## Imports

```
using Plots
plotlyjs()
using Unitful
using DataFrames
using Measurements
using Measurements: value, uncertainty
using CSV
```

Simulating the performance of an air propulsion system as an alternative to solid rocket motors.

Julia

Capstone

University

Code

Aerospace

For Capstone my team was tasked with designing a system capable of moving mining equipment and materials around the surface of the Moon using a propulsive landing. The system had to be tested on Earth with something feasible for our team to build in 2 semesters. One of the first considerations my capstone advisor wanted was to test the feasibility of an air propulsion system instead of the obvious solution that of using solid rocket motors. This document is just *napkin math* to determine if the system is even feasibly and is not meant to be a rigorous study of an air propulsion system that would easily keep a capstone team busy by itself.

I chose an off-the-shelf paintball gun tank for the pressure vessel. The primary consideration was the incredible pressure to weight ratio, and the fact that it is designed to be bumped around would be necessary for proving the safety of the system further into the project.

The nozzle diameter was changed until the air prop system had a *burn time* similar to a G18ST rocket motor. The propulsion system’s total impulse is not dependant on the nozzle diameter, so this was just done to make it plot nicely with the rest of the rocket motors since, at this time, it is unknown what the optimal thrust profile is.

These are just universal values for what a typical day would look like during the summer in Northern Arizona. (Çengel and Boles 2015)

The actual simulation is quite simple. The basic idea is that using the current pressure, you can calculate \(\dot{m}\), which allows calculating the Thrust, and then you can subtract the current mass of air in the tank by \(\dot{m}\) and recalculate pressure using the new mass then repeat the whole process.

The bulk of the equations in the simulation came from (Çengel and Boles 2015), while the Thrust and \(v_e\) equations came from (Sutton and Biblarz 2001, eq: 2-14).

\[ T = \dot{m} \cdot v_\text{Exit} + A_\text{Nozzle} \cdot (P - P_\text{Ambient}) \]

The initial pressure difference is 4190.0 ± 300.0 psi, which is massive, so the area of the nozzle significantly alters the thrust profile. The paintball tanks come with pressure regulators, in our case, 800 psi which is still a huge number compared to atmospheric pressure. While the total impulse of the system doesn’t change with different nozzle areas, the peak thrust and *burn time* vary greatly. One of the benefits of doing air propulsion and the reason it was even considered so seriously is that it should be possible to change the nozzle diameter in flight, allowing thrust to be throttled, making controlled landing easier to control.

```
t = 0.0u"s";
P = P0 |> u"Pa";
M = V * (P / (R * T)) |> u"kg";
ts = 1u"ms";
df = DataFrame(Thrust=(0 ± 0)u"N", Pressure=P0, Time=0.0u"s", Mass=M);
while M > 0.005u"kg"
# Calculate what is leaving tank
P = minimum([P, Pmax])
ve = sqrt((2 * γ / (γ - 1)) * R * T * (1 - P_amb / P)^((γ - 1) / γ)) |> u"m/s"
ρ = P / (R * T) |> u"kg/m^3"
ṁ = ρ * a_nozzle * ve |> u"kg/s"
Thrust = ṁ * ve + a_nozzle * (P - P_amb) |> u"N"
# Calculate what is still in the tank
M = M - ṁ * ts |> u"kg"
P = (M * R * T) / V |> u"Pa"
t = t + ts
df_step = DataFrame(Thrust=Thrust, Pressure=P, Time=t, Mass=M)
append!(df, df_step)
end
```

Below in figure 1, the result of the simulation is plotted. Notice the massive error once the tank starts running low. This is because the calculation for pressure has a lot of very uncertain variables. This is primarily due to air being a compressible fluid, making this simulation challenging to do accurately. The thrust being below 0 N might not make intuitive sense, but it’s technically possible for the pressure to compress, leaving the inside of the rocket nozzle with a pressure that’s actually below atmospheric pressure. The effect would likely last a fraction of a second, but the point stands that this simulation is wildly inaccurate and only meant to get an idea of what an air propulsion system is capable of.

```
thrust_values = df.Thrust .|> ustrip .|> value;
thrust_uncertainties = df.Thrust .|> ustrip .|> uncertainty;
air = DataFrame(Thrust=thrust_values, Uncertainty=thrust_uncertainties, Time=df.Time .|> u"s" .|> ustrip);
plot(df.Time .|> ustrip, thrust_values,
title="Thrust Over Time",
ribbon=(thrust_uncertainties, thrust_uncertainties),
fillalpha=0.2, label="Thrust",
xlabel="Time (s)",
ylabel="Thrust (N)",
)
```