6:11
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A canal with a gradient and a flow is a river. And navigatable rivers are pretty much flat. Taking a nearby river (the Medway) and measuring the change in height from the coast to the point at which it becomes essentially unnavigatable gives an average gradient of 0.0003 or about 30cm per kilometer. Wider rivers have lower gradients. The Mississippi has a gradient of about 1cm per km, yet still flows pretty quickly

If you have a flowing canal, you need a constant source of water. This is why canals use locks. Don't design canals as rivers, unless you already have a river to redirect through it.

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    $\begingroup$ And here I was going to suggest canals filled with high-viscosity fluid, thus greatly diminishing the flow rate. Just coat your barges with some magic fluid-phobic material so there's no friction resistance. $\endgroup$ – Carl Witthoft Apr 9 at 17:47
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    $\begingroup$ @CarlWitthoft Traditionally one uses the highly viscous fluid called asphalt for this, I'm just not sure if wheels count as magic friction reducing material... $\endgroup$ – mlk Apr 9 at 18:07
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    $\begingroup$ If you have two bodies of water at different heights, then the higher one will supply the water. $\endgroup$ – Acccumulation Apr 9 at 19:35
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    $\begingroup$ Having a flowing canal isn't a goal, it's an unfortunate consequence of having to move between two bodies of water at different heights. $\endgroup$ – Brilliand Apr 9 at 22:29
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    $\begingroup$ @Acccumulation And what happens when that body of water is empty? $\endgroup$ – immibis Apr 9 at 23:06
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The Suez Canal is an artificial waterway of about 190 km in length running across the Isthmus of Suez in northeastern Egypt which connects the Mediterranean Seas with the Gulf of Suez, an arm of the Red Sea. It has no locks, because the Mediterranean Sea and the Gulf of Suez have roughly the same water level and is thus the world's longest canal without locks. (Source)

Karakum Canal without locks in Russia is the longest canal in the world with a total length of 1400 kilometer. It was built in 1954 and completed in 1967 having a length of 840 kilometers and further extended in 1970 and 80s up to Caspian Sea coast making its total length of 1400 kilometers. (Source)

To answer your question, 0°.

The differences in "sea level altitude" between the sources and destinations of these bodies varies naturally according to the tide, but the canals are dug such that those variations are accommodated within the canal — which can only be done when the source and destination altitudes are fundamentally equal.

There is obviously some small amount of ascent/descent or the canals/rivers used for cargo transport wouldn't have a current. But that ascent/descent need not (and almost always is not) a condition of the path being traveled by the cargo ships. For example, the Mississippi has nearly all of its descent in its northern most reaches. By the time you get to the majority of shipping in the southern U.S., it's basically flat. The current is caused by water pressure from the north.

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    $\begingroup$ I don't think the OP is asking about sea canals; more the medieval style "river canals". They definitely have different water levels across the length of the canal, though obviously, they also used a lot of locks. 2° is the kind of grade that already gives trouble to rail traffic, I'd expect a navigable river/canal system would be substantially below 1°, especially without powered boats. Many navigable rivers have gradients in tens of centimeters per kilometer. And of course, even if a gradient is technically possible, at some point land traffic is going to be more economical anyway. $\endgroup$ – Luaan Apr 9 at 8:22
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    $\begingroup$ @jwenting uh, yeah you could. Literally all rivers have that. And what are canals if not redirected rivers or lakes? $\endgroup$ – Hobbamok Apr 9 at 8:49
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    $\begingroup$ @Hobbamok you could not, unless you want to drain one end of the canal into the other. A river has something where the water originates, or it'd run dry. Canals with a grade would drain whatever is at their high point into whatever is at their low point. Which can drain entire lakes. $\endgroup$ – jwenting Apr 9 at 11:34
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    $\begingroup$ 1 degree is a 1:60 gradient, or 16m per km - that's around 100 times steeper than most navigable rivers. $\endgroup$ – Robin Bennett Apr 9 at 11:52
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    $\begingroup$ "The current is caused by water pressure from the north." [citation needed] This isn't a pipeline. $\endgroup$ – Yakk Apr 9 at 14:48
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The gradient should be 0, UNLESS the high end of the canal has a flow rate high enough to compensate for the amount of water being drained from it by the canal into the low end.

