Basic Rocket Sizing and Performance: Saturn V Overview In this - PDF document

Basic Rocket Sizing and Performance: Saturn V Overview In this exercise we will perform a basic sizing and performance analysis for the Saturn V rocket. We will reverse engineer so that we have a way of checking our work. This is a 3-stage liquid-fuel rocket designed to carry up to 140,000 kg of payload into low earth orbit (LEO). Although the real rocket has three stages, we will only analyze the first stage. The methodology for the other stages is essentially the same. The sources I use are below, and in brackets is the shorthand I will use to refer to them throughout in the document: • [wiki]: https://en.wikipedia.org/wiki/Saturn_V (primarily the summary box on the right hand side) • [specs]: https://s3-eu-west-1.amazonaws.com/niavok/ps/saturn_v_first_stage.pdf The provided ‘=’ prompts below are mostly to help those who are computing by hand (as opposed to on a computer). If you’re on a computer you probably don’t need them. The only things you will report on your homework are the four metrics marked in red: length of the stage, empty mass of the stage, burnout velocity, burnout altitude. Basic Sizing Let’s start with the propellant mass. Normally this would be a design parameter and we’d iterate to figure out what we need. Since we are reverse engineering we will subtract the known empty mass from the gross mass, and see if this input leads us to comparable outputs [wiki: first stage]. M propellant = Next let’s determine the mass of the fuel and the oxidizer (propellant = fuel + oxidizer). To do this, we need to know the oxidizer:fuel mixture ratio. The first stage of the Saturn V uses a mixture of Rocket Propellant-1 (fuel) and Liquid Oxygen (oxidizer). You can find the oxidizer:fuel ratio by following the link at wiki: first stage: RP-1. M fuel = M oxidizer = Now that we know the masses of the fuel and oxidizer, let’s compute the volume we need in our tanks to hold them. We can compute this by using their known densities [wiki: first stage: fuel: links for RP-1 and LOX]. V olume fuel = V olume oxidizer = Now we can size the tanks. Take a look at the picture in [specs] and notice that the design choice is a stacked configuration. Thus, the diameter of each tank is (nominally) the diameter of the rocket [wiki: size]. With a known volume and diameter you can compute the required length of each tank (assume they are cylinders for simplicity). You’ll note in the picture that the lengths include bulkheads. There will always be 1

some additional structure for mounting and ullage volume so we need to apply a multiplier to our values. Based on the picture I estimate about a 40% increase in length. l fuel − tank = l ox − tank = Let’s pause and check our results. According to [specs] the volume of the actual fuel tank is 768 m 3 and its height is 13.4 m. The actual LOX tank is 1253 m 3 with a height of 19.5 m. Don’t expect exact matches using our simplified analysis, but they should be in the ballpark. The length of the stage is the length of the fuel tank + oxidizer tank + some additional structural markup ( ∼ 10%). Check that this is reasonably close to the actual first stage length (42.1 m) [wiki: first stage]. l stage = Let’s now compute the structural masses. The mass of the tank can be estimated most easily by assuming a thin shell cylindrical structure (as a rough approximation). According to this random internet citizen the thickness of the tank is about 5 mm. The density of aluminum is about 2810 kg/m 3 . M fuel − tank = M ox − tank = We also need to compute the mass of the rocket body for this stage (which is better approximated as a thin cylindrical shell, and is also made of aluminum). Assume that the shell thickness is 2 cm (I couldn’t find the real spec here, I just scaled this up from a known value on a smaller rocket). M body = The total dry mass is the mass of the two tanks, the body, and the engines. There are 5 engines and its mass can be found on wiki: first stage: engines. As a check you should see that your dry mass is reasonably close to the empty mass (130,000 kg) [wiki: first stage]. M dry (empty mass) = First stage (rough) sizing is complete! Let’s now turn our attention to overall performance. Basic Performance Normally, to compute the dry mass of the rest of the rocket we’d repeat the same process for the other stages. We will skip that and just use the known values. The dry masses [wiki] are 40,100 kg for stage 2 and 13,500 kg for stage 3. The nose mass I couldn’t find explicitly, but estimated it out from other known parameters as 60,800 kg. The total dry mass of the rocket is the dry mass of all three stages + mass of the nose + mass of the payload. Often a markup (say 5%) is applied to the masses (besides the payload) to account for other necessary structure. The estimated propellant mass for stage 2 and 3 is 456,100 kg and 109,500 kg. Total mass is the total dry mass + total mass of the propellants (check against wiki: mass). M total = Now we can compute the mass ratio (initial mass to final mass). Again, we will just do the first stage. λ (mass ratio) = The effective exhaust velocity can be found from the known I sp [wiki: engine]. C = 2

Then using the definition of thrust, and known thrust [wiki], we can find the mass flow rate. m = ˙ Now we can estimate the burn time of the propellant using the propellant mass and the mass flow rate (the mass flow rate in the nozzle is not exactly the same as the burn rate of the propellant because the density of gas in the chamber is variable, but it’s close enough for our purposes here). Burn times are listed on [wiki] if you want to check. t b = You can now compute the velocity at the end of the first stage using the “rocket equation”. According to [specs] it should be approximately 2680 m/s. V 1 = Also we can find the altitude we will reach at the end of the first stage. We didn’t show this in class, but you can derive an equation for altitude change by integrating the rocket equation from velocity to height. The result is � 1 − ln λ � − 1 2 gt 2 y = C t b (1) b λ − 1 where λ is the mass ratio. According to [specs] it should be approximately 61 km. altitude 1 = Again, don’t expect perfect accuracy. These latter equations are like the Brueguet range equation in that they use a simplified macro-level view. We’ve also ignored some important things like drag. These equations highlight what parameters matter and in what ratios. They should give a ball park answer, but to get a better answer you’d need to time march forward and numerically integrate. 3