58. A mathematical approach for creating a ‘nozzle’ for an open pipe flow, that takes into account the pipe size

[jovivier]

Please note that this method is not a method we have seen in any corroborating literature and therefore is only our opinion, it is up to you to decide if this is correct and check the results.

This method is only suitable for pipes that are discharging into atmosphere and not feeding into a dam or similar below the waterline.

Because the stream out of the pipe is inherently at atmospheric (i.e. 0m of pressure) this means that all the pressure has to end up as velocity head so:-
- the pressure just before the pipe end = V²/2g

You can substitute the velocity for Q/(Pi x D²/4) and then rearrange it so Q = K x Hⁿ (n will be 0.5), the expected form of the equation for outlet flow – the main trick is to allow for the units correctly. This equation currently has the flow in m/s.

For nozzles, using the equation form of Q = K x Hⁿ, IRRICAD expects flows in lph (and pressure in m) so the final equation would be:-
Q (lph) = (3600 x Pi x (2g)^0.5)/4000 x D² x H^0.5
Therefore Constant K = (3600 x Pi x (2g)^0.5)/4000 x D²
If you calculate this out then
Constant K = 12.524 x D²

Where:
Index n is assumed to be 0.5
D is the pipe internal diameter in mm
Pi is 3.141592
g gravity is 9.81m
H = pressure in metres.

You can use the above to create a nozzle for each pipe size, calculating out the relevant constant K for each internal pipe diameter. The index for all these nozzles can be assumed to be 0.5.

OpenPipeFlow.png

In theory there may be some minor edge effect that could be taken into account but we don’t think we have ever seen these documented and they will probably be quite small. However, there may be documentation somewhere about the pressure loss over a jagged edge, which you may also wish to account for in the equation.

The above formula seems to line up reasonably well with a couple of orifice equations we found in a quick Google search once you account for different units etc. e.g. (https://www.tlv.com/global/TI/calculato ... ifice.html).

Place the outlet and nozzle, setting a suitable nominal pressure (note, this cannot be 0).
Then use Detailed Analysis to design with this nozzle, to find the exact pressure and flow reaching the pipe end.
Note that if Detailed Analysis runs out of iterations select a different nominal outlet pressure and try again.