How Many stages ? Gary Snyder – June 2006
Question: How many stages should a launch vehicle use ?
Answer: As few as possible.
Okay thats a little trite. Let's limit the discussion to a low earth orbit and using somewhat conventional vehicles. I personally prefer pressure fed, metal tanked, vertical takeoff, expendable vehicles .... for this discussion.
An introductory class in rocket propulsion will show you a simple proof that a launch vehicle is optimized when each stage of the system produces the same Delta V (speed increase) as the one before it. This makes the assumptions that each stage has the same structural efficiency (Mass fraction) and each stages engines run at the same efficiency (ISP). Real rocket engines are more efficient in the vacuum of space and real launch vehicles rarely get much wind loading there either. With a pressure fed vehicle (heavy tanks) this is even more true because the tanks can be lower pressure and therefor lighter at higher altitudes.
Starting from a small 2 stage proposed launch vehicle, the PacAstro PA-1, we sat around one weekend and built a spreadsheet model of the system to see why the upper stage was so large. Turns out, with some theoretical engines, the upper stage can run on 1/3 the tank pressure (1/3 the tank weight) and produce almost half again the thrust on the same propellant flow (higher ISP). By letting goal seek function run on the proportional sizes of the two stages, the upper stage winds up being about 1/3 the size of the lower stage and produces about three times the Delta-V. The 'Proof' above suggested that it would be about 1/10th .and have an equal Delta-V.
The vehicle would deliver about 0.64% of it's takeoff mass to orbit (9700 m/s)
The next question was how about an upper stage? Well... designing a third stage, complete with different engines, tank diameters and everything seemed like large impediment and operational pain, so... Just add an additional second stage!
The obvious problem of a larger gross liftoff weight was solved by cutting both upperstages' tanks in half. I.E. Us two upper stages each holding half the propellants as the original upper stage. A little extra engine weight didn't affect the overall lift off weight (+0.5% GLOW). Payload delivered was 1.48% of gross take off mass. This wasn't even optimized for the upper stage size split.
Results: Same first stage, Same total propellant load, Same upper stage engines (but more of them), and over twice the payload delivered !
For this example, a three stage vehicle delivers twice the payload as the same sized two stage vehicle.