What size of tube should you choose? What size of coil? Does
it even make any difference?

Choosing Tube Material and Dimensions

Your projectile travels down a tube inside the coils. It provides an air gap,
which allows motion. (In fact, an air gap exists in all electromechanical machines. It is where
magnetic forces are converted into motion.)

Your tube cannot be a ferrous material such as iron, because it would guide the magnetic
flux away from the projectile. Bad idea.

Your tube must be some non-ferrous material, such as aluminum or copper or
bronze, or even paper or plastic. Note these materials conduct current, which is not a major
problem. However, it does cause two effects:

Internal eddy currents will arise during periods of changing magnetism; there are no eddy
currents during steady-state dc voltage. Periods of changing magnetism are extremely brief
during coil switching. We neglect it entirely in this application.

The projectile itself is magnetized and rushing through the tube. A conductive tube will
have induced eddy currents from the moving projectile, which tends to slow it down. This is
(I hope!) negligible.

So a non-conductive tube such as paper or glass might be slightly preferable, but it's
not that important. (Right? Send me any comments on this
subject.) Whatever it is, obviously you need a smooth inside surface to minimize friction. Duh.

Now, what about tube size? A popular approximation for the B-field (magnetic
flux density) in a solenoid is directly proportional to the coil current I and
to the number of turns per unit length n. The formula does not depend on the
coil radius. Why not? Because it assumes the coil length is significantly longer than
the coil radius. If that's not true, then your B-field will be compromised by
fringing effects.

Fringe effects are what happens when the magnetic flux disperses at the ends
of the coil. This tricky to compute. The path of the flux lines depends heavily on the parts
geometry. If you really want to know what happens, it might be easiest to just sprinkle iron
filings on a flat piece of paper near the coil and energize it. Can someone recommend a computer
modeling application for magnetic flux lines? I'd sure like to hear about
it! I found a free student's edition called QuickField by Tera
Analysis. It's fast and relatively easy to setup, and has lots of interesting colorized field
plots. You could also try Finite
Element Method Magnetics by David Meeker (another free download).

The bottom line is to make your coil as long or longer than its diameter. Otherwise
you may notice nonlinear effects, such as your projectile sticking to the side while inside the
tube.

It is essential to performance that the tube should be only as large as needed to barely accommodate
your chosen projectile.

Spin Stabilization

Also - if you have a slightly elliptical coil, and a slightly elliptical projectile, the projectile
will align itself with its major (widest) axis on the coil's minor axis. If you turn each succeeding
coil's minor axis a few degrees, the alignment of the projectile will put axial spin on the projectile
and stabilize it during its trajectory after leaving the tube.