Hello all, this is my first post here, but I've been following the N prize for a while, ever since I stumbled upon the half baked page quite a while ago.
I'm not affiliated with any team, and will probably never launch anything into space myself, primarily because I lack all of the necessary skills (not to mention disposable income) to do so. But that doesn't stop me from being interested in the various problems surrounding rocketry, in particular low budget rocketry.
This is probably where I should stop typing now, but I shall not. Fair warning, there are lots of words ahead, and some numbers.
Lately I've been thinking about what an optimal design would be, ideally using the simplest concepts and minimum of materials. I guess this could be rephrased as, "how to build a SMALL dumb booster." Initially I figured that one way would be to build several identical stages and simply stack them, and drop them when spent. All would be the same mass, diameter etc. However when I wrote some code to simulate this, it didn't take me long to realise that this was horribly inefficient.
So then I moved on to more traditional staging, heaviest first, then each stage getting progressively lighter. I also discovered pretty fast that if each stage had the same fuel mass fraction and specific impulse, then the optimal rocket was one where each stage provided the same delta-v to the remaining stack. However, it is pretty unlikely that some quite different sized stages operating at different altitudes would have the same fuel fraction and specific impulse. So then I started simulating 2, 3 and 4 stage rockets with varying parameters and collected a reasonable amount of data into a spread sheet. For the last few days I've been stuck, trying to find a simple way to calculate the optimal delta-v for each stage without having to run countless trial-and-error iterations on each possible combination.
However, I believe I have just found the answer (though I'm sure much smarter people have found it long before me), so for anyone who is interested (or still reading by this point), here is how I believe the optimal staging for a rocket can be calculated.
Suppose you want a 4 stage rocket, you want a total delta-v of 10,000 m/s, and the stages have the following fuel fractions and exhaust velocities at their corresponding operating altitudes:
stage 1 fuel mass fraction = 0.85, stage 1 v_e = 2900 m/s
stage 2 fuel mass fraction = 0.83, stage 2 v_e = 3000 m/s
stage 3 fuel mass fraction = 0.81, stage 3 v_e = 3100 m/s
stage 4 fuel mass fraction = 0.8, stage 4 v_e = 3200 m/s
I should point out that any interstage sections should be counted towards the mass of whichever stage they stay attached to after separation (usually the one being dropped).
Now calculate the maximum delta-v for each stage, (ie: as if each stage was launched solo, with no payload):
stage 1 max delta-v = 2900 * ln(1 / (1-0.85)) = 2900 * ln(1 / 0.15) = 5501.65 m/s
stage 2 max delta-v = 3000 * ln(1 / (1-0.83)) = 3000 * ln(1 / 0.17) = 5315.87 m/s
stage 3 max delta-v = 3100 * ln(1 / (1-0.81)) = 3100 * ln(1 / 0.19) = 5148.27 m/s
stage 4 max delta-v = 3200 * ln(1 / (1-0.8)) = 3200 * ln(1 / 0.2) = 5150.20 m/s
Subtract the lowest delta-v value from all the stages, I'll call this the "bonus" for each stage:
stage 1 bonus = 353.38 m/s
stage 2 bonus = 167.60 m/s
stage 3 bonus = 0
stage 4 bonus = 1.93 m/s
Add these up and subtract them from the desired delta-v:
remainder = 10000 - 522.91957831244939334702598397861 = 9477.08 m/s
Divide by 4 and add it on to the bonus for each stage and it will give you the optimum delta-v for each stage:
stage 1 optimum delta-v = 2722.65 m/s
stage 2 optimum delta-v = 2536.87 m/s
stage 3 optimum delta-v = 2369.27 m/s
stage 4 optimum delta-v = 2371.20 m/s
If you add these up you'll actually get 9999.99 m/s due to rounding errors, but you'll always want a little extra fuel in the tanks just in case anyway. From these optimum delta-v values, it is easy to calculate the mass and fuel required for each stage using the rocket equation, just remember to start at the last stage and work down.
Everything I have just said could be complete nonsense, if that's the case I hope someone will tell me so. I do not have any formal training in aerospace, I have merely followed where the numbers led me, I could easily have read them wrong.
The reason why I worked all this out is partly because I find it interesting, but also because I couldn't find any such information elsewhere. In fact one place I found simply said there wasn't any such method to calculate optimal staging and that you must simply run the aforementioned countless trial-and-error iterations to find it. So if this helps anyone in anyway, then I'm glad.
I have more thoughts on the matter, but I think this is probably long enough for 100 first posts already.