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...
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