HW 8
due 11/13/2024 before midnight via Learning Suite 15 possible points
We will be analyzing Saturn V: a 3-stage liquid-fuel rocket. We will only analyze the first stage, the others stages would follow essentially the same procedure. The first stage uses five Rocketdyne F-1 engines with RP-1 (rocket propellant-1) as the fuel and LOX (liquid oxygen) as the oxidizer.
| mass of other stages (+ payload) | 667,800 kg |
| rocket diameter | 10.1 m |
| ideal velocity change \(\Delta V_{ideal}\) | 3600 m/s |
| specific impulse \(I_{sp}\) | 283 s |
| oxidizer-to-fuel mass ratio | 2.27 |
| one engine mass (there are 5 total) | 8400 kg |
| density of RP-1 | 810 kg/m\(^3\) |
| density of LOX | 1141 kg/m\(^3\) |
| tank thickness | 5 mm |
| rocket body thickness | 2 cm |
| density of aluminum | 2810 kg/m\(^3\) |
The parameter \(\Delta V_{ideal}\) is the velocity change from Tsiolkovsky’s rocket equation ignoring all losses (gravity, drag, steering, etc.). The listed specific impulse is just an average of sea level and vacuum (which we’ll assume is constant for the entire stage).
The notional design for this stage is shown below to the left:
The fuel tanks and rocket body are made of aluminum. Note that a stacked configuration is used and that about 10 meters of extra length is required below the tanks for the engines. As shown on the right, it is easiest to estimate the required tank size by assuming that the tanks are cylinders and then add on about 3 meters per tank to account for the fact that the round end caps will need to be longer than a pure cylinder and that there will be some ullage volume (that means the tanks won’t be 100% filled). For simplicity, I’d treat the diameter of the tanks as the same as that of the rocket body (it’s a really small difference).
Because the structural mass depends on the propellant mass, and the propellant mass depends on the structural mass, an iterative process is required (i.e., root finding). If you’re struggling to know if you’re on the right track, because this is an actual rocket, you could start with the known propellant mass and work your way through the equations checking your numbers against the actual rocket, then fine tune from there.
Report the following for stage 1. Be sure to clearly show your work and assumptions.
- The propellant mass.
- The required body length for this stage.
- The dry mass (structural mass including engines, i.e., no propellant).