Surface area questions

[WaterWolf]

Just a small query:
Is pressure and surface area directly proportional for force?

EG: You have two chambers, both are half filled with water.
3" diameter chamber at 100 PSI
6" diameter chamber at 50 PSI.

Assuming a reduction down to 1" output, will they both have the same level of power?

I've been looking around for formulas on this but haven't had much luck, though from just guessing I would say that they would result in an about equal force of water out the nozzle.

[Ben]

It's pressure that matters, not force, so the 3 inch chamber will be stronger. It should have approximately 87% more water output from the empirical data I've measured combined with the Bernoulli equation.

Also, the answer the first question, Force = Pressure * Area, so pressure and surface area are proportional to force.

[SilentGuy]

The pressure determines the range. Thin tubing for the barrel and valve could reduce the flow coefficient, but that's not really an issue in the pressure chamber.

[isoaker_com]

Something just feels amiss here.

If one has the following situation:
A) 3" diameter chamber at 50 PSI
B) 6" diameter chamber at 50 PSI

wouldn't B offer better performance/power thanks to good ol' F=P*A? If diameter/volume of the PC were irrelevant to force, that'd mean builders would be better off making smaller PCs to maximize force, but I do not believe that to be the case. A larger PC should be able to yield greater stream force at lower total pressures thanks to increased area.

[Ben]

The increase in PC diameter would not change the performance dramatically. Yes, I used to be be all about force, but it's not what matters after reading into it. Once you have substantial volume, changing the diameter would not affect much.

I would suggest reading what was discussed in this thread even though I'm not done with it: http://forums.sscentral.org/t4795/

An article with a complete derivation of everything I did in my modified Bernoulli equation will be available eventually.

Drenchenator also was working with some other more advanced equations to approximate flow with turbulence approximations... but I do not know how far he went with that.

[Drenchenator]

From my understanding, the 6" chamber would only perform better in output because the cross-sectional area is far greater. This could also increase the range because the overall flow rate is increased, though it of course depends on the nozzle. It's not the force: A greater cross-section would aid performance because it has a greater potential for flow.

Edit: Well, all the above is good if the pressures are the same.

I was doing some rather simple calculations with the Navier-Stokes Equations, but not in the way that would actually model flow in a Super Soaker. I was just inputing in velocity fields with zero divergence (incompressible flow) and solving for the pressure, P(x,y,z,t) in this case. This is really the limit I can do; the only partial differential equations that I can solve are the ones involved in calculating a function from its gradient. You would have to do the exact opposite of what I did--input a pressure equation and solve for the velocity field--to solve for the overall flow; at that point, calculating the flow rate at certain times and points is straightforward with a "simple" flux integral. I can't solve the partial derivatives that way though.

[SilentGuy]

Re-read the first post...the 3" chamber is at 100 PSI, not at 50 PSI.

Yes, F = P * A, but F doesn't even matter. Flow analysis uses pressure, not an arbitrary force. (In this case, the force is arbitrary since it acts within the chamber, not at the nozzle.)

[isoaker_com]

Originally Posted by SilentGuy

Re-read the first post...the 3" chamber is at 100 PSI, not at 50 PSI.

Oh, I definitely read that WaterWolf asked a different question, stating 100PSI versus 50psi. I was just wanting to reduce it a touch to a simpler scenario to try to make sense of things (i.e. different diameter PCs, but with the same pressure). If flow is increased at all, force must be increased, otherwise where's the energy coming from to push more volume? I suppose that tubing inner diameter and length will, of course, affect the final stream quality. What remains unclear in my mind is how to go from gas pressure to liquid pressure and how much cross-sectional area between the pressurized gas will affect water flow, if at all. Does it become a matter of only pressure being most meaningful, but larger PCs offering longer shot times thanks to increased volume? Perhaps the larger cross-section of a larger PC is offset by the fact that one is also therefore pushing on a larger volume of liquid, thus the benefit of increased area is negated. Thus, CPS/elastic systems work at lower apparent pressures since the area upon which the chamber is acting upon is much larger than in an air-pressure system. This also makes me wonder; if you hold pressure constant, but tilt an air pressure chamber to expose more cross-sectional area, does one get increased flow?

Sidenote: my head hurts

Edit: forget my babble above. Pressure is all (sort of) that matters. Well, it's pressure versus size of PC opening. Only the pressure pointing in the direction of the opening is what will determine the available force/flow. If the exit pipe is larger, you can get more force and flow. If exit pipes are the same, a larger PC will have more volume at a given pressure, but a PC at higher pressure will have more available force through the opening.

[SilentGuy]

Yeah, for the most part, it doesn't matter how large the pressure chambers are because the flow is going to be constricted elsewhere anyway. It might help a bit though. And there's always the capacity issue.

Originally Posted by iSoaker

What remains unclear in my mind is how to go from gas pressure to liquid pressure and how much cross-sectional area between the pressurized gas will affect water flow, if at all.

What matters is that gas compresses when it's pressurized. If you want to think of it in terms of work and energy, the force comes from the pressure and the distance is the distance you push on the air to compress it. The chemical potential energy (from the compression) gets converted to kinetic energy.

Originally Posted by iSoaker

Perhaps the larger cross-section of a larger PC is offset by the fact that one is also therefore pushing on a larger volume of liquid, thus the benefit of increased area is negated.

Precisely. If you look at the entire system, you'll see that the only cross-sections that matter are that of the pump and that of the nozzle. Pump area determines the pressure based on the force you apply. More nozzle area does mean more force being used to push the water, but there's also more water to push; things cancel out until only pressure matters. Roughly.

Beyond that, you need to consider how the nozzle and tubing increases friction and drag, as well as how the stream will perform once it's actually in the air.