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I'm attempting to design a system that uses pressurised air to shoot tennis balls and am struggling to find the right mathematics to calculate the necessary air pressure and volume needed to propel the ball a certain distance.

The design involves an air compressor that feeds air into a chamber at say, f Litres/min, the chamber is a cylinder (let's say of radius r and height h) with a solenoid valve on one end.

Once there is enough air in the chamber at a high enough pressure the valve is opened - sending the air through it, and then out the other side into a barrel where the tennis ball sits.

The chamber and the barrel are pipes of equal diameter, however, the connections to the valve are smaller (let's say r1) on both sides.

Intuitively I believe that having the air coming out of the small hole of the valve, into the larger area of the barrel will cause it lose pressure very quickly as it expands sideways as well as going forwards to propel the ball - thus resulting in less force.

I'm looking for the right equations to model this system so that I can understand the relative tradeoffs in terms of increasing the valve size, increasing the air pressure, etc to maximise velocity.

All ideas appreciated, thanks!

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  • $\begingroup$ I would look up the Navier-Stokes and Euler equations as a start but the one thing to add would be an external, resistive frictional force between the ball and the barrel walls. $\endgroup$
    – honeste_vivere
    Commented Aug 8, 2017 at 12:55
  • $\begingroup$ Probably related to physics.stackexchange.com/q/228867/75633 $\endgroup$
    – stafusa
    Commented Aug 8, 2017 at 21:36

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You might be interested in checking the Wikipedia entry about flow through an orifice, or this answer to a similar question, if you want to start from first principles, but the calculations you'll find there are for steady flows, which is not your situation.

Most useful is probably the paper The exit velocity of a compressed air cannon, that gives as first approximation: enter image description here and then proceeds to consider the more realistic case with a valve.

Also probably relevant are the papers Internal Ballistics of a Spring-Air Pellet Gun and The Internal Ballistics of an Air Gun (pdf file here) and you can also get some basics from the comments in the question Calculating the theoretical velocity of a projectile.

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