RSE #38
Strangely I haven't as yet addressed the single most divisive issue of minimum cost design, the question of turbo pumps. Every rocket scientist from Robert Goddard to Von Braun has prescribed turbine driven pumps to increase the mass ratio and performance of their rockets. As a result, rocket engines are high tech, high pressure, and expensive. No one really questions the need for turbopumps.
For this analysis I have used the cost and performance specs for the Space Shuttle Main Engine, NASA's pride and joy, and used on the new Boeing EELV Delta IV, as the comparison to a very simple pressure fed booster. Costing about $140 million and weighting 7000 pounds, it produces an ISP of 363 at lift off. Combined with the cost per pound figures of the External Tank, and setting the mass ratio at a very good 20, the turbo pumped combination produces a delta vee of 92 ft/sec for every dollar. Setting the price of a pressure fed system at $10 per pound, an ISP of 260, and a mass ratio of 5, produces a delta vee of 6693 ft/sec for every dollar. I had expected the pressure fed to have some advantage, but that's a ratio of 72, a huge cost difference. Only NASA could ignore such a cost ratio.
Exploring a little further, I decided to LOWER the total delta vee of the pressure fed system to produce the same cost as the turbopumped system, to determine the range of values available. The results was a mass ratio of 1.02. Remember that the mass ratio of a common brick is 1.0. I could have also lowered the ISP to 3.6 to get the same result! Exploring further I reduced the ISP to 47, the figure used by Eugene Sanger for steam rockets, and got a mass ratio of 1.13. There is a lot of territory to explore here.
WHAT 72 MEANS
The above noted ratio of 72 means that you should only apply turbopumped rocket engines to a stage if you can save 72 times the weight difference in the lower stages. These growth factors are controlled by the staging ratio, which we have seen should be as low as possible. For a stage ratio of five, turbopumps should not appear until the second or possibly third stage, for a stage ratio of three wait until the fourth, for a stage early and often design you can get by with at least five stages before you should apply turbopumps to lower stage weight.