There is an actual mathematical formula for plunger to barrel ratio. I'm no mathematician here but I think it was that the barrel should be just under the same volume as the plunger tube. However this isn't always accurate due to imperfect seals, or to tight or loose dart fits.

*Scroll down to the bottom if you want to read the important part.*You could always figure it out for yourself.

I've figured it out for pneumatics and have began working on springers. It's not hard! I let my computer do the hard part; it ran for probably a total of two weeks straight to tabulate everything I have right now.

Start with first principles. The laws of conservation of mass, momentum, and energy go a long way. Once the dynamics of the spring gun system are described by differential equations, non-dimensionalize them, and numerically solve them for a large variety of situations, looking for a pattern. Specifically, you want to find where energy efficiency peaks.

(You also could do a lot of tests, non-dimensionalize the results, and figure out a correlation from there, but computers are far faster than people and the problem is relatively simple to describe mathematically.)

The non-dimensionalization is very important. Non-dimensionalization allows the results to be scaled and also reduces the number of variables. Note that plunger volume to barrel volume ratio has no dimensions. Obviously, that means that it's value is independent of any dimensions of the gun. If you test a gun that is twice as large but has the same plunger volume to barrel volume ratio, you can scale the results down. This is the same principle that allows wind tunnel testing at scale to work.

I'll detail how to start this for the low velocity isothermal gun case. Note that reality isn't isothermal. As the plunger adds energy to the plunger tube gas, the gas heats up. And as the dart takes energy from the barrel gas, the barrel gas cools. The isothermal assumption merely makes the equations more manageable. Any idiot can add the energy equation for the more accurate adiabatic case.

The low velocity part implies that the pressure waves travel far faster than the dart (so no darts over about Mach 0.3) and that dynamic pressure effects are negligible. We'll also make the assumption that the barrel gas has no inertia. This means we can use Newton's second law on the projectile with static pressures instead of the conservation of momentum.

Another assumption is that there are no leaks. This is good for the ideal case, but often what we deal with is less than ideal and leaks are unavoidable.

I'll now detail how to model spring guns. Be aware that I take the

lumped parameter approach. Also, I'd be happy to provide diagrams that should help explain what I'll detail below if necessary.

If you aren't familiar with at least the basic idea of calculus, skip to the bold part below. You might have to scroll a good bit.

Based on Newton's second law, the mass times the acceleration of the projectile is equal to the forces applied to the projectile. At this stage the friction force is assumed to act like a pressure force in the negative x direction for positive velocities for simplicity. This is somewhat unrealistic because it does not distinguish between dynamic and static friction, but that's not too big of a deal. Static friction is far more important than dynamic friction in spring guns.

m_d * x_d''(t) = A_b * (P_b(t) - P_atm - P_fd * sgn(x_d'(t))

m_d is the dart mass

x_d is the dart position

A_b is the barrel area

P_b is the pressure in the barrel

P_atm is atmospheric pressure

P_fd is the equivalent pressure of friction for the dart

Some logic is necessary to determine whether the applied forces cause the dart to move. If the applied forces aside from friction are lower than the static friction force and the dart is stationary, then the dart will remain stationary.

Similar logic applied to the piston returns the equation below.

m_p * x_p''(t) = A_p * (P_atm - P_p(t) - P_fd * sgn(x_p'(t)) + k * (L_s - x_p)

m_p is the piston mass

x_p is the piston position in the plunger (x_p = 0 is the back end of the plunger tube and x_p = L_p is the front of the plunger tube)

A_p is the piston area

P_atm is atmospheric pressure

P_fp is the equivalent pressure of friction for the piston

k is the spring constant

L_s is a constant that depends on the length of the spring and how it is positioned (generally it is the length of the spring)

How much air flows from the piston tube to the barrel (or vice-versa)? I could write a fairly long derivation, but instead I'll refer you all to any book on gas dynamics of fluid power control. Look for parts about flow through any generic restriction. I like the results provided in a book titled "Fluid Power Control" edited by John Blackburn and some of his associates on p. 214 to 217. This is a good old book that should be available in any decent engineering library. You can also use some simpler equations available on this page:

http://www.engineeri...ents-d_277.htmlSo, find an equation for the mass flow rate m_dot as a function of upstream pressure, downstream pressure, and some other parameters.

Apply the principle of mass conservation to the plunger tube and you'll end up with an equation like the one below.

A_p * d/dt(rho_p * (L_p - x_p)) = -m_dot * sgn(P_b - P_c)

rho_p is piston tube gas density

Mass conservation applied to the barrel returns something like the following.

d/dt(rho_b * (A_b * x_d + V_d)) = m_dot * sgn(P_b - P_c)

rho_b is barrel gas density

V_d is the "dead volume" between the piston tube and dart (Note that this isn't always bad for performance... some HELPS performance and this can be mathematically demonstrated.)

You'll also need an equation of state to relate gas pressure, density, and temperature. The

ideal gas law is very adequate here.

From the equations above, one could write them out more fully (using the product rule, algebraic manipulations, etc.), and from there non-dimensionalize them. To do so, define some non-dimensional parameters like x_d = L_b * x_d*. This basically means that x_d* varies from 0 to 1 where 0 is where the dart starts and 1 is where the dart exits the barrel.

Once the equations are fully non-dimensionalized, you can convert them to first-order ODEs and use whatever numerical scheme you desire to solve them. I'm not a math major, so I use

Euler's method. Don't make fun of me.

If done correctly, you can produce nice tables like this:

http://trettel.org/n...dm/pneu-eng.csvThat's the result of about 40 hours of continuous computing.

I hope those who read this far have an appreciation for how complicated this problem is. That's the easiest way to approach the problem of ideal barrel length. And I'll note that what I detail above isn't accurate for many situations.

**The point is that very few people here can take this approach to the problem. And if you don't take this approach, what you do is probably very wrong.**

If you couldn't figure out from my post above, I think most people would be better served by doing some rough tests, maybe following others' results for a good starting point. *Empirical testing is the way to go*. The problem of dart fit makes barrel lengths for springers fairly complicated. Not to mention the other factors that make the problem even more complicated.If you're a very mathematical person and you want to try some real applied math, have at it.

The reason you don't see any simple formulas for barrel length is that none exist. Those who want to figure out theoretical performance already can and they realize that any rules don't apply in general. And I'm convinced that if they were made, people would misuse them, so I'm not too keen on promoting them heavily.

**Edited by Doom, 08 May 2010 - 10:25 PM.**