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!