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Assume we have a pipe with a diameter of 10mm. Water is flowing through that pipe and we know that the water pressure at the 'open' end of the pipe is 2 atm. How would one go about calculating the flowrate of water? Reasonable assumptions are allowed.

I at first thought of using Poiseuille's equation, but that requires you to have a container length. Right now I have found no way to calculate flow in my situation.

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  • $\begingroup$ Do it for unit length ie 1. $\endgroup$
    – Solar Mike
    Jun 3, 2020 at 7:31
  • $\begingroup$ also see: waterprofessionals.com/learning-center/flow-rate-calculator $\endgroup$
    – Solar Mike
    Jun 3, 2020 at 8:16
  • $\begingroup$ Thank you for the reply @Solar Mike, unfirtunately the problem with those calculators is that they require the water velocity? which I don't have. Could you please elaborate a bit on why would using a unit length on 1 meter of pipe work? To me it still seems we are just calculating fluid flow rate through a pipe with 1 meter length, not an open ended pipe. I'm sorry if this should be obvious and I don't understand. $\endgroup$ Jun 3, 2020 at 10:39
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    $\begingroup$ A picture or a more clear definition of the inlet and outlet locations would be helpful. Can you describe why 1 meter of pipe is not the same as an open ended pipe to you? Alternatively, you can give us more details on what your "situation" is. As it stands, I would have to assume some length to use any equation that I know. $\endgroup$
    – J. Ari
    Jun 3, 2020 at 13:41
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    $\begingroup$ To calculate the flow rate, you need the inlet and exit pressures plus the pipe type and diameter. Then it's just a typical pressure head loss determination, probably best done with tables. It's the total system that matters, not a single cross-section. Obviously the flow rate is the same for the entire length of the pipe. $\endgroup$
    – Tiger Guy
    Jun 3, 2020 at 16:56

2 Answers 2

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There's Torricelli's equation for hydrodynamic calculations purpose and for calculating the velocity of perfect fluid flow at the $'open' end$ could be presented as: $$V = \sqrt{2gh}$$ $V $ is the velocity, $m/s$

$g $ is the acceleration due to gravity, $m/s^2$

$h $ is the liquid column height (above the 'open' end), $m$

The same conclusion could be drawn from the Bernoulli's incompressible flow equation

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  • $\begingroup$ Torricelli's and Bernoulli's equations only work if frictional losses are negligible. Unfortunately, the fluid mechanician's definition of the word "pipe" includes frictional losses not being negligible. $\endgroup$ Apr 1, 2021 at 16:02
  • $\begingroup$ @DanielHatton Barring entrance loss, what is the friction loss at an infinitesimal distance from the pipe inlet? Note, in the question the pipe length is not given. $\endgroup$
    – r13
    Jul 30, 2021 at 20:28
  • $\begingroup$ @r13 If you select the length of the "pipe" to be so short that pipe flow can't develop, it's not really a "pipe" at all - more like a sudden contraction followed by a sudden expansion (or if the length is really short, an orifice). You could probably, for that case, find an empirical lookup table of how the coefficient of discharge depends on aspect ratio and Reynolds number and use it to crank out the flow rate for a given pressure drop - but since it's not a "pipe", the result wouldn't be germane to the question. $\endgroup$ Aug 3, 2021 at 20:10
  • $\begingroup$ @DanielHatton 100' is a pipe, 1' or 6" are pipes too. Until the OP provides the pipe length, there is no base to consider "friction", but to estimate the discharge rate right at the outlet using the basic principle. $\endgroup$
    – r13
    Aug 3, 2021 at 20:23
  • $\begingroup$ @r13 Ah - so you're thinking that the "open end" is the downstream end, and proposing to analyse the free jet/vena contracta beyond the end of the pipe (which perhaps drops the pressure from 2atm to 1atm) in isolation? That could work, but still, the problem I'm having with it is that it makes the pipe a complete irrelevance, and the question mentions the word "pipe" an awful lot of times for that to be the intention. $\endgroup$ Aug 4, 2021 at 10:13
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Sorry, I don't have enough reputation to commenton Nikolay Jolshin's answer. Regardless of the pressure at the exit, if you know the difference in pressure you can use Torricelli's equation.

I just want to add that, depending on the application, you might also need to consider loses due to speed. For example adding a speed coeficient to Torricelli's eq. Many manufacturers provide such coeficient, which from my experience normally dances around 0.98

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