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Bernoulli equation with the appropriate adjustments

Posted: Tue Jun 12, 2007 12:21 am
by Kalogagatya
Hi guys!

i read the thread about fluid motion inside a watergun, and how it 'forces' the weapon back when the trigger is pulled..

without getting much into big mechanical details (otherwise this would be a neverending post..) the first thing you should know about is the Reynolds adimensional number.

this number, Re, is a ratio between the physical effect of fluid inercia vs. viscosity.

Re=(fluid density*fluid average speed*vessel diameter)/fluid viscosity

(fluid moves in a very complicated way, if you imagine a vessel, the fluid adjacent to the vessel has zero velocity, and the maximum velocity is in the axis of the tube. kind of a parabolic cone. that's why i said fluid average speed.)

this Reynolds number indicates whether your fluid motion is laminar or turbulent.

for Re<2000 (approximately) fluid motion is laminar. Above 4000 is turbulent.

why is Re important? because it is the simplest way to know if you are minimizing turbulence, which gives you more momentum at the nozzle exit

another important aspect:

the Bernoulli equation --> Bernoulli was a guy with a lot of time in his hands who came up with the theory that in ideal conditions, a stream of fluid maintains its mechanical energy from point A to B:

Pressure Energy Contribution + Dynamic Pressure Energy Contribution (fluid inercia,.. kind of.) + Gravity Energy Contribution = constant

in each point of fluid each one of the contribution type may vary, but its sum is a constant.

SuperSoakers ARE NOT built with ideal vessels :P you all know that

From point A to B:

tubes have rough surface that create turbulence hence dissipating some of the momentum mostly into heat. some of the turbulence creates localized back flow areas that reduce overall power in the fluid streamline;

intersections, nozzles entrances, etc.. all vessel junctions with a significant diameter ratio (D/d > 1.2 +\-) will create a special turbulence called 'vena contracta' effect. specially at the nozzle exit where you can find a lot of 'walls'. this is responsible for a water stream that easily loose its cylindrical shape.

while a localized turbulence with a 'vena contracta' effect has a contribution of K=0.5 to 3 or 5 to the overall power loss, a complicated valve can have a K value of 150 to 210. (don't ask me what is K, i mentioned it just as a way to compare, its way too complicated to write it all here :) )

the thing is, these turbulence effects are greatly amplified by a big Re value. :)

phew.. that's a lot of stuff here :)

oh and by the way, Navier-Stokes equations are best used in low Re values ;) , or in typical fluid motion such as laminar rotation or others. :)

i hope this is of some use here !


Posted: Tue Jun 12, 2007 2:30 am
by Silence
Wow...that was a very informative post! I'm really glad to see a fluid mechanics expert here! :)

I'm just wondering, would a vena contracta affect the Reynolds number itself? Doesn't sound like it.

Also, is there a way to calculate flow and stream velocity for a given system with a given pressure gradient? Obviously, we're not looking for a very precise formula, just a rough equation. All we know is that more pressure and a greater diameter help.

Posted: Tue Jun 12, 2007 2:40 am
by Kalogagatya
:) glad i can help!

vena contracta does not affect the reynolds number. vena contracta is a result of a high reynolds number (in fact more visible with a even higher reynolds number, since velocity is the main reason with this effect occurs..)

i will upload one of these days some images that will be really usefull

sure, it is really easy:

given the length of tubing, the material it is made of, the number and classification of valves (these first four aspects are not as important as the following), provided that i have the nozzle diameter and the water intake diameter, assuming that nozzle exit is in optimal atmospheric pressure conditions (PTN conditions, p=1atm and T=20ÂșC), i just need a pressure measure of the PC or PR, and i can actually calculate both flow, velocity, power (in Watt) and the maximum possible range of your water weapon :)

Roman water flow increase nozzle

Posted: Thu Jun 21, 2007 6:03 pm
by Steve Lepp
I started this note and lost it!

I recently read a note in a fluid mechanics text that Roman Householders who were charged for water usage by the size of pipes into the house, modified the pipe end to some sort of diffuser to increase the flow without increasing the pipe size. I'll try to find this info....if someone knows about this, would you please add it.


Steve Lepp

Posted: Thu Jun 21, 2007 6:59 pm
by Silence
Welcome to the forums! Sorry to hear you lost your post - it's quite frustrating when that happens.

Hmm, that hack sounds pretty clever - the most I could have done would be to reduce the pressure at the outlet by lowering it or something. Maybe the diffuser somehow reduced the number of eddies and amount of turbulence at the outlet or something.