# How would you determine the average velocity in a vertical cylindrical pipe that is attached to a reservoir that sits on top?

In this case, the circular pipe would have some length L, diameter D, and would be attached to the centre of some rectangular reservoir. There is no pump, just gravity causing the flow. How would you calculate the average velocity in the pipe? Also, in a second set-up with an identical reservoir but pipe of different diameter, is the average velocity the same?

The reservoirs are assumed to be identical, open and rectangular reservoirs in both situations. The pressures on the top of the reservoirs and orifice at the end of the pipes are assumed to be at atmospheric pressure (or simply equal pressures, I'm trying to understand how the diameter change affects the flow rate and flow velocities in the pipes themselves)

Edit: I believe the velocities will be the same (and since Q = AvK), so volumetric flow rates will be different, and the time to drain will be different. Someone else said that this is not the case and the volumetric flow rates will be equivalent, although I don't see why they would be. This isn't a homework question, I'm just trying to understand the concept here! This question came about after discussing the pipe diamteres used in some toilets in america vs europe.

Thank you!

• Hi, I have down voted you because this is not a "do my homework site". Homework, or homework type questions need to show some effort, or research, rather than just presenting the question for someone else to do. – user140606 Jan 8 '17 at 8:46
• Hi this isn't a homework question, I'm just curious what approach would be taken. I thought of using Q=AvK but am not sure if a head loss approach is more appropriate. I'm looking to understand the concept. – masiewpao Jan 8 '17 at 8:58
• Is the pipe attached to the bottom of the rectangular reservoir, away from the vertical edges of the reservoir? Do you want to include the effect of the frictional pressure drop on the flow rate, or are you willing to assume that the fluid is ideally inviscid? Are you familiar with the Bernoulli equation? – Chet Miller Jan 8 '17 at 12:38
• Hi, I am assuming it was in the centre. The fluid being ideally invisic works well. I am somewhat! – masiewpao Jan 8 '17 at 15:21
• Is the pipe long and narrow or short and wide? If it is long and narrow viscosity predominates. Otherwise Bernoulli predominates. If it's in-between, you have to consider both. – Mike Dunlavey Jan 8 '17 at 16:46

When you say just gravity causing the flow, do you mean that there is only gravity acting on it? No pressure difference or Friction? In that case,

Acceleration on each water molecule $= g$
Initial Velocity of each molecule, $V_I= 0$
Final velocity, $V_F= \sqrt{V_I^2 + 2gL} = \sqrt{2gL}$
For uniform acceleration, average velocity, $V_{avg}= {V_I + V_F \over 2}$
$V_{avg} = {0 + \sqrt{2gL} \over 2}$

$V_{avg} = {\sqrt2 . \sqrt{gL} \over 2}$

$V_{avg} = {\sqrt{gL} \over \sqrt2}$

• Sorry, I think I have phrased it badly. I mean to say there is no external pump, but that the reservoir is an open one, and the atmospheric and pressure due to water are included. I will add that in. – masiewpao Jan 8 '17 at 11:23