A Bunch of Math Associated with a 9 mm big bore gun

I just got done building a 9mm bigbore homebuild.
In the process, I used some physics to calculate (estimate):

1 the lock time
2 the time required to open the valve
3 the force holding the valve open

These are only aproximations of the real world values, but the helped me figure out how to get the best performance out of the spring, hammer, breach, and valve.

Lock time:

.17 kg hammer
2 inch hammer stroke = .0508 meter

Standard formula for motion:

x = 1/2 A T squared

In our case:
x = hammer stroke = 2 inch = .0508 meter
A = acceleration of the hammer
T = lock time (time for the hammer to hit the breach, from when it is released)

To get acceleration of the hammer we need newtons first law:

F = Mass times Acceleration
Or:
Acceleration = Force / Mass

In our case:
Acceleration = acceleration of the hammer
Force = spring force on the hammer = 20 pounds force = 88 newtons
Mass = Mass of the hammer = .17 kilograms

A = 88 / .17 = 517 meters / second squared

Plugging that number into the first equation:

.0508 = .5 715 T squared

T = .014
Lock time = 14 milliseconds

Velocity of hammer as it strikes the breach:

V = A T = 517 meters/second sq .014 seconds = 7.23 meters/second

Time required to open valve:
When the hammer hits the breach, momentum must be conserved

M1 V1 = M2 V2

M1 = breach mass + hammer mass = .17 kg + .045 = .215
M2 = hammer mass = .17 kg
V1 = breach + hammer velocity =
V2 = hammer velocity = 7.23 meters / second

V1 is the velocity of the combined mass of the hammer and breach
And is the initial velocity of the valve as it first starts to open.

After calculation V1 = 5.7 meters / second

If the valve opens up by .2 inch = .0051 meter
When the valve is fully open its velocity is zero:

.0051 / 5.7 X 2 = .0018 seconds to open the valve

This is just under two millisecond to open the valve.

After the valve is knocked open it is held open by the piston
effect of the tophat in the breach

For my 9MM bigbore, the diameter of the breach is .5 inch.
The force holding the valve open is the area of the .5 inch breach opening times the back preassure in the barrel.

F = pie R squared X back pressure = 3.14 X .25 sq X 1000PSI = .196 X 1000 = 196 pounds

196 pounds holding the valve open (if the breach pressure is 1000 PSI)
The valve will not close until the back pressure drops, as when the pellet leaves the barrel.

This last number (196 pounds holding the valve open) convinced me to install a very heavy valve return spring, but cut it so there is practally no preload on it. That did wonders for performance. It allowed the valve to open easily (no preload) and wide, but prevented tank dump. The previous spring was lighter but had a lot of preload, so it was hard to open, then was too weak to close the valve quickly after the pellet left the barrel (I was experiancing a partial tank dump, because the valve didnt close quick enough).

The formula for conservation of momentum: M1 V1 = M2 V2 , tells us to get the fastest valve opening, you need to make the hammer heavy and the breach as light as you can. Use brass for the hammer and use aluminum for the breach. If you make the breach out of steel or brass, you will need a very, very heavy hammer and a very strong spring.

14 milliseconds sounds long for a lock time??? Also, its another 5 milliseconds for the projectile to clear the barrel . Total time from when you pull the trigger till the pellet clears the barrel is about 20 milliseconds. To me that sounds like enough time to pull the shot!??

Mark