Isentropic Efficiency of a Paintball Gun

Have you ever come across an old paintball gun while cleaning your basement and think “I wonder what the true efficiency of these things is anyway?” How many paintballs would an “ideal” paintball gun be able to shoot on a given tank of compressed air, and how does that compare to a real-world one? No? Well, I’m gonna tell ya anyways…

First, how should we model this? Some paintball guns work by having a fixed volume of gas stored in a chamber behind the barrel, which is released to propel the paintball when the trigger is pulled. The chamber is refilled for the next shot to complete the cycle. As a decent first approximation, we can model this as an isentropic expansion of that initial volume of gas as it pushes the paintball down the barrel, which seems at first glance to be pretty simple to model.

We’ll start with the paintball at the back end of the barrel, with essentially no space behind it between it and the trapped high pressure gas. We’ll assume there’s a valve behind the paintball that is instantly slammed wide open to put the paintball in contact with the high pressure gas, which begins accelerating it down the barrel due to the pressure difference between the high pressure gas behind the paintball and the much lower atmospheric pressure in front of the paintball. Our system will be closed, consisting of only the expanding gas doing work on the paintball (and subsequently on the atmosphere as it moves it out of the way).

 

We can write an energy balance for the expanding gas as it gives some of its internal energy to the paintball and to the atmosphere as it pushes the air in front of the paintball out of the way.

 

Here we assume the process happens so fast that no heat is transferred between the gas and any surroundings, and that the change in potential energy of the gas is negligible, and the change in kinetic energy of the gas is negligible. This last assumption may be a bad one! The typical limit for paintball speed is about 300 feet per second, which is about 30% of the speed of sound in air at room temperature. We’ll check it at the end, and maybe explore what happens if we don’t make that assumption.

For an ideal gas, there is an exact relation for the change in internal energy for an isentropic process where the pressure is changed. This is derived from the fact that the change in internal energy is only a function of the change in temperature of the gas and the heat capacity of the gas. We can also use the isentropic relation for an ideal gas relating the temperature, pressure, and ratio (k) of constant pressure (C_p) and constant volume (C_v) specific heats. Here T_o and P_o are the initial temperature and pressure of the initial high pressure gas volume (V_o), and T and P are the temperature and pressure of the gas after it has expanded some amount down the barrel.

 

But what is the pressure (P) behind the paintball going to be when the paintball leaves the barrel? We can think of three possible scenarios:

1)      The pressure is somewhere above atmospheric pressure.

2)      The pressure is exactly atmospheric pressure.

3)      The pressure is somewhere below atmospheric pressure.

Intuitively, I would guess that #2 is the most efficient, as in case #1, the gas is still exerting a net positive force on the paintball when it leaves and could have pushed a bit more, and in #3, at the end the atmosphere will be pushing harder on the paintball than the gas behind it is at the end, actually slowing it down. Let’s make no assumptions and just see where the math takes us…

We can use the isentropic relation for pressure and volume to relate the initial pressure and volume of gas to the final pressure (P) and volume (Vo, the initial chamber volume + Vb, the volume of the barrel)

 

Solving for the volume of the barrel, Vb, and substituting it in to the above equation, doing much rearranging, we solve for either the velocity as a function of the final pressure and initial volume, or the initial volume as a function of the desired velocity and final pressure. 

 

Or

 

We can also relate the volume of the barrel to its cross section area and length, to find the barrel length as a function of the initial gas volume and final pressure.

 

Well, now we can just go ahead and plot some stuff!

For a gas pressure of about 115 psia (which is about the pressure that many spool-style paintball guns operate at) a desired velocity of 280 feet per second (a typical velocity target to make sure you are under a 300 feet per second field limit), we get this plot:

Sure enough, the required gas volume reaches a minimum of 2.02 x 10^-5 cubic meters right at the atmospheric pressure I used in the inputs, 14.7 psia. The required barrel length is also about 11 inches, which is just about the length of many actual paintball barrels. But interestingly, the required volume is pretty flat – even if you shorten the barrel by 30% to 8 inches, the required air volume is only increased by about 5%, so having a somewhat shorter barrel doesn’t really impose that much of an efficiency penalty.

So how many paintballs can this ideal paintball gun shoot on a given tank of gas? Since we now know the pressure, temperature, and volume of gas required to accelerate a single paintball, we can figure out how many moles of gas this would be. At 115 psia, the ideal gas law should be pretty accurate.

However, a typical paintball air tank stores gas at 4,500 psi, which is so far above atmospheric pressure that the ideal gas law is going to start breaking down under pressure. Sure enough, if you look up the density of nitrogen at 300K and 4,500 psia in the NIST database, it’s 10,787 moles per cubic meter. For a 77 cubic inch air tank (assuming the same properties as pure nitrogen):

The actual tank is only going to hold about 86% of what we calculated from the ideal gas law. Also, we aren’t going to be able to use every single molecule of gas in the tank to propel paintballs. Once the pressure falls below the operating pressure of the paintball gun (115 psia in our example), the chamber won’t be pressurized enough to propel the ball, so we will assume that what is left in the tank when it reaches 115 psia is not usable.

Well there we have it, an “ideal” paintball gun should manage 2058 shots at 280 feet per second on a paintball gun operating at 115 psia (~100 psi gauge pressure) with a 77 cubic inch compressed air tank filled to 4,500 psia. Here’s a youtube video of someone trialing an actual spool valve paintball gun in exactly that configuration – he manages 1690 shots, not bad at 82% of ideal! Here’s another one with the exact same paintball gun and tank with a bit higher velocity – he manages 1393 shots, while my model predicts 1844 shots would be possible at that higher velocity (76% efficient).

This model doesn’t account for the energy needed to cycle the bolt back and forth each shot, or the friction of the paintball as it travels down the barrel, or the kinetic energy imparted to the gas as it accelerates down the barrel. On the kinetic energy of the gas, the 0.00642 moles of air per shot works out to 0.19 grams of air per shot. While much less than the 3 grams of paintball accelerated to speed, there is a good bit of kinetic energy imparted into the air that we haven’t accounted for. Also, as gases are accelerated to a significant fraction of the speed of sound, their static pressure (and consequently their ability to push on objects to accelerate them) is reduced, requiring more air per shot than I’ve calculated with this simplified model. All in all, I’m surprised that actual guns even managed 70-80% efficiency. Of course, we don’t know the actual pressure the tank was filled to, or the exact weight of the paintballs they were using, both of which would affect the results by several percent if they deviated from what I assumed.

There’s another class of paintball gun designs that works somewhat differently. Instead of releasing a fixed volume of pressurized gas for each shot, they have a valve that releases gas from a high pressure reservoir into the barrel for a short period of time. Anecdotally the internet seems to have the idea that this design is more efficient – I’ll have to work it out and see if there is fundamental reason that is.

Back in my teenage years I got way into paintball, but not in the usual way. I started with a cheap Brass Eagle Stingray. It was a terrible, but I soon discovered and joined a dedicated early internet subculture of enthusiasts who modified their Stingrays to improve its numerous shortcomings. This early experience sparked an interest in how gases behave that will surely bore my kids to tears once I can get them to sit in one place long enough to listen to me drone on about it. And now apparently you have too, so there you go.