Optimal Barrel Length

[Silence]

What is the optimal barrel length in WBLs? Clearly, longer barrels generally allow for more time and for more speed to be transferred to the balloon, but at the extremes, the air expansion decreases and friction becomes a bigger factor.

I'm thinking about using a few formulas to find the best length, although that is also affected by portability and the like. Any suggestions? I'll take into account the barrel size and the pressure, but it probably doesn't need to be any more complicated than that.

Also, in the automatic mini-balloon launcher I've detailed before (as opposed to a sponge shooter), you might as well use a longer barrel to make maximum use of the air. Otherwise, there would be a lot of time in between shots where there's nothing inside the barrel and when you're just wasting air.

IMO, for that situation, low pressure from the regulator and a long barrel would be the most efficient.

[Ben]

Optimal barrel length only exists when the pressure is dropping, in this case due to the movement of the projectile. When the force of pressure equals the force of friction, acceleration equals zero and the barrel should stop there. After that point, the force of friction is higher than the force of pressure and the projectile decelerates.

Calculating where that occurs is difficult however. You first have to calculate the coefficient of kinetic friction (not really too hard) and then doing a lot of math to get an equation for the acceleration of the projectile. We tried to calculate this in AP Physics and believe me, it's simply not going to happen for most people unless you know differential equations. We asked a math teacher for help and even what we finished with after that didn't seem right.

The final equation didn't feel right likely because after that you have to optimize the velocity of the projectile due to the drag. Terminal velocity does matter a lot. Sometimes you'll exceed the terminal velocity and the projectile will slow down a lot for that reason. So you should first calulate the terminal velocity and then find the barrel length that will yield that velocity in reality.

The best method is really trial and error for that reason. Keep cutting and record the length and the average range of a few shots on that length. That's the easiest way to get every variable working at the same time.

You can avoid the problem of optimal barrel length by using low friction materials for barrels, a pressure regulator so that the projectile never decelerates, and an aerodynamic water balloon of sorts.

Edit: I'm also quite sure that there's programs or spreadsheets that will calculate this for you, but you'd have to measure everything as accurately as possible. That's another reason I'd suggest for trial-and-error - accuracy of measurements can be sketchy without good quality equipment that most don't have available to them.

[Silence]

Thanks for the reply. I assumed it would be difficult, but I guess I'll have to return to the problem in a few years--that would be an interesting challenge, at least.

For an automatic WBL, it would probably just be worth it to use a barrel that is short enough to be practical, but to increase the strength of the magazine springs. That should give a higher rate of fire and wouldn't waste air, but of course I would keep the pressure down.

Alternatively, I could just cut a barrel of the length I desire and then change the pressure to see at which point there's a major decrease in performance.

[Drenchenator]

The only guaranteed way to do this is through testing. As Ben said, the math is beyond most people. Optimization is a major section of calculus, and the differential equations alone would be over most people's heads. Ben and I did attempt to calculate this during our AP Physics class, but I do not remember finishing the equation.

[joannaardway]

The GGDT program does have an optimiser that predicts optimum barrel length.

However, I suggest as long as you can make it, and the chamber at about 1:1 C:B ratio (beyond this you hit the point of dimishing returns - forget 2:1, that's a maximum in my mind)

[Jonas]

Originally Posted by Ben

Optimal barrel length only exists when the pressure is dropping, in this case due to the movement of the projectile. When the force of pressure equals the force of friction, acceleration equals zero and the barrel should stop there. After that point, the force of friction is higher than the force of pressure and the projectile decelerates.

Calculating where that occurs is difficult however. You first have to calculate the coefficient of kinetic friction (not really too hard) and then doing a lot of math to get an equation for the acceleration of the projectile. We tried to calculate this in AP Physics and believe me, it's simply not going to happen for most people unless you know differential equations. We asked a math teacher for help and even what we finished with after that didn't seem right.

you should get it without calculus
friction = -F_friction find this out by pushing it in the tube and measuring force
force of the pressure = F_p = 2Pi*r^2, r = diameter of the barrel/2
Boyles law applied: p*(l*Pi*r^2 + V_chamber) = p_0*V_chamber, l = length of barrel

put those together we have F(l), a first degree equation.

d/dt(v(l)) = a(l) = F(l)/m
if you want, you can calculate the integral and get v(l), find out it's a nice parabola and has its maximum when d/dt(v(l)) = 0 aka F(l) = 0

[joannaardway]

It's far more complex than that.

A de-pressurizing gas cools, reducing it's pressure further. Flow needs to be taken into account as well, because in a split second situation like that, the gasses are not isobaric.

The figure you've quoted will be higher than the optimum. Besides, optimum barrel lengths are either so long that the thing is unusable, or the chamber is so short, it's not worth using either.

[Jonas]

didn't think about the cooling lol

Yes I agree 'optimum' barrel lengths are too long to be useful in real life.

[Drenchenator]

you should get it without calculus
friction = -F_friction find this out by pushing it in the tube and measuring force
force of the pressure = F_p = 2Pi*r^2, r = diameter of the barrel/2
Boyles law applied: p*(l*Pi*r^2 + V_chamber) = p_0*V_chamber, l = length of barrel

put those together we have F(l), a first degree equation.

d/dt(v(l)) = a(l) = F(l)/m
if you want, you can calculate the integral and get v(l), find out it's a nice parabola and has its maximum when d/dt(v(l)) = 0 aka F(l) = 0

This is quite confusing. First of all, your force of pressure equation excludes the actual pressure. The way that you used Boyle's Law is correct yet incorrect because you did not add the pressure of the atmosphere into the equation. Boyle's Law, like all the other gas laws, uses absolute pressure which is the pressure of the atmosphere plus the pressure that you would read on a pressure gauge (gauge pressure). Lastly, you can't just integrate anything, and in this case you can't really. a=dv/dt, so when this is treated as a differencial equation, you would have to integrate with respect to time. The equation in this case is a velocity versus distance one, and to integrate this correctly it should be a velocity versus time one. This is probably what you were trying to do:



The end result is a force of pressure equation with the independent variable being distance. This makes it an ideal candidate for integration into a work equation which then can be turned into a velocity versus distance equation (K=.5mv^2). But using the same force of pressure equation, you can substitute in the F_F for F and solve for X. This should work because when the projectile is in equilibrium (a=0), the force of friction would equal the force of propulsion (pressure in this case). Obviously, both can't work, so one would have to be incorrect (or they could both end up with the same solution, but I doubt that that would happen). During my physics class we tried to make an "optimal barrel length" equation and used a similar method to this one. But I don't think that we ever completed it (or at least I don't remember completing it). I also believe that it would just be easier and more accurate to use the empirical method to find the optimal length. It just gets very confusing because of the many "methods" of doing it.

Ideally, you would want to find the projectile's distance traveled at the maximum of a velocity versus distance equation or find the distance at the zero of an acceleration versus time equation. This can also be expressed as finding the distance which the projectile travels in the barrel when the force of friction equals the force of propulsion. At all of these points, the projectile is no longer accelerating so the velocity will begin to drop after this point. For that reason it is the optimal barrel length for range because range is maximized when the exit velocity is maximized as well.