What i am looking for:

I am looking for a formular to calculate the water output in $l/s$. My goal is to match a pump (pumping the water back up) with the following setup, to keep it going with as few corrections as possible.

Where i am at the moment:

Currently i have Torricelli's law for an ideal system.

But with this i am not sure if i have to take $h = h_2 - h_3$ or $h = h_2 - h_3 + h_1$?


The pipe is flexible.

I don't know:

  • The friction of the pipe.
  • The angles.

My setup: Sketch of the setup.

  1. An upper water tank with a water level $h_2$ mm (which is consistant!).
  2. A lower water tank.
  3. A pipe with it's intake $h_3$ mm above the upper Tank bottom. And it's outlet $h_1$ mm below the upper tank bottom.
  4. The pipes diameter is $d$ mm.
  5. The pipes length $l_1$ mm.
  6. The water flow is started by sucking (like stealing fuel).
  • $\begingroup$ What you want is the Hazen-Williams equation. You will need the coefficient for the type of pipe, pipe ID, and also equivalent pipe lengths for each bend in the system, which you will add to the pipe length. Flexible pipe will make this difficult, but you will have to know or estimate the curves. Set the head loss to h4 + h3 and solve for flow rate. $\endgroup$
    – Tiger Guy
    Commented Feb 13, 2020 at 15:57

1 Answer 1


Currently i have Torricelli's law for an ideal system.

You could have started out with Bernoullis Equation, but it doesn't really matter here.

But with this i am not sure if i have to take h=h2−h3 or h=h2−h3+h1?

Neither, if the free jet exiting the pipe is not continous, then I'd say you would use $\Delta h = h_1+h_2$. I don't see any reason to place the pipe so far above the surface level of the lower container that this would be a problem, so I will answer that the height is:

enter image description here

It can even dip into the lower container. The height difference is always the difference between the two water surfaces, it does not matter where exactly you place the pipe.

I studied the so called steam filament in fluid dynamics, so we image a streamline (green) from point 0 to 3 and follow it. (hence I recommended Bernoulli as a starting point)

enter image description here

Now I'd like to imagine the parts that cancel each other out, e.g. the inverted U shaped top section, the fluid needs to go up a little and go down the same distance. Overall there is no height difference then.

So I marked all parts that cancel out each other and I get: enter image description here

Which is consistent with what I said earlier.

I know this is a very over simplyfied way to regard the problem, however I thought the visualization may help here. You can calculate everything with bernoulli and get the same result.

The velocity of the surface.

What surface? You said they are constant, i.e. the containers are rather large (one of the assumptions of Toricelli btw). If not you need to modify your calculations bearing in mind that:

$v = \frac{\partial h}{\partial t}$

and need to use the version of bernoulli for instationary flow.

The angles.

They don't matter, see above.

edit in regards to Scott Dunnington's comment, I need to clarify that my approach of course neglects friction in the pipe (as you did with your assumptions). However you can extend the Bernoulli equation with a term that accounts for these additional pressure drops. It probably won't matter, but just for the sake of complecity.

  • $\begingroup$ Thanks for you answer, it has been real helpful! 'The velocity of the surface.' has been a misspelling what i ment was the friction in the pipe. $\endgroup$
    – Valanthor
    Commented Feb 19, 2020 at 12:15
  • $\begingroup$ @valanthor Glad to help. If your question is answered, don't forget to mark the answer as selected. $\endgroup$
    – idkfa
    Commented Feb 19, 2020 at 14:45

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