A less-ideal model of paintball gun efficiency

At the end of the last post I rattled off a long list of possibly bad assumptions I'd made when analyzing the efficiency of a paintball gun. Let's jettison some of the ideal gas assumptions and switch to a more accurate equation of state model and see if it has any big effect on the results. The result will be a ... less ideal model?

First, we didn't really give any thought to what happened to the gas that pushed the paintball after it expanded. The energy to push the paintball and the atmosphere out of the way had to come from somewhere, and our initial energy balance showed that it came from the internal energy of the gas. 

 

Which is another way of saying that it came from the heat energy of the gas. Meaning if we take that energy away, the gas is going to get cold.  

For an ideal gas expanding isentropically, the ratio of the temperature before (T0) and after (T)  expansion can be related to the pressure before (P0) and after (P) expansion and the ratio of specific heats (γ). 

 

Sure enough, an ideal gas model anticipates that our gas will get down to 167K, or -159 degrees F, if it started at 300K before the first shot. Cold! 

Some of this gas will remain in the chamber where the gas originated behind the barrel. When the chamber is closed off to the barrel and re-pressurized, that original gas will be compressed by the high pressure gas entering the chamber. We can perform a mass and energy balance on the incoming gas compressing and mixing with the remaining gas (System 2 below), as well as a mass and energy balance on the high pressure tank (System 1 below). But, especially at the pressure of the high pressure storage tank, the gas is not going to adhere very closely to the ideal gas law. We can use the CoolProp equation of state package to more accurately relate the pressure, temperature, and density of a real gas. This equation of state can also use the specific internal energy and specific enthalpy as an input or output, meaning if we solve our material and energy balance for the final internal energy and assume a small transfer of gas from the tank to the chamber (Δn), we can directly calculate the final density. From the known internal energy and density, we can calculate the temperature and pressure of the final state.

 

We can perform this iteration until our calculated pressure matches our target final pressure for the volume chamber. If we overshot, we can even interpolate how much we overshot and adjust our last Δn step accordingly.

 

For a single shot, we can then calculate the temperature and pressure of the re-pressurized chamber and the slightly depleted tank. We can also assume an isentropic expansion of the chamber after the shot to find the temperature of the volume chamber's gas after it has expanded to 1 atmosphere. If we repeat this over and over, we can get a better estimate of how many shots we can get out of tank.

I made a program that performs this iteration for each shot. Running the program, we get 1791 shots before the tank pressure falls to the repressurization pressure of the volume chamber. Here's the variation in the temperature of the tank and the temperature of the volume chamber after it is repressurized and after it is vented to atmosphere to shoot the paintball.

 

The initial conditions were 300 K for the tank and (depressurized) volume chamber, 4500 psia for the tank, and 1 atmosphere for the volume chamber. Nitrogen was assumed for the gas properties. With the first shot, the air entering the volume chamber increases the temperature to 358K. When the volume chamber is vented to shoot the paintball, the temperature drops to about 200K. After this first shot, the temperature follows a steady decline, as does the temperature of the tank. At about shot 1700, though, the temperature of the volume chamber after the shot flatlines at 77K. Which of course is the boiling point of nitrogen! That means that after the temperature of the volume chamber before the shot falls to about 150K, an isentropic expansion causes some of the nitrogen to condense. In reality, the shot velocity of the paintball will probably start to fall dramatically at this point, meaning that 1700 shots is probably all that you can expect.

However! This model calculates that the temperature of the tank will fall to 100K, but neglects any heat transfer from the material that makes up the tank to the gas contained in the tank. Over the course the time it takes to shoot a couple of thousand shots, the tank material is definitely going to transfer heat into the gas. To model this, we can assume that the tank material and gas come into thermal equilibrium (reach the same temperature) after each shot. The amount of heat transferred from the tank will equal the amount of heat absorbed by the gas at a constant volume. 

 

The CoolProp equations of state can conveniently calculate the constant volume heat capacity of the gas after each shot. After the gas absorbs a bit of heat from tank, it will also increase in pressure. This final pressure can be calculated from the known equilibrium pressure and density (moles remaining in the tank divided by the tank volume).

Assuming a 1 kilogram tank with a heat capacity of 900 J/kg K, we get the following profile, with 2,811 total shots before the tank pressure falls to the repressurization pressure of the volume chamber:

 

A few interesting things here - first, the tank temperature falls to about 261 K, a bit below the normal freezing point of water but not insanely cold, so neglecting heat transfer from the surroundings into the tank probably isn't that big of a deal. The initial temperature of the volume chamber never falls below 312K. Since this temperature is higher than the 300K assumed from each shot in the last post, less actual gas is required from each shot, as long as the shot is fired before the volume chamber cools back off! The heat transfer from the tank also netted us an additional 1,100 shots, about 2/3 more than if no heat transferred occurred.

All in all, this additional refinement led to a shot count within 6% of the previous assumption of constant tank and volume chamber temperature and ideal gas behavior for the expanding volume chamber gas. In a more realistic paintball game scenario, some heat would be lost from the volume chamber between shots, requiring slightly more gas than assumed in this model (more like the ideal gas model) and some heat would be absorbed by the tank and its gas from the surroundings between shots (leading to a higher volume chamber temperature after repressurization and less gas needed if some shots are made in quick succession). These effects will counteract each other, so neglecting them probably isn't that bad of an assumption.

Where to go next? I'm curious how this shot count for a spool valve style configuration compares to a poppet valve style configuration like used by the indestructible Tippmann 98. Next time...

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, we have neglected the fact that some gas that remains in the valve at atmospheric pressure before it is cycled and recharged before the next shot. Having some gas in there to start will reduce the amount of gas needed to fully re-charge it. The only gas actually lost from the process is the gas that ends up in the barrel. For these conditions, this turns out to be about 77% of the gas.

 

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 2672 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 63% 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 2407 shots would be possible at that higher velocity (58% 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 or gas 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 involved 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. We've also assumed pure ideal gas behavior except for the density of air in the high pressure tank, and we assume that the tank and paintball gun stay at 300 Kelvin the entire time - as high pressure gas is vented, the tank will tend to cool down, while the paintball gun itself will heat up each shot as the high pressure gas repressurizes the "Vo" volume. Of course, we also don’t know the actual pressure the tank was filled to, the exact operating pressure of the gun, or the exact weight of the paintballs they were using, all 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.