# Pressure vs. Velocity Problem

This isn't a homework question; however, it is probably similar to one.

I am doing a design project that involves a hail cannon. Basically I have created an air gun that propels ice balls.

I have a compressed air tank, with an electronic quick release valve and then a barrel made of PVC.

My question is this,

I have tested the gun with different size hail. I have a chronograph to measure velocity, and I have a pressure gauge in the air tank. We tested different air pressures until we found one that resulted in a particular velocity. (In this case its the terminal velocity of falling hail.)

Anyway, I now have to calculate what these pressures should be theoretically. Ive tried using Bernoulli's equation with little luck. I get answers but the theoretical pressures come out higher than the empirical pressures. Obvsiously this is impossible, so either I'm doing it wrong or I'm not seeing the over all picture.

Can anyone give me some suggestions on what equation I can use or maybe where to start.

For the record, empirically we have 6.5psi shoots .5" diameter hail at a velocity we want. When doing it theoretically, the same diameter hail shoots the same velocity at 9.5 psi.

Hence the reason why I think im going wrong somewhere.

Any help is appreciated.

## Answers and Replies

Kurdt
Staff Emeritus
Gold Member
This is not my area of expertise but from what I remember Bernoulli's equation gives the velocity of the fluid that is flowing not the object it may be pushing. In this case I would recommend using Force= pressure x area and work out the impulse on the hale that you are firing. Impulse is the change in momentum of a particle caused by the force, so I guess you know how long the jet of air acts upon the hail and you'd have an equation something like.

P x A x T = m x v

Where T is the time the force is acting m is the mass and A is the area its acting upon. Anyway like i say this is not my area of expertise but maybe posting first might get the ball rolling on more replies.

Good luck!

FredGarvin
I actually have experience with a large compressed air gun, shooting hail balls at jet engines.

The situation is rather complex for a few reasons. First, you are dealing with a rapid expansion of the gas. It is hardly steady state and incompressible and good luck finding a streamline, which throws Bernoulli right out the widow. Secondly, you are expanding across a time varying orifice (your valve). Next, you have the inconsistancies of your hail ball and the leakage of the compressed air around the hail ball. Also, unless you have a high speed camera, chances are you are deforming your hail balls during the shot, so you can't rely on the idea of having a sphere or whatever shape you started with.

Honestly, I think that your 9.5 psi guess isn't too bad considering what you have to deal with. I know the percentage difference sucks, but this is not an easy thing to solve. There are a lot of variables involved.

Perhaps if Clausius or Brewnog see this thread they will comment as well.

So am I to assume that the pressure needed to propel the ice a certain velocity cannot be solved theoretically without taking a long time to do?

My professor wants me to do it, but I don't know if it can be done. I have talked to 2 other professors who haven't lead me in the right direction yet. I'm starting to think it can't be modeled because of so many assumptions and variables.

At least in my opinion, if it can be modeled, I don't see how it can be that accurate.

I used P-Po = 1/2*rho*V^2

However, I think I am wrong because I used the density of ice and the velocity it needs to travel. This is part of Bernoulli's equation and that is for fluid flow. Ice obviously isnt a fluid so I think this would be wrong to use. Also when I graph several velocities, the graph is way off.

Any more help would be appreciated.

Kurdt
Staff Emeritus