Which for any decent sized canal would be a lot indeed and probably make the upstream end of the canal hazardous for shipping because of the rapids.

This is simple physics. Water (as any liquid really) want to establish equilibrium, meaning the surface wants to be flat and level. Thus it flows from high to low until equilibrium is established, both vessels having the same water level.

The only way to thus have a canal with one end higher than the other without an artificial barrier (a lock) in it is to ensure that the high end gains water as fast as the canal can drain it.

In theory of course you could calculate the exact drainage and pump water out of the low end at that rate, back into the high end. But locks would be much easier, cheaper, and more reliable.

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  • $\begingroup$ * where "flat" and "level" are non-technical terms referring to the imperceptible curve that water follows to maintain an equal gravitational potential... $\endgroup$ – JCRM Apr 9 at 12:55
  • $\begingroup$ You don't need a pump, if natural processes are filling it up. And even if there's a net flow, it can still take years to level off, if the flow is small enough and the volume is large enough. $\endgroup$ – Acccumulation Apr 9 at 19:38
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    $\begingroup$ @Acccumulation A canal dependent civilisation, requires functional canals. Investing in a Canal that would suffer at the hands of intermittent weather, or potentially would ruin the already established network of canals is a bad investment. It would be like investing in a rail line which would work fine for the next 10 years except for droughts, and will prevent any trains on linked rails from running after that. Requiring either a massive refurbishment program or new technology to fix. $\endgroup$ – Kain0_0 Apr 10 at 6:40
  • $\begingroup$ @Kain0_0 I'm taking the the desire for flow to be motivated by out-of-universe plot reasons, and not an in-universe desire of the characters. And according to this article, it would take the Amazon six years to empty a body of water the size of the Mediterranean Sea. So if you had a canal with a flow one hundredth of the Amazon, it would take six hundred years. And with rain, it wouldn't suffer at the hand of intermittent weather, since it would take six hundred years of drought to empty it. $\endgroup$ – Acccumulation Apr 10 at 14:50
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Your only limiter is how much your beasts of burden can drag up river. This is less dependent on slope than it is on the speed of flow and shape of the hull. Personally I'd favour double hulled boats if your channel is wide enough, they cut through water pretty well while giving you width for cargo space. Whereas a similar single hull would probably need to be a long thin boat in a heavy current.

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    $\begingroup$ So, the more hydrodynamic the boat hull is, the easier it is to be pulled through the water. That sort of makes sense. $\endgroup$ – GiruŠatuku Apr 9 at 12:00
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    $\begingroup$ The limiter is that the water will flow out of your canal at the bottom end leaving it empty of water, unless it is refilled at the top, in which case it is a river. $\endgroup$ – rghome Apr 9 at 12:52
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    $\begingroup$ If you're talking about a catamaran, they're generally referred to as twin-hulled. "Double hull" generally refers to a hull inside a hull. $\endgroup$ – Acccumulation Apr 9 at 19:44
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    $\begingroup$ @Acccumulation maybe where you come from, over here no, and I'm Polynesian, we invented these vessels over 2000 years ago. $\endgroup$ – Kilisi Apr 10 at 4:14
  • $\begingroup$ @Kilisi "Hull" is not a Polynesian term. While "double hull" can refer to catamarans, "twin-hulled" is significantly less ambiguous. $\endgroup$ – Acccumulation Apr 10 at 14:53
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I wonder if Tesla valve-shaped channel would work for you. It can make flow 10-100 times slower than unimpeded one, so I guess in theory 1/10 gradient would become possible. But it works best with pulsed flow and when covered so I am not sure if it would work in this case.

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If your canal empties into the Bay of Fundy, you've got fifty feet of vertical displacement to work with, thanks to the tides.

You can achieve whatever angle you might fancy by varying the length of your canal.

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