This program is made to help potato gun constructors to find a proper design for their gun. The physics, thermodynamics and combustion kinetics are as near to reality as needed, without using multidimensional fluid and combustion simulation. This software provides everything to calculate the pressure, muzzle velocity and other needful data of a potato gun. 

For simplicity some assumptions were made, if your design cannot fulfil these (at least nearly), the figures will differ:

1. The gun is perfectly sealed; this means no gas will exit the combustion chamber and barrel until the projectile reaches the muzzle. Even small leakages (projectile fit!) in mm scale will alter the results dramatically. 

2. The pressure distribution in the entire combustion chamber plus barrel is perfect, because sonic speed (>300m/s) is much higher than flame velocity or projectile velocity. I think there will be no problem interfering with this assumption.

3. The weight of the air in the barrel is neglegted.

4. Acceleration of flame velocity because of heating due to compression of the remaining air/butane gas because of pressure increase is neglected. It will have an effect in the later combustion phase at high pressure (>300000Pa). (I sit here in southern India and do not have any papers about flame velocity versus temperature available, please excuse that) 

5. The air/butane mixture is optimal. 


The model bases on a potato gun with a cylindrical combustion chamber of bigger length than diameter, in which the plain flame front(s) propagate(s) from the ignition to the end(s) of the chamber (see 'manual II'). The three dimensional expansion of the combustion zone directly after ignition is not considered. Only the one dimensional expansion is calculated. 
For the simulation some figures have to be entered; One by one are discussed in the following text.


Flame velocity
Normally a mixture of butane and air is used to propel the projectile by its combustion. Such a mixture possesses a distinct laminar flame velocity: The velocity a flame moves forward the undisturbed (not turbulent) gas mixture. This velocity depends on pressure, temperature, components and their quantity. The pressure dependency is small (therefore neglected). The flame velocity does not vary a lot at 0C to 100C; it varies extremely near the temperature of self ignition. 
The mixture of combustion gas and air can be lean (more O2 than butane) or rich (more butane than O2). The highest combustion temperatures and flame velocities are reached at stoechiometric mixture (exact proportion to burn all O2 and butane). The laminar flame velocity has a peak there and drops rapidly towards the rich and lean extinguish borders. 
For butane/air the stoechiometric mixture is at 0.068 kg/kg butane/mixture (that is 9.5 Vol.% butane in mixture); the extinguish borders are somewhere at 0.033 (5%) and 0.100 kg/kg (15Vol.%); the flame velocity is between 0.3 and 0.37 m/s.
Propane will provide similar figures. In regular potato gun combustion chambers the laminar flame velocity can be assumed as flame velocity (the vorticity there is almost zero).

Barrel area
This is the cross section area of the barrel. The force on the projectile is calculated by pressure times area; the volume occupied by the propulsion gas increases with the forward movement of the projectile times barrel area (V=Vchamber+Xpro*A). 

Projectile weight
The acceleration of the projectile is calculated by a = F/m (equals force divided by mass). Usually the weight can be estimated by multiplying the barrel area, length of projectile (somewhere between barrel diameter and double barrel diameter) and the density of potatoes (900-1000 kg/m). 

Pressure gain
This is the factor the pressure multiplies by combustion minus one, if the volume is constant (be cautious! Calculations often use figures of combustion at constant pressure). The formula is N = (p1-p0)/p0; it depends of mixture temperature,  components and their quantities. Calculations show a pressure gain of 8.7, stoechiometric butane/air at 27C (9.6 at 0C). If the mixture differs from stoechiometric, the pressure gain drops. 

Barrel length
This does not need further explanations (I hope). 

Combustion chamber length
This is the length from the ignition point to the end of the combustion chamber. If you did previous testing and/or research, you already consider an ignition point in the middle of the chamber to be best. In this case simply enter the length from ignition to the ends, which should be half of the total combustion chamber length. 

Volume of the combustion chamber
This should be understood without explanations, too. 

Projectile friction
This is the friction in Newton between the projectile and the barrel. 

Combustion velocity deviation
When the flame zone expands, the pressure rises in the combustion chamber. Of course the remaining volumina of butane/air mixture are compressed (they do not have the heat release like the combusted gas to prevent them from). As mentioned before, the flame velocity does not change, but the compression results in shortening the length the flame front has to pass (v=x/t). The heat release (with it pressure raise) increases per time due to the formula 
Energy / time = velocity*Area*density*(combustion energy per kg). 
Instead of modelling this density variance the flame velocity is adapted. The density change is calculated by:
(dens1/dens0) = (p1/p0)exp(1/1.4) = (v1/v0) due to isentropic compression (compression without heat transfer). 


The numeric simulation takes account of pressure variation during combustion, increase of the volume occupied by the propelling gas caused by the movement of the projectile in the barrel and optionally the acceleration of combustion rate caused by pressure rise. For simplifications see the upper text. 


Results

Projectile time in barrel
This is the time between ignition and projectile exit. 

Muzzle velocity
This is the velocity of the projectile at the muzzle.

Maximum pressure 
This is the maximum pressure of the propelling gas (0 Pa is ambient pressure, not vacuum; Pa = N/m). 

Burnout until projectile exit
This is the percentage of combusted gas compared to the total amount of mixture (mass fraction).

Projectile impulse
This is the impulse of the projectile exiting the barrel (m/s). 

Projectile energy
This is the kinetic energy of the projectile exiting the barrel (J).


On the final page the pressure, burnout and velocity of the projectile is plotted versus position of the projectile in the barrel. 


This simulation shows, that only a fraction of the butane/air mixture is burned in common gun designs. The projectile leaves before the whole energy of the propulsion gas can be released. It also shows that a bigger combustion chamber is not always better than a smaller one. 

Possible advantages can be achieved by turbulating the butane/air mixture before or while combusting. The flame velocity in technical combustors are raised up to 100 m/s by turbulant combustion zones. Unfortunately measures are hard to realize and will complicate the handling of the gun. 


I wish everyone, who obtains and uses this software successful research, development and testing on advanced types.

B. Wagner